Visual Perception of Longitudinal Waves: Theory and Observations

Visual Perception of Longitudinal Waves: Theory and Observations | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Visual Perception of Longitudinal Waves: Theory and Observations Christopher W. Tyler, Joshua A. Solomon, Stuart M. Anstis This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7802344/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 23 Mar, 2026 Read the published version in Scientific Reports → Version 1 posted 9 You are reading this latest preprint version Abstract We introduce the study of the visual perception of longitudinal travelling wave motion, which as a physical phenomenon forms the basis of acoustics and some forms of seismic transmission. A theoretical analysis of Physical longitudinal wave motion reveals that it exhibits profound nonlinearities that have been almost entirely neglected by the physics community. We simulated longitudinal motion in visual form with a random-dot field in which each dot particle oscillates sinusoidally about a fixed position at the same frequency but with a phase advance proportional to its distance from the origin. The resultant longitudinal density wave well-approximates a sinusoidal function at very low oscillation amplitudes, becoming progressively distorted as oscillation amplitude increases. When the maximum velocity of each particle equals that of the propagation, the density function approximates a spike, which splits into two at even greater amplitudes. Perceptually, the motion splits into forward motion of the crests and backward for the troughs. Adding a single (‘rigid’) velocity component can eliminate either the forward or backward percept. Remarkably, the speed needed for perceptual cancellation scaled with oscillation amplitude. Longitudinal waves evoke no motion aftereffect. These unexpected results underline the emergent, or higher-order, nature of the longitudinal travelling wave motion. Biological sciences/Neuroscience Physical sciences/Physics Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction Any travelling wave can be decomposed into longitudinal and transverse oscillations. In mechanical waves, those oscillations are applied to the positions of particles within the solid, liquid, and/or gaseous medium through which the wave travels. (Electromagnetic waves, on the other hand, can travel through a vacuum.) Sound waves in gases are wholly longitudinal. Particles are displaced away from and back toward the origin of the sound. These oscillations can be illustrated with a matrix of horizontal moving dots, as in Movie 1a (also Fig. 1 a). Transverse waves can also be illustrated with a matrix of moving dots. Note that the longitudinal wave depicted in Movie 1a and the transverse wave depicted in Movie 1b (also Fig. 1 b) both propagate rightward, but whereas each individual dot in Movie 1a oscillates horizontally, each individual dot in Movie 1b oscillates vertically. In both cases, what is propagating is not the local elements per se but the phase of the oscillations, which advances from left to right. Whereas some perceptual qualities of transverse waves travelling through visual texture (Zanker 1994 ; Lu & Sperling 1995 ) have been studied using psychophysical paradigms like those used with drifting luminance (e.g. Graham, 1972 ) and chromatic (e.g. Cavanagh, Tyler & Favreau, 1984 ) gratings, our focus here is on the properties of longitudinal waves, in which local oscillations are parallel to the wave’s propagation. Movie 1. Animation of Fig. 1 . Here and for subsequent movies, double-click to activate. Formal specification Each particle in a longitudinal wave oscillates around a fixed position \(\:{x}_{0}\) , called its equilibrium position. At any time \(\:t\) , its position \(\:x\) can be described as $$\:x={x}_{0}+a\:\text{s}\text{i}\text{n}\left[\omega\:\left(\frac{{x}_{0}}{c}-t\right)\right],\:\:\:\:\:\:\:\:\:\:(\text{E}\text{q}.\:1)$$ where \(\:a\) is the amplitude, \(\:\omega\:\) is the angular frequency, and \(\:c\) is the propagation speed. (Note that longitudinal waves in a 3D medium propagate three-dimensionally (though anisotropically) from the (1D) source of vibration, whereas transverse waves can only be two-dimensional in a 3D world. For convenience, we will assume that fixed positions are randomly sampled from a 1D uniform distribution. Wavelength is defined as $$\:\lambda\:=\frac{2\pi\:c}{\omega\:}.\:\:\:\:\:\:\:\:\:\:(\text{E}\text{q}.\:2)$$ Each panel in Fig. 2 shows the probability density function of the travelling wave over space for \(\:n=2\) wavelengths. There seems to be no closed-form solution for this function. Instead, we fixed \(\:c=\omega\:,\) \(\:t=0,\) and used numerical methods (Mathematica’s NIntegrate) to compute the cumulative distribution function over \(\:n\) wavelengths of \(\:{x}_{0}\) , differentiated with respect to \(\:x\) , and plotted this differential, i.e., $$\:{f}_{X}\left(x\right)=\frac{d}{dx}\:\frac{{\int\:}_{0}^{n\lambda\:}H\left[x+\lambda\:-{x}_{0}-a\:\text{s}\text{i}\text{n}\left({x}_{0}\right)\right]\:d{x}_{0}}{{\int\:}_{0}^{n\lambda\:}H\left[n\lambda\:-{x}_{0}-a\:\text{s}\text{i}\text{n}\left({x}_{0}\right)\right]\:d{x}_{0}},\:\:\:\:\:\:\:\:\:\:(\text{E}\text{q}.\:3)$$ where \(\:H\) denotes the Heaviside step function. The behavior of Eq. 3 is that increasing the oscillation amplitude does not merely increase the amplitude of the density modulation, it also changes the shape of the modulation from quasi-sinusoidal at low amplitudes (e.g. \(\:a=0.003\lambda\:\) ) to spiky at moderate amplitudes (e.g. \(\:a=0.1\lambda\:\) ) to a waveform with two peaks per wavelength (e.g. when \(\:a=0.3\lambda\:\) ), as seen in Fig. 2 . The maximum and minimum values of these function vary as nonlinear functions of the oscillation amplitude (Fig. 3 A). Nevertheless, using the Michelson ratio [(maximum – minimum)/(maximum + minimum)] to index the density modulation, we find that modulation increases linearly with oscillation amplitudes less than \(\:0.16\lambda\:\) , where it hits a Michelson ratio of 0.95 (see Fig. 3 ). From this point, the amplitude appears to drift down slightly, although this portion of the curve is not defined at high accuracy due to the sampling limitations of the numerical methods used for this assessment. We note that these nonlinear relationships between oscillation amplitude and the density measure seems essentially absent from the literature on the physics of sound. Note that the ratio between each particle’s maximum speed and the propagation speed of the medium is 2πa⁄λ. Thus, an individual particle will briefly move faster than the longitudinal wave it forms whenever a > λ/(2π). We suggest that the resultant distortion can be considered the cyclic equivalent of shockwaves, such as the sonic boom created when a jet moves faster than the speed of sound. From the viewpoint of the usual sinusoidal approximation to this nonlinear function, it is important to know the levels at which it is applicable. A distortion criterion of < 1% root mean square deviation is exceeded at the \(\:0.003\) amplitude and the criterion of < 10% is exceeded at the \(\:0.03\) amplitude, the latter being depicted in Fig. 2 for visual reference. For comparison, these levels correspond to acoustic vibrations of about 0.1 and 1 mm in amplitude, respectively, for sound at 10 kHz travelling in air, which are well within the range of high-volume audio loudspeakers. Methods The stimuli are presented as movies throughout the text. Mathematica or MATLAB was used to create each movie. The code for each movie can be downloaded from the website http://www.staff.city.ac.uk/~solomon/LongitudinalWaves.zip . While \(\:a\lambda\:/\left(2\pi\:\right)\) , the peaks double and odd multiples of \(\:\pi\:\) radians become local minima. In that case, local maxima were found using Mathematica's FindMaximum routine, which necessarily fails at the singular point when \(\:a=\lambda\:/\left(2\pi\:\right)\) . For all other oscillation amplitudes, sampling density was as close to as possible to 1024 phases per wavelength, subject to the constraints that all local maxima and local minima should be sampled, and all samples would be equally spaced. Note that derivative d/dx in Eq. 3 can be written \(\:\underset{h\to\:0}{\text{lim}}\left[r\left(x+h\right)-r\left(x\right)\:\right]/h\) , where \(\:r\left(x\right)\) is the ratio of that equation’s two integrals, expressed as a function of \(\:x\) . This derivative was approximated using \(\:\left[r\left(x+h\right)-r\left(x\right)\right]/h\) , with \(\:h={10}^{-10}\) . Results Unipolar dot travelling waves Stimuli Inspired by the on-line resource created by Zeleny et al. ( 2011 ), we created visual renditions of annular longitudinal waves from Eq. 1, using a substrate of 600 oscillating random-dot samples whose equilibrium positions were uniformly distributed throughout an annular region of a 2:1 radial extent. Movie 2 contains 6 annular wave motions of increasing amplitude corresponding to the 6 panels in Fig. 2 . The amplitudes are specified as its proportion of a wavelength. Movie 2. Dynamic illustrations of longitudinal waves, propagating clockwise through annuli of randomly placed dots. The annular format is designed to allow fixation at the center of each annulus to eliminate tracking eye movements. The angle of each dot oscillates sinusoidally in place with an amplitude ranging from \(\:0.003\lambda\:\) (top left) to \(\:a=0.3\lambda\:\) (bottom right), as labeled. The red dot near the top is designed to aid verification that each dot is oscillation in place. The motion conditions match those diagrammed in Fig. 2 . In all panels, the phase propagation speed is 42 polar deg/s. Wave motion is visible for oscillation amplitudes above ~ \(\:0.01\lambda\:\) . Whereas rectangular dot arrays with horizontal or vertical oscillations like those illustrated in Fig. 1 tend to encourage eye movements in the direction of propagation (or possibly in the opposite direction), we opted to make perceptual judgments with annular dot arrays, because fixation at the center of an annulus discourages eye-movement tracking in any particular direction, keeping the dot array rotating at a fixed retinal eccentricity. Observations With fixation at the annulus center to avoid foveal tracking, it is difficult to appreciate that each dot is merely oscillating around a stationary position in the annulus. This oscillation can, however, be verified by foveating any individual dot within the annular band. With central fixation, wave propagation is immediately apparent as the clockwise rotation of the eight compression regions (“crests”) around each of the annuli with sufficient amplitude. The rarefaction regions (“troughs”) between crests appear to cohere into a uniform texture that rotates backwards (here, counterclockwise), even though the phase propagation direction is clockwise. This reverse motion thus represents the intrusion of the local motion of the individual dots for half the phase of the local oscillations, though perceived as a coherent reverse motion of the half-phase patches. At the intermediate amplitudes, most observers can discern a subtle impression of depth, in which the compression regions appear closer to the viewer than the rarefaction regions. With focal attention, the relative salience of individual crests and troughs may fluctuate, but the opposing directions of motion they convey can be experienced simultaneously. The perceived speeds of rotation of the crests and troughs appear to increase with oscillation amplitude. This percept is robust, but it is an illusion because each crest (and trough) requires exactly 8.53 s to complete a full revolution around the annulus center. Density-luminance reciprocity Stimuli The high-density peak regions of our black-dot stimuli necessarily have a lower average luminance than the low-density trough regions. To determine if and how the visual perception of longitudinal motion was contingent upon this “reciprocity” between density and average luminance (Mulligan & MacLeod, 1988 ), we created stimuli in which dot luminance was proportional to the average density of dots in each phase of the longitudinal wave. This manipulation eliminates the (expected) luminance contrast between crests and troughs, virtually eliminating their ability to stimulate standard motion-energy mechanisms, including the Reichardt detector (Reichardt, 1987 ; van Santen & Sperling, 1985 ). Observations With luminance equated, the crests still appear to rotate forwards (clockwise) and the troughs backwards (counterclockwise), as in the original version. The wave motion is fully visible for levels of 0.03λ and above. The depth impression is similar to that for the original version with uncompensated luminance modulation. This luminance-balanced control makes clear that the percept of the bidirectional wave motion is undiminished from the level of 0.03λ and above, suggesting that it can be conveyed by something other than standard, luminance-based motion-energy mechanisms. Movie 3. Luminance-balanced version of Movie 2, in which dot luminance varies in proportion to dot density. Wave motion is easily visible from about \(\:\:0.03\lambda\:\) . Polarity-randomized dots Stimuli Randomly selecting the polarity (black or white) of each dot is guaranteed to reduce any contrast between the average luminances of crest and trough, consequently minimizing the contribution from standard, luminance-based motion-energy mechanisms to the wave-motion percept. Examples of these “drift-balanced” stimuli (Chubb & Sperling, 1988 ) have been provided in Movie 4. Observations All the observations made with black-dot stimuli (discussed in the section on unipolar dot travelling waves) apply equally to the polarity-randomized stimuli. Evidently, luminance contrast is not required for the perception of longitudinal wave motion, or for the visual segregation of the clockwise-propagating crests from the troughs, which again appear to rotate in the counterclockwise direction. Density-contrast reciprocity Stimuli The high-density peaks of our polarity-randomized stimuli necessarily have a more contrast energy than the low-density troughs. To determine if and how the visual perception of longitudinal motion was contingent upon this “reciprocity” between density and contrast energy (Morgan et al. 2022 ), we created stimuli in which the absolute value of each dot’s Weber contrast was proportional to the average density of dots in each phase of the longitudinal wave. This manipulation reduces the angular modulation of contrast energy around each annulus, consequently reducing its ability to stimulate the “2nd -order” motion system, putatively responsible for computing the direction of spatiotemporal amplitude modulations (Chubb & Sperling, 1989 ). Observations Contrast balancing weakens both clockwise and counterclockwise apparent motions, but both motions remain visible at moderate oscillation amplitudes ( \(\:0.03\lambda\:\:-0.1\lambda\:\) ). 2. At high amplitudes ( \(\:>0.16\lambda\:\) ), clockwise propagation of the crests becomes very hard to see, and the counterclockwise rotation of the trough regions dominate perception. Movie 5. Contrast-balanced version of Movie 2, in which dot contrast varies with dot density. Wave motion is easily visible \(\:0.03\lambda\:\:-0.1\lambda\:\) . The basic result that the transparent counter-rotating wave motion survives both luminance balancing and contrast balancing suggests that the wave motion percept derives neither from the spatiotemporal modulations of luminance nor from spatiotemporal modulations of contrast energy. What about spatiotemporal modulations of dot density? Note that several lines of evidence (e.g., Morgan et al., 2014 ; Morgan et al., 2022 ; Zeljic et al., 2024 ) suggest that texture density is computed locally, early in the hierarchy of visual computations. Density waves Stimuli One inescapable feature of longitudinal waves is the periodic pile-up of particles in the direction of propagation. We test its ability to convey propagation when other features of the longitudinal wave have been removed. Although our contrast-balanced stimuli (described in the section on density-contrast reciprocity) have spatiotemporal modulations of neither luminance nor contrast energy, they still contain spatiotemporal modulations of motion: the average motion of crest dots is clockwise, while the average motion of trough dots is counterclockwise. We virtually eliminated this local motion without changing the overall structure of our annuli by randomly re-assigning the radial position of each dot within the annulus on each frame, thus removing the local oscillatory dot motion but retaining the density modulation waves. The version of this stimulus in Movie 6. tests whether waves of dot density in the absence of coordinated local motions of the dots provides a motion cue when compensated for the ancillary contrast modulation. Movie 6. Contrast-balanced density waves as in Movie 5, but modified to eliminate local dot motions by randomizing the radial position of each dot. Propagation of individual crests can be seen with effort at high amplitudes. Observations Having disrupted each dot’s trajectory, counterclockwise motion remains at neither the local level nor the global level. No dot moves clockwise either, but the relatively dense, low-contrast crests continue to rotate physically in a clockwise direction around the annulus. That motion is very hard to see, but it does seem possible to track the motion of any individual crest with local attention. There is a lot of flicker due to resampling the radial location of each dot in the annular region, including content at high temporal frequencies. We conclude that waves of pure dot density with nulling of the consequent contrast modulation are almost invisible. The slight residual motion is likely attributable to an imperfect match of the dot density and compensatory contrast modulation functions. Flickering particles Did the high temporal frequency content in our density waves simply mask their density modulation, or was the local motion of individual dots (absent from our density waves in previous sections) important for the appearance of contrast-balanced longitudinal waves? Stimuli To assess the impact of flicker on the visibility of contrast-balanced longitudinal waves, we created a new version of Movie 5 in which dot polarity was reassigned randomly on each frame. Note that this manipulation doesn’t necessarily produce the same amount of flicker inherent in the density waves, but it does produce equally high temporal frequencies. Movie 7. Flickering version of Movie 5, in which dot polarity is randomly reassigned on each frame. Observations Flicker eliminates any appearance of coherent motion from the low amplitude ( \(\:0.1\lambda\:\) ) stimuli, but individual crests can be seen to rotate clockwise with effort. When the oscillation amplitude \(\:a=0.1\lambda\:\) , all 8 crests can be seen to rotate coherently, but the counterclockwise rotation of the troughs dominates (as it does with higher oscillation amplitudes). The introduction of random polarity reassignment unquestionably served to mask the apparent propagation of some of the contrast-balanced stimuli. It is conceivable that the apparent propagation of all our contrast-balanced stimuli could have been masked with more flicker. Consequently, it is uncertain whether the spatiotemporal modulation of dot density is sufficient to convey the impression of propagation amongst longitudinal waves, or whether a contribution from each dot’s oscillating trajectory is required. A contribution from those oscillations seems to be required for the impression of coherent, retrograde motion from the troughs. Cancellation of wave motion by opposing directions of rigid motion The question arises how the apparent motions of crests and troughs are related to the local motions of the dots throughout the waveform. Clearly, neither direction of apparent motion corresponds to the average dot motion, because each dot is merely oscillating in place. Its motion is neither forward (clockwise) nor backward (counterclockwise) on average. We addressed this question by adding a rigid rotation to the travelling wave stimuli to determine at what speed it would be perceived to cancel either the clockwise crest motion or the counterclockwise trough motion. The default hypothesis is that the apparent speed is controlled by the rate of phase propagation, which is held constant in the following cancellation tests. Another possibility is that the apparent forward and backward motions correspond to the dots’ fastest forward and backward motions (with velocities \(\:\pm\:2\pi\:ac/\lambda\:\) ), respectively. Alternatively, the apparent velocities could correspond to the dots’ average forward or backward motions, respectively (with velocities \(\:\pm\:4ac/\lambda\:\) ). (Note that, here and subsequently, we use the term ‘velocity’ to specify angular velocity relative to the center of the annular stimuli.) Stimuli To test between these alternatives, stimuli were generated with a range of rigid motions added to all the dots in the annular longitudinal waves. Movie 8A-C illustrates a consensus when the oscillation amplitude had the high amplitude \(\:0.25\) (i.e., beyond the linear range of Fig. 3 B) In this case we judged the crests to be static when a rigid counterclockwise motion having a speed equal to 105% of the wave’s propagation (i.e. \(\:1.05c\) ) was added to each dot. The troughs were judged to be static when a rigid clockwise motion having a speed equal to 150% of the wave’s propagation was added to each dot. The longitudinal wave in the central annulus has no additional rigid motion. Movie 8. Different levels of rigid rotational motion added to each dot of high-amplitude ( \(\:0.25\lambda\:\) ) longitudinal waves. Motion of the frames is keyed to the added motion to give a clear indication of its velocity. A: With matching reverse added motion (speed \(\:1.05c\) ). B,E: No added motion. C: Matching forward added motion (speed \(\:1.50c\) ). D,F: Equal but non-matching intermediate speed (speed \(\:1.30c\) ). Note that the crests appear stationary in panel A and the trough regions appear stationary in panel C. In panels D and F, neither the crests nor the troughs appear stationary (both appear to rotate counterclockwise) but the frames rotate at the same speed. Panels 8D and F illustrate the same high-amplitude longitudinal wave with equal but opposite rigid velocities of 1.3 c . This intermediate speed proves too fast to cancel the crests’ motion (they appear to rotate counterclockwise in 8D) and too slow to cancel the troughs’ (they appear to rotate counterclockwise in 8F). Cancellation speeds for lower-velocity waves are plotted in Fig. 4 . Movie 9 contains the same longitudinal waves shown in Movie 2, with the wave’s actual propagation velocity subtracted from each dot. This is approximately the correct speed for cancelling the apparent propagation of the crests only when the oscillation amplitude \(\:a\approx\:0.2\lambda\:\) . For larger amplitudes, it is insufficient, and the crests appear to rotate clockwise.) For smaller amplitudes, it is overkill, and any discernible crests appear to rotate counterclockwise. These speed-cancellation observations reveal the remarkable result that the perceived propagation speed is some form of local average that lies between the particles’ average speed in that direction and their maximum speed. This is also true for the transparent, retrograde motion of the troughs, which seem to move a little faster. This difference in apparent speed increases with oscillation amplitudes > \(\:0.1\lambda\:\) , whereas the difference between the apparent widths of crest and trough is maximal when \(\:a=0.16\lambda\:\) . Movie 9. Identical levels of counterclockwise motion with propagation speed c added to Movie 2 modified by adding identical levels of counterclockwise motion at its phase propagation speed of \(\:c\) = 42 deg/s. This is too slow to cancel the apparent clockwise propagation when \(\:a=0.3\lambda\:\) (bottom right) but too fast to cancel apparent propagation when \(\:a<0.2\lambda\:\) . Observations (see Fig. 4 ) The cancellation speeds are not constant but scale monotonically with the oscillation amplitudes. Setting the cancellation velocity to the average velocity of the crest or trough regions, respectively fails to cancel the perceived motion (except in one incidental case of the crest velocity at 0.3 lambda), invalidating the averaging hypothesis that the dot motion in whole of each half-cycle contributes to the perceived velocity. Effective cancellation requires added velocities closer to the maxima within each crest or trough. For higher-amplitude waves, the trough-cancellation speed substantially exceeds the crest-cancellation speed. These speed-cancellation observations reveal the remarkable result that the perceived wave motion depends on some property of the local dot motions whose oscillation is essentially invisible without guided scrutiny. This property is some form of local average that lies between their average speed and their maximum speed, separately and differently within the crest and trough regions \(\:.\) Motion aftereffect Stimuli A question that does not seem to have been previously addressed in the perceptual literature is whether travelling wave motion generates a motion aftereffect. This may be viewed in a cyclic adaptation paradigm of 10-s adapting and 2-s test periods, as seen in Movie 10. Note here that the high-amplitude longitudinal wave in panel B is identical to the one in Movie 8B. The rotating texture in panel A has a velocity equal to the sum of the rigid velocities required to cancel the apparent motions of crests and troughs, as described in the previous section. Movie 10. Motion aftereffect and lack thereof. A 2.13-s static test period follows 10.67 s of motion adaptation with a uniform texture rotating at 6.3 deg/s (panel A) and a high-amplitude ( \(\:0.25\lambda\:\) ) longitudinal wave propagating at 42 deg/s (panel B). The angular velocity in panel A is set to the sum of the rigid velocities required to cancel the two directions of transparent motion in panel B (see Movie 8A–C). A strong motion aftereffect is seen during the static test period with centered adaptation in panel A, but not for the same in panel B. Observations Even though the sum of (oppositely signed) velocities required to cancel the longitudinal wave’s opposite directions of apparent motion is small, that sum nonetheless elicits a strong motion aftereffect (Movie 9A). The longitudinal wave itself does not elicit any motion aftereffect (Movie 9B). These observations are consistent with evidence (Kohn & Movshon, 2003 ) that the adaptation underlying the motion aftereffect occurs primarily in the directionally selective neurons of cortical area V1, where receptive fields are relatively small. Such neurons favoring clockwise motion would not be expected to receive any stronger stimulation than similarly positioned neurons favoring counterclockwise motion when individual dots are merely oscillating back and forth in their receptive fields. On the other hand, such neurons would receive stronger stimulation from rigid, clockwise motion. Consequently, the rigid motion should produce a significant aftereffect whereas the longitudinal motion should not. Discussion Longitudinal waves are created from ensembles of locally oscillating dots with a progressive phase advance as a function of position. These oscillations create regions of compression and rarefaction whose propagation and retrograde motions so dominate perception that the local motion of individual dots is all but impossible to discern. These global, transparent motions thus arise by some process of motion integration. Investigation of physical longitudinal wave motion reveals that it exhibits profound nonlinearities that are largely neglected in physics community. Longitudinal motion is almost universally treated with the small amplitude approximation, where a sinusoidal driving function is assumed to generate an essentially sinusoidal travelling wave, in the longitudinal wave case as in the transverse wave case. A form of high amplitude nonlinearity for pressure shock waves (impulse responses) was recognized early in the C19th by Poisson (1808) and elaborated by Challis ( 1848 ) and Stokes ( 1848 ) to show that the longitudinal shock waves approximated an asymmetric sawtooth form that could become multivalued at high amplitude, violating the assumptions of the differential equations. This nonlinear behavior is due to the adiabatic properties of the transmission fluid and does not apply to the basic ideal gas that obeys Boyle’s Law, in which the transmission speed is invariant (Blackstock et al., 2020 ). We show that, even for locally linear transmission through the medium at constant speed, the density of the longitudinal wave motion becomes notably non-sinusoidal at oscillation amplitudes beyond about 2% of the wavelength, and progressively piles up into a narrow cyclic density spike around 16% of the wavelength, beyond which the peak splits into a double spike as the density accretion overtakes the maximum velocity of the local medium being perturbed. This nonlinearity is of entirely different character than the adiabatic shock wave nonlinearities that were the subject of contentious analyses in the C19th. Those turned out to be time-asymmetric in the direction of a sawtooth wave, whereas our analysis applies to speed-invariant Boylean gases, in which the density singularity for a sinusoidal input remains time-symmetric but becomes a double singularity at higher oscillation amplitudes. Having established the full nature of longitudinal waves, we then turned to the perceptual appreciation of visual depictions of wave motion, such as are often used in acoustics courses. Instead of the usual linear wave of Fig. 1 , we use rotating ring configurations of travelling wave motion to control eye movements. These show that the local dot motion in the wave is virtually invisible, but is subsumed under the global impression of motion, in which each cycle splits into two opposing regions: a dominant forward motion of the crest regions and a residual backward motion of the trough regions. This percept is obtained even when the oscillation amplitude is small enough that the wave motion remained sinusoidal. At higher amplitudes, the forward motion of the crests is enhanced by the nonlinear pile-up of the dot density there, which results in them being perceived as narrow ‘walls’ between broad ‘fields’ of opposing motion. In the rotating ring configuration, the individual cycles then integrated into transparent counterrotating motion fields. Perceptual binding of all 8 crests (or all 8 troughs), see Movie 2 into a single coherent texture necessarily requires neurons having receptive fields large enough to be stimulated by them. We then asked whether the appearance of the forward component of the longitudinal wave motion was attributable to the difference between the average luminances of the crests and troughs. This hypothesis was tested in two ways. First, we reduced the crests’ average luminance to match that of the troughs (Movie 3). Second, we randomized dot polarity so that the expected luminance within each phase of the wave was equal to that of the mid-gray background (Movie 4). Neither of these manipulations eliminated the impression of transparent, global rotations in opposite directions. The impression of global motion in either direction can be eliminated by the addition of a uniform velocity to each particle. This cancelling velocity is close to the peak oscillation velocity (Movie 8), but we have found that it varies with oscillation amplitude and direction (i.e., consistent with or opposite to that of propagation). The distinction between a global property and the local components from which it is formed was highlighted by the Gestalt psychologists at the beginning of the twentieth century. The visual system’s ability to integrate local motion signals into a global percept has been recognized (e.g. by Smith et al., 1994 ) since the end of the twentieth century. Just as the Smith group sought to eliminate spatial frequency components from their random-dot stimulus that could potentially excite motion-energy detectors on a global scale, we too examined various “drift-balanced” (Chubb & Sperling, 1988 ) modifications of our black-dot stimuli that were designed to hide their emergent properties from “1st -order” motion detectors (Cavanagh & Mather, 1989 ; Chubb & Sperling, 1989 ). As previously noted, the mere elimination of global-scale motion energy (via luminance cancellation or polarity randomization) proved ineffective at eliminating the impression of transparent, global motion in opposite directions. However, reduction of the crests’ contrast did make them harder to see (Movie 5). In this case, the retrograde motion of the troughs was perceptually dominant. One manipulation that did succeed in eliminating the impression of retrograde motion was the cancellation of the local motion from the troughs (Movie 6). The propagation of high-density crests in these latter movies was just perceptible. The perceptual results were extended by testing for a motion aftereffect generated by longitudinal wave motion. In fact, motion aftereffects are essentially non-existent for the pure longitudinal wave motion, even though strong motion aftereffects are seen for rigidly moving textures of the same format (Movie 9). Given our contention that the impression of transparent global motions arises from large-scale mechanisms that integrate local motion information from individual dots, this negative result may not seem surprising. After all, the inability of other non-Fourier motion stimuli to elicit an aftereffect in static test stimuli has been well established (Banks & Kane, 1972 ; Nishida & Sato, 1995 ). That being said, it is important to remember that black-dot longitudinal waves would indeed be expected to stimulate motion-energy detectors sensitive to the difference between the average luminances of crest and trough. It is therefore surprising to find that these longitudinal waves elicit no motion aftereffect. Declarations Additional Information No competing interests Funding Declaration This study was not supported by external funding Author Contribution Initial observations and background: CWTConceptual analysis: CWT, JAS, SMAMathematical analysis and stimulus programming: JASNarrative description: CWT, JAS, SMA Data Availability The code for each movie can be downloaded from the website http://www.staff.city.ac.uk/~solomon/LongitudinalWaves.zip. References Banks, W. P., & Kane, D. A. (1972). Discontinuity of seen motion reduces the visual motion aftereffect. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 290, 153-168. Blackstock, D. T., Cormack, J. M., & Hamilton, M. F. (2020). Early history of nonlinear acoustics. In Proceedings of Meetings on Acoustics 2019. Acoustical Society of America, 36:1, 045007. Cavanagh, P., Arguin, M., & von Grünau, M. (1989). Interattribute apparent motion. Vision Research 29, 1197-1204. Cavanagh, P., & Mather, G. (1989). Motion: the long and short of it. Spatial Vision, 4, 103-129. Cavanagh, P., Tyler, C. W., & Favreau, O. E. (1984). Perceived velocity of moving chromatic gratings. Journal of the Optical Society of America A 1, 893-899. Ceperley, P. (2016). The Lagrangian approach to simple waves – several common waves that lack momentum. Resonances, Waves and Fields. https://resonanceswavesandfields.blogspot.com/2016/01/the-lagrangian-approach-to-simple-waves.html (Accessed 10/10/2024). Challis, J. (1848) On the velocity of sound, in reply to the remarks of the Astronomer Royal. Phil. Mag. Ser. 3, 32, 494–499. Chubb, C., & Sperling, G. (1988). Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception. Journal of the Optical Society of America A, 5, 1986-2007. Chubb, C., & Sperling, G. (1989). Two motion perception mechanisms revealed through distance driven reversal of apparent motion. Proceedings of the National Academy of Sciences USA, 86, 2985-2989. Graham, N. (1972). Spatial frequency channels in the human visual system: Effects of luminance and pattern drift rate. Vision Research 12 , 53-68. Kohn, A., & Movshon, J. A. (2003). Neuronal adaptation to visual motion in area MT of the macaque. Neuron 39, 681. Ledgeway, T., & Smith, A. T. (1994). The duration of the motion aftereffect following adaptation to first-order and second-order motion. Perception 23, 1211-1219. Lu, Z.-L., & Sperling, G. (1995). The functional architecture of human visual motion perception. Vision Research 35, 2697. Morgan, M. J., Macleod, D. I. A., & Solomon, J. A. (2022) The channel for detecting contrast modulation also responds to density modulation (or vice versa ). Vision Research, 192, 107948. Mulligan, J., & MacLeod, D. I. A. (1988). Reciprocity between luminance and dot density in the perception of brightness. Vision Research 28, 503-519. Morgan, M. J., Raphael, S., Tibber, M. S., & Dakin, S. C. (2014). A texture processing model of the ‘visual sense of number’. Proc. Roy. Soc. B., 281, 20141137. Nishida, S., & Sato, T. (1995). Motion aftereffect with flickering test patterns reveals higher stages of motion processing. Vision Research, 35, 477-490. Petersik, J. T., Hicks, K. I. & Pantle, A. J. (1978). Apparent movement of successively generated subjective figures. Perception, IO, 563-572 Reichardt, W. (1987) Evaluation of optical motion information by movement detectors. Journal of Comparative Physiology A. 161(4):533-47. Seiffert, A. E., & Cavanagh, P. (1998). Position displacement, not velocity, is the cue to motion detection of second-order stimuli. Vision Research 38, 3569-3582. Smith, A. T., Snowden, R. J., & Milne, A. B. (1994). Is global motion really based on spatial integration of local motion signals? Vision Research, 34, 2425-2430. Sperling, G., Solomon, J. A., Lu, Z., & Chubb, C. (1994). First and second-order processes in the perception of motion and texture. In J.M. Zurada, R.J. Marks II, & C.J. Robinson, Computational Intelligence: Imitating Life, 223-236. Stokes, G.G. (1848) On a difficulty in the theory of sound. Phil. Mag. Ser. 3, 33, 349–356. Whitney, David, & Bressler, D.W. (2007). Second-order motion without awareness: Passive adaptation to second-order motion produces a motion aftereffect. Vision Research 47, 569-579. Van der Smagt, M. J., Verstraten, F. A., Vaessen, E. B., van Londen, T., & van de Grind, W. A. (1999). Motion aftereffect of combined first-order and second-order motion. Perception 28, 1397-1411. Van Santen J.P., Sperling G. (1985) Elaborated Reichardt detectors. Journal of the optical Society of America A. 2(2):300-21. Zanker, J. M. (1993). Theta motion: A paradoxical stimulus to explore higher order motion extraction. Vision Research, 33, 553-569. Zanker, J. M. (1994). What is the elementary mechanism underlying secondary motion processing? Investigative Ophthalmology & Visual Science (Supplement), 35, 1405. Zeleny, E., Bengtsson, M. G., & Russell, D. A. (2011) Longitudinal and Transverse Waves. https://demonstrations.wolfram.com/LongitudinalAndTransverseWaves/ (Accessed 16/10/2024). Zeljic, K., Morgan, M. J., & Solomon, J. A. (2024). Monocular and binocular mechanisms detect modulations of dot density and dot contrast. Vision Research, 215, 108347. Movies Movies 1 to 10 are available in the Supplementary Files section. Additional Declarations No competing interests reported. Supplementary Files Movies.docx Cite Share Download PDF Status: Published Journal Publication published 23 Mar, 2026 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 19 Nov, 2025 Reviewers agreed at journal 26 Oct, 2025 Reviews received at journal 17 Oct, 2025 Reviewers agreed at journal 14 Oct, 2025 Reviewers invited by journal 14 Oct, 2025 Editor invited by journal 14 Oct, 2025 Editor assigned by journal 11 Oct, 2025 Submission checks completed at journal 11 Oct, 2025 First submitted to journal 07 Oct, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7802344","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":535189375,"identity":"1f04f15a-b47c-484f-87cb-147002a2144c","order_by":0,"name":"Christopher W. Tyler","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAlElEQVRIiWNgGAWjYHACxgcMByRI0sHMbECyFjYJhgOkaOCfkX+s8scZCwZ+6eMXiNMicSOZ7TbPDQkGyb6cAuK0GEgDtTB8kGAwOMOTQLyWwh8ka2EAOczgDPsB4rRI3H9sLM1zRoJHsoeHOB0M/D0HH378caxOjp+H/QGReqAAaAWPAWlagIBUW0bBKBgFo2DEAABNSSX7QRUgbQAAAABJRU5ErkJggg==","orcid":"","institution":"Smith-Kettlewell Eye Research Institute","correspondingAuthor":true,"prefix":"","firstName":"Christopher","middleName":"W.","lastName":"Tyler","suffix":""},{"id":535189377,"identity":"cb5ed747-def4-480b-9dc3-121d04c14fff","order_by":1,"name":"Joshua A. Solomon","email":"","orcid":"","institution":"University of London","correspondingAuthor":false,"prefix":"","firstName":"Joshua","middleName":"A.","lastName":"Solomon","suffix":""},{"id":535189378,"identity":"3044f67c-d141-4826-a03a-ef78315e53ce","order_by":2,"name":"Stuart M. Anstis","email":"","orcid":"","institution":"University of California","correspondingAuthor":false,"prefix":"","firstName":"Stuart","middleName":"M.","lastName":"Anstis","suffix":""}],"badges":[],"createdAt":"2025-10-07 19:53:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7802344/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7802344/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-026-36204-y","type":"published","date":"2026-03-23T16:09:13+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":94589885,"identity":"f2ff8370-71f1-4f3c-a192-f0dce56fcf7a","added_by":"auto","created_at":"2025-10-28 18:20:45","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":2904589,"visible":true,"origin":"","legend":"","description":"","filename":"LongitudinalWaveMotionFinal.docx","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/39278af963e6abcc839af976.docx"},{"id":94590376,"identity":"04904b12-0887-4fdc-9403-aa7d5d475ace","added_by":"auto","created_at":"2025-10-28 18:21:08","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":4960,"visible":true,"origin":"","legend":"","description":"","filename":"d1bbf84b0b464806bc8b1f6c2a794918.json","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/6631b3c1954fc12cdc71951c.json"},{"id":94590118,"identity":"f09fb8f3-db00-4d88-bfe9-0cb537bed79c","added_by":"auto","created_at":"2025-10-28 18:20:57","extension":"xml","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":92621,"visible":true,"origin":"","legend":"","description":"","filename":"d1bbf84b0b464806bc8b1f6c2a7949181enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/f5454391e99c3dee6d2bd184.xml"},{"id":94589374,"identity":"e46918ad-d0df-42f7-b0a3-8178379aa499","added_by":"auto","created_at":"2025-10-28 18:20:11","extension":"jpeg","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":333854,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/a364ca2cc530452fcaf7d23a.jpeg"},{"id":94590408,"identity":"5cf89b53-5c5c-4964-820c-b88d0aef73e2","added_by":"auto","created_at":"2025-10-28 18:21:11","extension":"png","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":250844,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/85e36534f8eb4ed0090d1596.png"},{"id":94588899,"identity":"781f6d5b-af07-4af3-b08e-ff7fbee389e7","added_by":"auto","created_at":"2025-10-28 18:19:53","extension":"png","order_by":5,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":200008,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/4b30e7db3e4690bb46961e88.png"},{"id":94589895,"identity":"df846cec-14f6-42c7-be59-fdb997e838e0","added_by":"auto","created_at":"2025-10-28 18:20:45","extension":"png","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":271413,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage12.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/4d526e24eda912a5bbfa51d1.png"},{"id":94590395,"identity":"17bd639d-0dd6-47f9-b992-6b87e9e03fdf","added_by":"auto","created_at":"2025-10-28 18:21:10","extension":"emf","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":112072,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage13.emf","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/5c0e35f9d88055600a79210a.emf"},{"id":94590456,"identity":"ee73d878-77d3-481f-962d-6610de63bb03","added_by":"auto","created_at":"2025-10-28 18:21:13","extension":"png","order_by":8,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":205862,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage14.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/60af66eed5096e9de0d67e0d.png"},{"id":94589865,"identity":"bc0bcb2e-3b30-455e-9e38-07c067de723f","added_by":"auto","created_at":"2025-10-28 18:20:42","extension":"png","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":173646,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/03f6c7dd374a741cda7b1806.png"},{"id":94589237,"identity":"323447cc-ad94-43ed-9de9-b937768b97bc","added_by":"auto","created_at":"2025-10-28 18:20:03","extension":"jpeg","order_by":10,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":192553,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage3.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/120004b7a4421ac5d3c7b433.jpeg"},{"id":94590455,"identity":"37d4f9f9-6e40-4e67-af50-90f40c62af8d","added_by":"auto","created_at":"2025-10-28 18:21:13","extension":"jpeg","order_by":11,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":160314,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage4.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/b5ec2cc52fe93b1931c2bb61.jpeg"},{"id":94590446,"identity":"e28521e3-9fdd-4e82-b02e-c999bad92728","added_by":"auto","created_at":"2025-10-28 18:21:13","extension":"png","order_by":12,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":286322,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/99c916ba640fa0d421ed9363.png"},{"id":94589906,"identity":"8c57f80e-5659-4d48-92e8-fde477fbd879","added_by":"auto","created_at":"2025-10-28 18:20:47","extension":"png","order_by":13,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":284137,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/562609a7f2548b349ba8071b.png"},{"id":94595614,"identity":"609b5be6-6535-4cc8-92ff-cd75add2098d","added_by":"auto","created_at":"2025-10-28 18:35:30","extension":"png","order_by":14,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":273677,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/406e08d441f389f88ce42970.png"},{"id":94589894,"identity":"fd670186-652e-4055-bad2-adaaa7afa88c","added_by":"auto","created_at":"2025-10-28 18:20:45","extension":"png","order_by":15,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":250698,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/0236200cc7aacc725f8195c7.png"},{"id":94589413,"identity":"4202e9da-560a-4591-9d7e-83c907689c0e","added_by":"auto","created_at":"2025-10-28 18:20:14","extension":"png","order_by":16,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":253813,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/13157348d6473bd087e5c045.png"},{"id":94589554,"identity":"f6630b58-e58a-4247-9ef1-f52341701f66","added_by":"auto","created_at":"2025-10-28 18:20:27","extension":"png","order_by":17,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":296325,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/1408410bb29f8c404dd9f8d6.png"},{"id":94589890,"identity":"f47d2c69-ae3b-4508-8f97-86aace39353d","added_by":"auto","created_at":"2025-10-28 18:20:45","extension":"png","order_by":18,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":128349,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage10.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/03d457ce06446e6761c16f40.png"},{"id":94590448,"identity":"b8d8f3bd-21af-4881-851f-3e9306aa0456","added_by":"auto","created_at":"2025-10-28 18:21:13","extension":"png","order_by":19,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":40953,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage11.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/cf24ca6b67a430a6228f0523.png"},{"id":94589414,"identity":"91550a70-929d-4cbf-98fc-6fcb848cc6dc","added_by":"auto","created_at":"2025-10-28 18:20:14","extension":"png","order_by":20,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":134855,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage12.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/5b6ab600a5a4b538e56e39b6.png"},{"id":94589633,"identity":"dc7a810f-c0f7-403f-8446-d1b5c040159a","added_by":"auto","created_at":"2025-10-28 18:20:29","extension":"png","order_by":21,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":5026,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage13.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/cdfa8cf6e551695c8087ba84.png"},{"id":94589522,"identity":"0b31045d-32fa-4f78-a813-a4f2f3e3ed55","added_by":"auto","created_at":"2025-10-28 18:20:24","extension":"png","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":36505,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage14.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/7e741a983e7e8a7841fd85fa.png"},{"id":94589794,"identity":"253c64f8-a1b7-498a-8cc3-e3783dd97bd7","added_by":"auto","created_at":"2025-10-28 18:20:37","extension":"png","order_by":23,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":87873,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage2.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/960d42b50fffe036be57c591.png"},{"id":94589796,"identity":"c5580ec1-afb3-46e6-add0-d714ced9ee2d","added_by":"auto","created_at":"2025-10-28 18:20:37","extension":"png","order_by":24,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":64865,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage3.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/8cb58e0ef2267456ce76a4d1.png"},{"id":94589649,"identity":"082d5c99-bb3c-401a-95fb-9ee02a4c16f1","added_by":"auto","created_at":"2025-10-28 18:20:31","extension":"png","order_by":25,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":38340,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage4.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/30c0c0aa21a4b04de0a703f7.png"},{"id":94589571,"identity":"83e2e5cc-001c-412a-b9b6-a1abd3fb312e","added_by":"auto","created_at":"2025-10-28 18:20:28","extension":"png","order_by":26,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":55703,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage5.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/ce0de83a701c2c71e5d47714.png"},{"id":94590194,"identity":"e772204e-df3d-409f-b65d-319abb1f048d","added_by":"auto","created_at":"2025-10-28 18:21:00","extension":"png","order_by":27,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":138273,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/9c3e27430aa7ceec3792b52b.png"},{"id":94596052,"identity":"2cc5794f-42ec-40a8-a418-3cf756e53c59","added_by":"auto","created_at":"2025-10-28 18:38:15","extension":"png","order_by":28,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":134761,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage7.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/8f195e09e1b6efd82c8247b1.png"},{"id":94589438,"identity":"83ec9ec3-d44f-43bf-9443-9a48139980de","added_by":"auto","created_at":"2025-10-28 18:20:14","extension":"png","order_by":29,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":128517,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage8.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/7b72cda005022e27b82a85f5.png"},{"id":94589005,"identity":"55e832e7-39db-479a-8e1d-dbf42b22b3a5","added_by":"auto","created_at":"2025-10-28 18:19:55","extension":"png","order_by":30,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":131001,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/d418f45a9423797f4dcac841.png"},{"id":94589376,"identity":"4961ed6e-5135-4502-ae6a-03e1fd303589","added_by":"auto","created_at":"2025-10-28 18:20:11","extension":"xml","order_by":31,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":90908,"visible":true,"origin":"","legend":"","description":"","filename":"d1bbf84b0b464806bc8b1f6c2a7949181structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/8da05dade2cd82e4f5e14db0.xml"},{"id":94590184,"identity":"71cc8946-92bc-45bb-ac4e-f8340bb35b79","added_by":"auto","created_at":"2025-10-28 18:21:00","extension":"html","order_by":32,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":107264,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/b945564f9c9c90ad068ac830.html"},{"id":94590390,"identity":"ba80078d-d129-4e77-aa2f-440e1c4773cc","added_by":"auto","created_at":"2025-10-28 18:21:09","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":202193,"visible":true,"origin":"","legend":"\u003cp\u003eStatic illustrations of (a) longitudinal and (b) transverse waves, propagating rightward through a grid of dots. (See Movie 1 for a dynamic version of this figure.) For illustrative purposes, equilibrium positions of the dots are equally spaced. In all subsequent movies, equilibrium positions were selected at random from a uniform distribution.) In each panel, 10 successive temporal samples of the wave have been arranged vertically. Each dot has been replaced by an arrow indicating the direction and speed of its motion.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/d0eaaad090daf2f85fca4fa1.png"},{"id":94590385,"identity":"8d415f82-2083-4967-a356-5f788dd35585","added_by":"auto","created_at":"2025-10-28 18:21:09","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":153268,"visible":true,"origin":"","legend":"\u003cp\u003eLongitudinal-wave density functions (blue curves) with wavelength λ=2π and oscillation amplitudes ranging from a=0.003λ to \u0026nbsp;a=0.3λ. The green curve at 0.03λ is a true sinusoid whose peak-to-peak amplitude equals that of the density. \u0026nbsp;Note that the waveform at higher amplitudes resembles the double-peaked form at 0.3λ but with wider bars.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/0e9dbd2bb148c2459b33583d.png"},{"id":94590116,"identity":"35fc3a1c-c303-4cd8-815c-c1e2bc92c152","added_by":"auto","created_at":"2025-10-28 18:20:57","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":146148,"visible":true,"origin":"","legend":"\u003cp\u003eSee image above for figure legend\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/a592cd1750ffc4e112877d67.png"},{"id":94590458,"identity":"7fe83a1e-d6d5-4cf9-9194-211e94ade116","added_by":"auto","created_at":"2025-10-28 18:21:13","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":87193,"visible":true,"origin":"","legend":"\u003cp\u003eCancellation speeds for the crest maxima and the trough minima of rotating longitudinal waves at a constant propagation speed of c = 42 deg/s. The only variable is the oscillation amplitude. The solid line plots the maximum dot speed without any additional rigid motion, while the dashed line plots their average speed. In each case, the clockwise and counterclockwise speeds of each dot are equal.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/0ab78fe10f869e5112bde508.png"},{"id":105755869,"identity":"140fc86b-fe5a-44f2-af4f-73803ead913e","added_by":"auto","created_at":"2026-03-30 16:32:01","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1146803,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/be7b85e1-37e5-47b7-8444-8538332d9a5f.pdf"},{"id":94589893,"identity":"c90fcda1-b1fc-4782-91e9-663d44123e0e","added_by":"auto","created_at":"2025-10-28 18:20:45","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":2469731,"visible":true,"origin":"","legend":"","description":"","filename":"Movies.docx","url":"https://assets-eu.researchsquare.com/files/rs-7802344/v1/15db17c3a84115b80aa0cee0.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Visual Perception of Longitudinal Waves: Theory and Observations","fulltext":[{"header":"Introduction","content":"\u003cp\u003eAny travelling wave can be decomposed into longitudinal and transverse oscillations. In mechanical waves, those oscillations are applied to the positions of particles within the solid, liquid, and/or gaseous medium through which the wave travels. (Electromagnetic waves, on the other hand, can travel through a vacuum.) Sound waves in gases are wholly longitudinal. Particles are displaced away from and back toward the origin of the sound. These oscillations can be illustrated with a matrix of horizontal moving dots, as in Movie 1a (also Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003ea). Transverse waves can also be illustrated with a matrix of moving dots. Note that the longitudinal wave depicted in Movie 1a and the transverse wave depicted in Movie 1b (also Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eb) both propagate rightward, but whereas each individual dot in Movie 1a oscillates horizontally, each individual dot in Movie 1b oscillates vertically. In both cases, what is propagating is not the local elements \u003cem\u003eper se\u003c/em\u003e but the phase of the oscillations, which advances from left to right.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eWhereas some perceptual qualities of transverse waves travelling through visual texture (Zanker \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e1994\u003c/span\u003e; Lu \u0026amp; Sperling \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e1995\u003c/span\u003e) have been studied using psychophysical paradigms like those used with drifting luminance (e.g. Graham, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1972\u003c/span\u003e) and chromatic (e.g. Cavanagh, Tyler \u0026amp; Favreau, \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e1984\u003c/span\u003e) gratings, our focus here is on the properties of longitudinal waves, in which local oscillations are parallel to the wave’s propagation.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eMovie 1. Animation of Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Here and for subsequent movies, double-click to activate.\u003c/p\u003e\n\u003ch3\u003eFormal specification\u003c/h3\u003e\n\u003cp\u003eEach particle in a longitudinal wave oscillates around a fixed position \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{x}_{0}\\)\u003c/span\u003e\u003c/span\u003e, called its equilibrium position. At any time \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:t\\)\u003c/span\u003e\u003c/span\u003e, its position \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:x\\)\u003c/span\u003e\u003c/span\u003e can be described as\u003c/p\u003e\u003cdiv id=\"Equa\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equa\" name=\"EquationSource\"\u003e\n$$\\:x={x}_{0}+a\\:\\text{s}\\text{i}\\text{n}\\left[\\omega\\:\\left(\\frac{{x}_{0}}{c}-t\\right)\\right],\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:(\\text{E}\\text{q}.\\:1)$$\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a\\)\u003c/span\u003e\u003c/span\u003e is the amplitude, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\omega\\:\\)\u003c/span\u003e\u003c/span\u003e is the angular frequency, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:c\\)\u003c/span\u003e\u003c/span\u003e is the propagation speed. (Note that longitudinal waves in a 3D medium propagate three-dimensionally (though anisotropically) from the (1D) source of vibration, whereas transverse waves can only be two-dimensional in a 3D world. For convenience, we will assume that fixed positions are randomly sampled from a 1D uniform distribution. Wavelength is defined as\u003c/p\u003e\u003cdiv id=\"Equb\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equb\" name=\"EquationSource\"\u003e\n$$\\:\\lambda\\:=\\frac{2\\pi\\:c}{\\omega\\:}.\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:(\\text{E}\\text{q}.\\:2)$$\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eEach panel in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e shows the probability density function of the travelling wave over space for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:n=2\\)\u003c/span\u003e\u003c/span\u003e wavelengths. There seems to be no closed-form solution for this function. Instead, we fixed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:c=\\omega\\:,\\)\u003c/span\u003e\u003c/span\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:t=0,\\)\u003c/span\u003e\u003c/span\u003e and used numerical methods (Mathematica’s NIntegrate) to compute the cumulative distribution function over \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:n\\)\u003c/span\u003e\u003c/span\u003e wavelengths of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{x}_{0}\\)\u003c/span\u003e\u003c/span\u003e, differentiated with respect to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:x\\)\u003c/span\u003e\u003c/span\u003e, and plotted this differential, i.e.,\u003c/p\u003e\u003cdiv id=\"Equc\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equc\" name=\"EquationSource\"\u003e\n$$\\:{f}_{X}\\left(x\\right)=\\frac{d}{dx}\\:\\frac{{\\int\\:}_{0}^{n\\lambda\\:}H\\left[x+\\lambda\\:-{x}_{0}-a\\:\\text{s}\\text{i}\\text{n}\\left({x}_{0}\\right)\\right]\\:d{x}_{0}}{{\\int\\:}_{0}^{n\\lambda\\:}H\\left[n\\lambda\\:-{x}_{0}-a\\:\\text{s}\\text{i}\\text{n}\\left({x}_{0}\\right)\\right]\\:d{x}_{0}},\\:\\:\\:\\:\\:\\:\\:\\:\\:\\:(\\text{E}\\text{q}.\\:3)$$\u003c/div\u003e\u003c/div\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:H\\)\u003c/span\u003e\u003c/span\u003e denotes the Heaviside step function.\u003c/p\u003e\u003cp\u003eThe behavior of Eq.\u0026nbsp;3 is that increasing the oscillation amplitude does not merely increase the amplitude of the density modulation, it also changes the shape of the modulation from quasi-sinusoidal at low amplitudes (e.g. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a=0.003\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e) to spiky at moderate amplitudes (e.g. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a=0.1\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e) to a waveform with two peaks per wavelength (e.g. when \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a=0.3\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e), as seen in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The maximum and minimum values of these function vary as nonlinear functions of the oscillation amplitude (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eA). Nevertheless, using the Michelson ratio [(maximum – minimum)/(maximum + minimum)] to index the density modulation, we find that modulation increases linearly with oscillation amplitudes less than \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.16\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e, where it hits a Michelson ratio of 0.95 (see Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). From this point, the amplitude appears to drift down slightly, although this portion of the curve is not defined at high accuracy due to the sampling limitations of the numerical methods used for this assessment. We note that these nonlinear relationships between oscillation amplitude and the density measure seems essentially absent from the literature on the physics of sound.\u003c/p\u003e\u003cp\u003eNote that the ratio between each particle’s maximum speed and the propagation speed of the medium is 2πa⁄λ. Thus, an individual particle will briefly move faster than the longitudinal wave it forms whenever a \u0026gt; λ/(2π). We suggest that the resultant distortion can be considered the cyclic equivalent of shockwaves, such as the sonic boom created when a jet moves faster than the speed of sound.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFrom the viewpoint of the usual sinusoidal approximation to this nonlinear function, it is important to know the levels at which it is applicable. A distortion criterion of \u0026lt; 1% root mean square deviation is exceeded at the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.003\\)\u003c/span\u003e\u003c/span\u003e amplitude and the criterion of \u0026lt; 10% is exceeded at the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.03\\)\u003c/span\u003e\u003c/span\u003e amplitude, the latter being depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e for visual reference. For comparison, these levels correspond to acoustic vibrations of about 0.1 and 1 mm in amplitude, respectively, for sound at 10 kHz travelling in air, which are well within the range of high-volume audio loudspeakers.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"Methods","content":"\u003cp\u003eThe stimuli are presented as movies throughout the text. Mathematica or MATLAB was used to create each movie. The code for each movie can be downloaded from the website \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttp://www.staff.city.ac.uk/~solomon/LongitudinalWaves.zip\u003c/span\u003e\u003cspan address=\"http://www.staff.city.ac.uk/~solomon/LongitudinalWaves.zip\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eWhile \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a\u0026lt;\\lambda\\:/\\left(2\\pi\\:\\right)\\)\u003c/span\u003e\u003c/span\u003e, Eq.\u0026nbsp;3 places local maxima and minima at odd and even multiples of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\pi\\:\\)\u003c/span\u003e\u003c/span\u003e radians, respectively. When \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a\u0026gt;\\lambda\\:/\\left(2\\pi\\:\\right)\\)\u003c/span\u003e\u003c/span\u003e, the peaks double and odd multiples of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\pi\\:\\)\u003c/span\u003e\u003c/span\u003e radians become local minima. In that case, local maxima were found using Mathematica's FindMaximum routine, which necessarily fails at the singular point when \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a=\\lambda\\:/\\left(2\\pi\\:\\right)\\)\u003c/span\u003e\u003c/span\u003e. For all other oscillation amplitudes, sampling density was as close to as possible to 1024 phases per wavelength, subject to the constraints that all local maxima and local minima should be sampled, and all samples would be equally spaced. Note that derivative \u003cem\u003ed/dx\u003c/em\u003e in Eq.\u0026nbsp;3 can be written \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\underset{h\\to\\:0}{\\text{lim}}\\left[r\\left(x+h\\right)-r\\left(x\\right)\\:\\right]/h\\)\u003c/span\u003e\u003c/span\u003e, where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:r\\left(x\\right)\\)\u003c/span\u003e\u003c/span\u003e is the ratio of that equation’s two integrals, expressed as a function of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:x\\)\u003c/span\u003e\u003c/span\u003e. This derivative was approximated using \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\left[r\\left(x+h\\right)-r\\left(x\\right)\\right]/h\\)\u003c/span\u003e\u003c/span\u003e, with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:h={10}^{-10}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e"},{"header":"Results","content":"\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003eUnipolar dot travelling waves\u003c/h2\u003e\u003cdiv id=\"Sec6\" class=\"Section3\"\u003e\u003ch2\u003eStimuli\u003c/h2\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eInspired by the on-line resource created by Zeleny et al. (\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2011\u003c/span\u003e), we created visual renditions of annular longitudinal waves from Eq.\u0026nbsp;1, using a substrate of 600 oscillating random-dot samples whose equilibrium positions were uniformly distributed throughout an annular region of a 2:1 radial extent. Movie 2 contains 6 annular wave motions of increasing amplitude corresponding to the 6 panels in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The amplitudes are specified as its proportion of a wavelength.\u003c/p\u003e\u003cp\u003eMovie 2. Dynamic illustrations of longitudinal waves, propagating clockwise through annuli of randomly placed dots. The annular format is designed to allow fixation at the center of each annulus to eliminate tracking eye movements. The angle of each dot oscillates sinusoidally in place with an amplitude ranging from \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.003\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e (top left) to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a=0.3\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e (bottom right), as labeled. The red dot near the top is designed to aid verification that each dot is oscillation in place. The motion conditions match those diagrammed in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. In all panels, the phase propagation speed is 42 polar deg/s. Wave motion is visible for oscillation amplitudes above ~\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.01\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eWhereas rectangular dot arrays with horizontal or vertical oscillations like those illustrated in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e tend to encourage eye movements in the direction of propagation (or possibly in the opposite direction), we opted to make perceptual judgments with annular dot arrays, because fixation at the center of an annulus discourages eye-movement tracking in any particular direction, keeping the dot array rotating at a fixed retinal eccentricity.\u003c/p\u003e\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eObservations\u003c/span\u003e\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eWith fixation at the annulus center to avoid foveal tracking, it is difficult to appreciate that each dot is merely oscillating around a stationary position in the annulus. This oscillation can, however, be verified by foveating any individual dot within the annular band.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eWith central fixation, wave propagation is immediately apparent as the clockwise rotation of the eight compression regions (\u0026ldquo;crests\u0026rdquo;) around each of the annuli with sufficient amplitude.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe rarefaction regions (\u0026ldquo;troughs\u0026rdquo;) between crests appear to cohere into a uniform texture that rotates backwards (here, counterclockwise), even though the phase propagation direction is clockwise. This reverse motion thus represents the intrusion of the local motion of the individual dots for half the phase of the local oscillations, though perceived as a coherent reverse motion of the half-phase patches.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eAt the intermediate amplitudes, most observers can discern a subtle impression of depth, in which the compression regions appear closer to the viewer than the rarefaction regions.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eWith focal attention, the relative salience of individual crests and troughs may fluctuate, but the opposing directions of motion they convey can be experienced simultaneously.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe perceived speeds of rotation of the crests and troughs appear to increase with oscillation amplitude. This percept is robust, but it is an illusion because each crest (and trough) requires exactly 8.53 s to complete a full revolution around the annulus center.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\n\u003ch3\u003eDensity-luminance reciprocity\u003c/h3\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003eStimuli\u003c/h2\u003e\u003cp\u003eThe high-density peak regions of our black-dot stimuli necessarily have a lower average luminance than the low-density trough regions. To determine if and how the visual perception of longitudinal motion was contingent upon this \u0026ldquo;reciprocity\u0026rdquo; between density and average luminance (Mulligan \u0026amp; MacLeod, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1988\u003c/span\u003e), we created stimuli in which dot luminance was proportional to the average density of dots in each phase of the longitudinal wave. This manipulation eliminates the (expected) luminance contrast between crests and troughs, virtually eliminating their ability to stimulate standard motion-energy mechanisms, including the Reichardt detector (Reichardt, \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e1987\u003c/span\u003e; van Santen \u0026amp; Sperling, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e1985\u003c/span\u003e).\u003c/p\u003e\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eObservations\u003c/span\u003e\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eWith luminance equated, the crests still appear to rotate forwards (clockwise) and the troughs backwards (counterclockwise), as in the original version.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe wave motion is fully visible for levels of 0.03λ and above.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe depth impression is similar to that for the original version with uncompensated luminance modulation.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eThis luminance-balanced control makes clear that the percept of the bidirectional wave motion is undiminished from the level of 0.03λ and above, suggesting that it can be conveyed by something other than standard, luminance-based motion-energy mechanisms.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eMovie 3. Luminance-balanced version of Movie 2, in which dot luminance varies in proportion to dot density. Wave motion is easily visible from about\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\:0.03\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003c/div\u003e\n\u003ch3\u003ePolarity-randomized dots\u003c/h3\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\u003ch2\u003eStimuli\u003c/h2\u003e\u003cp\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eRandomly selecting the polarity (black or white) of each dot is guaranteed to reduce any contrast between the average luminances of crest and trough, consequently minimizing the contribution from standard, luminance-based motion-energy mechanisms to the wave-motion percept. Examples of these \u0026ldquo;drift-balanced\u0026rdquo; stimuli (Chubb \u0026amp; Sperling, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1988\u003c/span\u003e) have been provided in Movie 4.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\u003ch2\u003eObservations\u003c/h2\u003e\u003cp\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eAll the observations made with black-dot stimuli (discussed in the section on unipolar dot travelling waves) apply equally to the polarity-randomized stimuli. Evidently, luminance contrast is not required for the perception of longitudinal wave motion, or for the visual segregation of the clockwise-propagating crests from the troughs, which again appear to rotate in the counterclockwise direction.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\u003ch2\u003eDensity-contrast reciprocity\u003c/h2\u003e\u003cdiv id=\"Sec13\" class=\"Section3\"\u003e\u003ch2\u003eStimuli\u003c/h2\u003e\u003cp\u003eThe high-density peaks of our polarity-randomized stimuli necessarily have a more contrast energy than the low-density troughs. To determine if and how the visual perception of longitudinal motion was contingent upon this \u0026ldquo;reciprocity\u0026rdquo; between density and contrast energy (Morgan et al. \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2022\u003c/span\u003e), we created stimuli in which the absolute value of each dot\u0026rsquo;s Weber contrast was proportional to the average density of dots in each phase of the longitudinal wave. This manipulation reduces the angular modulation of contrast energy around each annulus, consequently reducing its ability to stimulate the \u0026ldquo;2nd -order\u0026rdquo; motion system, putatively responsible for computing the direction of spatiotemporal amplitude modulations (Chubb \u0026amp; Sperling, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1989\u003c/span\u003e).\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\u003ch2\u003eObservations\u003c/h2\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eContrast balancing weakens both clockwise and counterclockwise apparent motions, but both motions remain visible at moderate oscillation amplitudes (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.03\\lambda\\:\\:-0.1\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003e2. At high amplitudes (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\u0026gt;0.16\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e), clockwise propagation of the crests becomes very hard to see, and the counterclockwise rotation of the trough regions dominate perception.\u003c/p\u003e\u003cp\u003eMovie 5. Contrast-balanced version of Movie 2, in which dot contrast varies with dot density. Wave motion is easily visible \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.03\\lambda\\:\\:-0.1\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e.\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eThe basic result that the transparent counter-rotating wave motion survives both luminance balancing and contrast balancing suggests that the wave motion percept derives neither from the spatiotemporal modulations of luminance nor from spatiotemporal modulations of contrast energy. What about spatiotemporal modulations of dot density? Note that several lines of evidence (e.g., Morgan et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Morgan et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2022\u003c/span\u003e; Zeljic et al., \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) suggest that texture density is computed locally, early in the hierarchy of visual computations.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\u003ch2\u003eDensity waves\u003c/h2\u003e\u003cdiv id=\"Sec16\" class=\"Section3\"\u003e\u003ch2\u003eStimuli\u003c/h2\u003e\u003cp\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eOne inescapable feature of longitudinal waves is the periodic pile-up of particles in the direction of propagation. We test its ability to convey propagation when other features of the longitudinal wave have been removed. Although our contrast-balanced stimuli (described in the section on density-contrast reciprocity) have spatiotemporal modulations of neither luminance nor contrast energy, they still contain spatiotemporal modulations of motion: the average motion of crest dots is clockwise, while the average motion of trough dots is counterclockwise. We virtually eliminated this local motion without changing the overall structure of our annuli by randomly re-assigning the radial position of each dot within the annulus on each frame, thus removing the local oscillatory dot motion but retaining the density modulation waves. The version of this stimulus in Movie 6. tests whether waves of dot density in the absence of coordinated local motions of the dots provides a motion cue when compensated for the ancillary contrast modulation.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eMovie 6. Contrast-balanced density waves as in Movie 5, but modified to eliminate local dot motions by randomizing the radial position of each dot. Propagation of individual crests can be seen with effort at high amplitudes.\u003c/p\u003e\u003cp\u003eObservations\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eHaving disrupted each dot\u0026rsquo;s trajectory, counterclockwise motion remains at neither the local level nor the global level.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eNo dot moves clockwise either, but the relatively dense, low-contrast crests continue to rotate physically in a clockwise direction around the annulus. That motion is very hard to see, but it does seem possible to track the motion of any individual crest with local attention.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThere is a lot of flicker due to resampling the radial location of each dot in the annular region, including content at high temporal frequencies.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eWe conclude that waves of pure dot density with nulling of the consequent contrast modulation are almost invisible. The slight residual motion is likely attributable to an imperfect match of the dot density and compensatory contrast modulation functions.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv id=\"Sec17\" class=\"Section2\"\u003e\u003ch2\u003eFlickering particles\u003c/h2\u003e\u003cp\u003eDid the high temporal frequency content in our density waves simply mask their density modulation, or was the local motion of individual dots (absent from our density waves in previous sections) important for the appearance of contrast-balanced longitudinal waves?\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec18\" class=\"Section2\"\u003e\u003ch2\u003eStimuli\u003c/h2\u003e\u003cp\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eTo assess the impact of flicker on the visibility of contrast-balanced longitudinal waves, we created a new version of Movie 5 in which dot polarity was reassigned randomly on each frame. Note that this manipulation doesn\u0026rsquo;t necessarily produce the same amount of flicker inherent in the density waves, but it does produce equally high temporal frequencies.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eMovie 7. Flickering version of Movie 5, in which dot polarity is randomly reassigned on each frame.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec19\" class=\"Section2\"\u003e\u003ch2\u003eObservations\u003c/h2\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eFlicker eliminates any appearance of coherent motion from the low amplitude (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\u0026lt;0.1\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e) annuli.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eFlicker all but eliminates the appearance of propagation from the high-amplitude (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\u0026gt;0.1\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e) stimuli, but individual crests can be seen to rotate clockwise with effort.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eWhen the oscillation amplitude \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a=0.1\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e, all 8 crests can be seen to rotate coherently, but the counterclockwise rotation of the troughs dominates (as it does with higher oscillation amplitudes).\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eThe introduction of random polarity reassignment unquestionably served to mask the apparent propagation of some of the contrast-balanced stimuli. It is conceivable that the apparent propagation of all our contrast-balanced stimuli could have been masked with more flicker. Consequently, it is uncertain whether the spatiotemporal modulation of dot density is sufficient to convey the impression of propagation amongst longitudinal waves, or whether a contribution from each dot\u0026rsquo;s oscillating trajectory is required. A contribution from those oscillations seems to be required for the impression of coherent, retrograde motion from the troughs.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec20\" class=\"Section2\"\u003e\u003ch2\u003eCancellation of wave motion by opposing directions of rigid motion\u003c/h2\u003e\u003cp\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eThe question arises how the apparent motions of crests and troughs are related to the local motions of the dots throughout the waveform. Clearly, neither direction of apparent motion corresponds to the average dot motion, because each dot is merely oscillating in place. Its motion is neither forward (clockwise) nor backward (counterclockwise) on average. We addressed this question by adding a rigid rotation to the travelling wave stimuli to determine at what speed it would be perceived to cancel either the clockwise crest motion or the counterclockwise trough motion. The default hypothesis is that the apparent speed is controlled by the rate of phase propagation, which is held constant in the following cancellation tests. Another possibility is that the apparent forward and backward motions correspond to the dots\u0026rsquo; fastest forward and backward motions (with velocities \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\pm\\:2\\pi\\:ac/\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e), respectively. Alternatively, the apparent velocities could correspond to the dots\u0026rsquo; average forward or backward motions, respectively (with velocities \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\pm\\:4ac/\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e). (Note that, here and subsequently, we use the term \u0026lsquo;velocity\u0026rsquo; to specify angular velocity relative to the center of the annular stimuli.)\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec21\" class=\"Section2\"\u003e\u003ch2\u003eStimuli\u003c/h2\u003e\u003cp\u003eTo test between these alternatives, stimuli were generated with a range of rigid motions added to all the dots in the annular longitudinal waves. Movie 8A-C illustrates a consensus when the oscillation amplitude had the high amplitude \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.25\\)\u003c/span\u003e\u003c/span\u003e (i.e., beyond the linear range of Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003eB) In this case we judged the crests to be static when a rigid counterclockwise motion having a speed equal to 105% of the wave\u0026rsquo;s propagation (i.e. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:1.05c\\)\u003c/span\u003e\u003c/span\u003e) was added to each dot. The troughs were judged to be static when a rigid clockwise motion having a speed equal to 150% of the wave\u0026rsquo;s propagation was added to each dot. The longitudinal wave in the central annulus has no additional rigid motion.\u003c/p\u003e\u003cp\u003eMovie 8. Different levels of rigid rotational motion added to each dot of high-amplitude (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.25\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e) longitudinal waves. Motion of the frames is keyed to the added motion to give a clear indication of its velocity. A: With matching reverse added motion (speed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:1.05c\\)\u003c/span\u003e\u003c/span\u003e). B,E: No added motion. C: Matching forward added motion (speed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:1.50c\\)\u003c/span\u003e\u003c/span\u003e). D,F: Equal but non-matching intermediate speed (speed \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:1.30c\\)\u003c/span\u003e\u003c/span\u003e). Note that the crests appear stationary in panel A and the trough regions appear stationary in panel C. In panels D and F, neither the crests nor the troughs appear stationary (both appear to rotate counterclockwise) but the frames rotate at the same speed.\u003c/p\u003e\u003cp\u003ePanels 8D and F illustrate the same high-amplitude longitudinal wave with equal but opposite rigid velocities of 1.3\u003cem\u003ec\u003c/em\u003e. This intermediate speed proves too fast to cancel the crests\u0026rsquo; motion (they appear to rotate counterclockwise in 8D) and too slow to cancel the troughs\u0026rsquo; (they appear to rotate counterclockwise in 8F). Cancellation speeds for lower-velocity waves are plotted in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eMovie 9 contains the same longitudinal waves shown in Movie 2, with the wave\u0026rsquo;s actual propagation velocity subtracted from each dot. This is approximately the correct speed for cancelling the apparent propagation of the crests only when the oscillation amplitude \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a\\approx\\:0.2\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e. For larger amplitudes, it is insufficient, and the crests appear to rotate clockwise.) For smaller amplitudes, it is overkill, and any discernible crests appear to rotate counterclockwise.\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eThese speed-cancellation observations reveal the remarkable result that the perceived propagation speed is some form of local average that lies between the particles\u0026rsquo; average speed in that direction and their maximum speed. This is also true for the transparent, retrograde motion of the troughs, which seem to move a little faster. This difference in apparent speed increases with oscillation amplitudes \u0026gt;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.1\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e, whereas the difference between the apparent widths of crest and trough is maximal when \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a=0.16\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eMovie 9. Identical levels of counterclockwise motion with propagation speed c added to Movie 2 modified by adding identical levels of counterclockwise motion at its phase propagation speed of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:c\\)\u003c/span\u003e\u003c/span\u003e = 42 deg/s. This is too slow to cancel the apparent clockwise propagation when \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a=0.3\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e (bottom right) but too fast to cancel apparent propagation when \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:a\u0026lt;0.2\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eObservations (see\u003c/span\u003e Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003e)\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe cancellation speeds are not constant but scale monotonically with the oscillation amplitudes.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eSetting the cancellation velocity to the average velocity of the crest or trough regions, respectively fails to cancel the perceived motion (except in one incidental case of the crest velocity at 0.3 lambda), invalidating the averaging hypothesis that the dot motion in whole of each half-cycle contributes to the perceived velocity. Effective cancellation requires added velocities closer to the maxima within each crest or trough.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eFor higher-amplitude waves, the trough-cancellation speed substantially exceeds the crest-cancellation speed.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003eThese speed-cancellation observations reveal the remarkable result that the perceived wave motion depends on some property of the local dot motions whose oscillation is essentially invisible without guided scrutiny. This property is some form of local average that lies between their average speed and their maximum speed, separately and differently within the crest and trough regions\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:.\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec22\" class=\"Section2\"\u003e\u003ch2\u003eMotion aftereffect\u003c/h2\u003e\u003cdiv id=\"Sec23\" class=\"Section3\"\u003e\u003ch2\u003eStimuli\u003c/h2\u003e\u003cp\u003eA question that does not seem to have been previously addressed in the perceptual literature is whether travelling wave motion generates a motion aftereffect. This may be viewed in a cyclic adaptation paradigm of 10-s adapting and 2-s test periods, as seen in Movie 10. Note here that the high-amplitude longitudinal wave in panel B is identical to the one in Movie 8B. The rotating texture in panel A has a velocity equal to the sum of the rigid velocities required to cancel the apparent motions of crests and troughs, as described in the previous section.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eMovie 10. Motion aftereffect and lack thereof. A 2.13-s static test period follows 10.67 s of motion adaptation with a uniform texture rotating at 6.3 deg/s (panel A) and a high-amplitude (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.25\\lambda\\:\\)\u003c/span\u003e\u003c/span\u003e) longitudinal wave propagating at 42 deg/s (panel B). The angular velocity in panel A is set to the sum of the rigid velocities required to cancel the two directions of transparent motion in panel B (see Movie 8A\u0026ndash;C). A strong motion aftereffect is seen during the static test period with centered adaptation in panel A, but not for the same in panel B.\u003c/p\u003e\u003cp\u003e\u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\u003eObservations\u003c/span\u003e\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eEven though the sum of (oppositely signed) velocities required to cancel the longitudinal wave\u0026rsquo;s opposite directions of apparent motion is small, that sum nonetheless elicits a strong motion aftereffect (Movie 9A).\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003eThe longitudinal wave itself does not elicit any motion aftereffect (Movie 9B).\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eThese observations are consistent with evidence (Kohn \u0026amp; Movshon, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2003\u003c/span\u003e) that the adaptation underlying the motion aftereffect occurs primarily in the directionally selective neurons of cortical area V1, where receptive fields are relatively small. Such neurons favoring clockwise motion would not be expected to receive any stronger stimulation than similarly positioned neurons favoring counterclockwise motion when individual dots are merely oscillating back and forth in their receptive fields. On the other hand, such neurons would receive stronger stimulation from rigid, clockwise motion. Consequently, the rigid motion should produce a significant aftereffect whereas the longitudinal motion should not.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e"},{"header":"Discussion","content":"\u003cp\u003eLongitudinal waves are created from ensembles of locally oscillating dots with a progressive phase advance as a function of position. These oscillations create regions of compression and rarefaction whose propagation and retrograde motions so dominate perception that the local motion of individual dots is all but impossible to discern. These global, transparent motions thus arise by some process of motion integration.\u003c/p\u003e\u003cp\u003eInvestigation of physical longitudinal wave motion reveals that it exhibits profound nonlinearities that are largely neglected in physics community. Longitudinal motion is almost universally treated with the small amplitude approximation, where a sinusoidal driving function is assumed to generate an essentially sinusoidal travelling wave, in the longitudinal wave case as in the transverse wave case.\u003c/p\u003e\u003cp\u003eA form of high amplitude nonlinearity for pressure shock waves (impulse responses) was recognized early in the C19th by Poisson (1808) and elaborated by Challis (\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e1848\u003c/span\u003e) and Stokes (\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e1848\u003c/span\u003e) to show that the longitudinal shock waves approximated an asymmetric sawtooth form that could become multivalued at high amplitude, violating the assumptions of the differential equations. This nonlinear behavior is due to the adiabatic properties of the transmission fluid and does not apply to the basic ideal gas that obeys Boyle\u0026rsquo;s Law, in which the transmission speed is invariant (Blackstock et al., \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eWe show that, even for locally linear transmission through the medium at constant speed, the density of the longitudinal wave motion becomes notably non-sinusoidal at oscillation amplitudes beyond about 2% of the wavelength, and progressively piles up into a narrow cyclic density spike around 16% of the wavelength, beyond which the peak splits into a double spike as the density accretion overtakes the maximum velocity of the local medium being perturbed. This nonlinearity is of entirely different character than the adiabatic shock wave nonlinearities that were the subject of contentious analyses in the C19th. Those turned out to be time-asymmetric in the direction of a sawtooth wave, whereas our analysis applies to speed-invariant Boylean gases, in which the density singularity for a sinusoidal input remains time-symmetric but becomes a double singularity at higher oscillation amplitudes.\u003c/p\u003e\u003cp\u003eHaving established the full nature of longitudinal waves, we then turned to the perceptual appreciation of visual depictions of wave motion, such as are often used in acoustics courses. Instead of the usual linear wave of Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, we use rotating ring configurations of travelling wave motion to control eye movements. These show that the local dot motion in the wave is virtually invisible, but is subsumed under the global impression of motion, in which each cycle splits into two opposing regions: a dominant forward motion of the crest regions and a residual backward motion of the trough regions. This percept is obtained even when the oscillation amplitude is small enough that the wave motion remained sinusoidal. At higher amplitudes, the forward motion of the crests is enhanced by the nonlinear pile-up of the dot density there, which results in them being perceived as narrow \u0026lsquo;walls\u0026rsquo; between broad \u0026lsquo;fields\u0026rsquo; of opposing motion. In the rotating ring configuration, the individual cycles then integrated into transparent counterrotating motion fields. Perceptual binding of all 8 crests (or all 8 troughs), see Movie 2 into a single coherent texture necessarily requires neurons having receptive fields large enough to be stimulated by them.\u003c/p\u003e\u003cp\u003eWe then asked whether the appearance of the forward component of the longitudinal wave motion was attributable to the difference between the average luminances of the crests and troughs. This hypothesis was tested in two ways. First, we reduced the crests\u0026rsquo; average luminance to match that of the troughs (Movie 3). Second, we randomized dot polarity so that the expected luminance within each phase of the wave was equal to that of the mid-gray background (Movie 4). Neither of these manipulations eliminated the impression of transparent, global rotations in opposite directions.\u003c/p\u003e\u003cp\u003eThe impression of global motion in either direction can be eliminated by the addition of a uniform velocity to each particle. This cancelling velocity is close to the peak oscillation velocity (Movie 8), but we have found that it varies with oscillation amplitude and direction (i.e., consistent with or opposite to that of propagation).\u003c/p\u003e\u003cp\u003eThe distinction between a global property and the local components from which it is formed was highlighted by the Gestalt psychologists at the beginning of the twentieth century. The visual system\u0026rsquo;s ability to integrate local motion signals into a global percept has been recognized (e.g. by Smith et al., \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e1994\u003c/span\u003e) since the end of the twentieth century. Just as the Smith group sought to eliminate spatial frequency components from their random-dot stimulus that could potentially excite motion-energy detectors on a global scale, we too examined various \u0026ldquo;drift-balanced\u0026rdquo; (Chubb \u0026amp; Sperling, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e1988\u003c/span\u003e) modifications of our black-dot stimuli that were designed to hide their emergent properties from \u0026ldquo;1st -order\u0026rdquo; motion detectors (Cavanagh \u0026amp; Mather, \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e1989\u003c/span\u003e; Chubb \u0026amp; Sperling, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1989\u003c/span\u003e).\u003c/p\u003e\u003cp\u003eAs previously noted, the mere elimination of global-scale motion energy (via luminance cancellation or polarity randomization) proved ineffective at eliminating the impression of transparent, global motion in opposite directions. However, reduction of the crests\u0026rsquo; contrast did make them harder to see (Movie 5). In this case, the retrograde motion of the troughs was perceptually dominant. One manipulation that did succeed in eliminating the impression of retrograde motion was the cancellation of the local motion from the troughs (Movie 6). The propagation of high-density crests in these latter movies was just perceptible.\u003c/p\u003e\u003cp\u003eThe perceptual results were extended by testing for a motion aftereffect generated by longitudinal wave motion. In fact, motion aftereffects are essentially non-existent for the pure longitudinal wave motion, even though strong motion aftereffects are seen for rigidly moving textures of the same format (Movie 9). Given our contention that the impression of transparent global motions arises from large-scale mechanisms that integrate local motion information from individual dots, this negative result may not seem surprising. After all, the inability of other non-Fourier motion stimuli to elicit an aftereffect in static test stimuli has been well established (Banks \u0026amp; Kane, \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1972\u003c/span\u003e; Nishida \u0026amp; Sato, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e1995\u003c/span\u003e). That being said, it is important to remember that black-dot longitudinal waves would indeed be expected to stimulate motion-energy detectors sensitive to the difference between the average luminances of crest and trough. It is therefore surprising to find that these longitudinal waves elicit no motion aftereffect.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003ch2\u003eAdditional Information\u003c/h2\u003e\u003cp\u003eNo competing interests\u003c/p\u003e\u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e\u003cp\u003eDeclaration\u003c/p\u003e\u003cp\u003eThis study was not supported by external funding\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eInitial observations and background: CWTConceptual analysis: CWT, JAS, SMAMathematical analysis and stimulus programming: JASNarrative description: CWT, JAS, SMA\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eThe code for each movie can be downloaded from the website http://www.staff.city.ac.uk/~solomon/LongitudinalWaves.zip.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eBanks, W. P., \u0026amp; Kane, D. A. (1972). Discontinuity of seen motion reduces the visual motion aftereffect. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 290, 153-168.\u003c/li\u003e\n\u003cli\u003eBlackstock, D. T., Cormack, J. M., \u0026amp; Hamilton, M. F. (2020). Early history of nonlinear acoustics. In Proceedings of Meetings on Acoustics 2019. Acoustical Society of America, 36:1, 045007.\u003c/li\u003e\n\u003cli\u003eCavanagh, P., Arguin, M., \u0026amp; von Gr\u0026uuml;nau, M. (1989). Interattribute apparent motion. Vision Research 29, 1197-1204. \u003c/li\u003e\n\u003cli\u003eCavanagh, P., \u0026amp; Mather, G. (1989). Motion: the long and short of it. Spatial Vision, 4, 103-129.\u003c/li\u003e\n\u003cli\u003eCavanagh, P., Tyler, C. W., \u0026amp; Favreau, O. E. (1984). Perceived velocity of moving chromatic gratings. Journal of the Optical Society of America A 1, 893-899. \u003c/li\u003e\n\u003cli\u003eCeperley, P. (2016). The Lagrangian approach to simple waves \u0026ndash; several common waves that lack momentum. Resonances, Waves and Fields. https://resonanceswavesandfields.blogspot.com/2016/01/the-lagrangian-approach-to-simple-waves.html (Accessed 10/10/2024).\u003c/li\u003e\n\u003cli\u003eChallis, J. (1848) On the velocity of sound, in reply to the remarks of the Astronomer Royal. Phil. Mag. Ser. 3, 32, 494\u0026ndash;499.\u003c/li\u003e\n\u003cli\u003eChubb, C., \u0026amp; Sperling, G. (1988). Drift-balanced random stimuli: a general basis for studying non-Fourier motion perception. Journal of the Optical Society of America A, 5, 1986-2007.\u003c/li\u003e\n\u003cli\u003eChubb, C., \u0026amp; Sperling, G. (1989). Two motion perception mechanisms revealed through distance driven reversal of apparent motion. Proceedings of the National Academy of Sciences USA, 86, 2985-2989.\u003c/li\u003e\n\u003cli\u003eGraham, N. (1972). Spatial frequency channels in the human visual system: Effects of luminance and pattern drift rate. Vision Research \u003cem\u003e12\u003c/em\u003e, 53-68. \u003c/li\u003e\n\u003cli\u003eKohn, A., \u0026amp; Movshon, J. A. (2003). Neuronal adaptation to visual motion in area MT of the macaque. Neuron 39, 681.\u003c/li\u003e\n\u003cli\u003eLedgeway, T., \u0026amp; Smith, A. T. (1994). The duration of the motion aftereffect following adaptation to first-order and second-order motion. Perception 23, 1211-1219.\u003c/li\u003e\n\u003cli\u003eLu, Z.-L., \u0026amp; Sperling, G. (1995). The functional architecture of human visual motion perception. Vision Research 35, 2697.\u003c/li\u003e\n\u003cli\u003eMorgan, M. J., Macleod, D. I. A., \u0026amp; Solomon, J. A. (2022) The channel for detecting contrast modulation also responds to density modulation (or \u003cem\u003evice versa\u003c/em\u003e). Vision Research, \u003cem\u003e192, \u003c/em\u003e107948.\u003c/li\u003e\n\u003cli\u003eMulligan, J., \u0026amp; MacLeod, D. I. A. (1988). Reciprocity between luminance and dot density in the perception of brightness. Vision Research 28, 503-519.\u003c/li\u003e\n\u003cli\u003eMorgan, M. J., Raphael, S., Tibber, M. S., \u0026amp; Dakin, S. C. (2014). A texture processing model of the \u0026lsquo;visual sense of number\u0026rsquo;. Proc. Roy. Soc. B., 281, 20141137.\u003c/li\u003e\n\u003cli\u003eNishida, S., \u0026amp; Sato, T. (1995). Motion aftereffect with flickering test patterns reveals higher stages of motion processing. Vision Research, 35, 477-490.\u003c/li\u003e\n\u003cli\u003ePetersik, J. T., Hicks, K. I. \u0026amp; Pantle, A. J. (1978). Apparent movement of successively generated subjective figures. Perception, IO, 563-572\u003c/li\u003e\n\u003cli\u003eReichardt, W. (1987) Evaluation of optical motion information by movement detectors. Journal of Comparative Physiology A. 161(4):533-47. \u003c/li\u003e\n\u003cli\u003eSeiffert, A. E., \u0026amp; Cavanagh, P. (1998). Position displacement, not velocity, is the cue to motion detection of second-order stimuli. Vision Research 38, 3569-3582. \u003c/li\u003e\n\u003cli\u003eSmith, A. T., Snowden, R. J., \u0026amp; Milne, A. B. (1994). Is global motion really based on spatial integration of local motion signals? Vision Research, 34, 2425-2430.\u003c/li\u003e\n\u003cli\u003eSperling, G., Solomon, J. A., Lu, Z., \u0026amp; Chubb, C. (1994). First and second-order processes in the perception of motion and texture. In J.M. Zurada, R.J. Marks II, \u0026amp; C.J. Robinson, Computational Intelligence: Imitating Life, 223-236.\u003c/li\u003e\n\u003cli\u003eStokes, G.G. (1848) On a difficulty in the theory of sound. Phil. Mag. Ser. 3, 33, 349\u0026ndash;356.\u003c/li\u003e\n\u003cli\u003eWhitney, David, \u0026amp; Bressler, D.W. (2007). Second-order motion without awareness: Passive adaptation to second-order motion produces a motion aftereffect. Vision Research 47, 569-579. \u003c/li\u003e\n\u003cli\u003eVan der Smagt, M. J., Verstraten, F. A., Vaessen, E. B., van Londen, T., \u0026amp; van de Grind, W. A. (1999). Motion aftereffect of combined first-order and second-order motion. Perception 28, 1397-1411.\u003c/li\u003e\n\u003cli\u003eVan Santen J.P., Sperling G. (1985) Elaborated Reichardt detectors. Journal of the optical Society of America A. 2(2):300-21. \u003c/li\u003e\n\u003cli\u003eZanker, J. M. (1993). Theta motion: A paradoxical stimulus to explore higher order motion extraction. Vision Research, 33, 553-569.\u003c/li\u003e\n\u003cli\u003eZanker, J. M. (1994). What is the elementary mechanism underlying secondary motion processing? Investigative Ophthalmology \u0026amp; Visual Science (Supplement), 35, 1405.\u003c/li\u003e\n\u003cli\u003eZeleny, E., Bengtsson, M. G., \u0026amp; Russell, D. A. (2011) Longitudinal and Transverse Waves. https://demonstrations.wolfram.com/LongitudinalAndTransverseWaves/ (Accessed 16/10/2024).\u003c/li\u003e\n\u003cli\u003eZeljic, K., Morgan, M. J., \u0026amp; Solomon, J. A. (2024). Monocular and binocular mechanisms detect modulations of dot density and dot contrast. Vision Research, 215, 108347.\u003c/li\u003e\n\u003c/ol\u003e"},{"header":"Movies","content":"\u003cp\u003eMovies 1 to 10 are available in the Supplementary Files section.\u003c/p\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-7802344/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7802344/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe introduce the study of the visual perception of longitudinal travelling wave motion, which as a physical phenomenon forms the basis of acoustics and some forms of seismic transmission. A theoretical analysis of Physical longitudinal wave motion reveals that it exhibits profound nonlinearities that have been almost entirely neglected by the physics community. We simulated longitudinal motion in visual form with a random-dot field in which each dot particle oscillates sinusoidally about a fixed position at the same frequency but with a phase advance proportional to its distance from the origin. The resultant longitudinal density wave well-approximates a sinusoidal function at very low oscillation amplitudes, becoming progressively distorted as oscillation amplitude increases. When the maximum velocity of each particle equals that of the propagation, the density function approximates a spike, which splits into two at even greater amplitudes. Perceptually, the motion splits into forward motion of the crests and backward for the troughs. Adding a single (\u0026lsquo;rigid\u0026rsquo;) velocity component can eliminate either the forward or backward percept. Remarkably, the speed needed for perceptual cancellation scaled with oscillation amplitude. Longitudinal waves evoke no motion aftereffect. These unexpected results underline the emergent, or higher-order, nature of the longitudinal travelling wave motion.\u003c/p\u003e","manuscriptTitle":"Visual Perception of Longitudinal Waves: Theory and Observations","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-10-28 16:43:34","doi":"10.21203/rs.3.rs-7802344/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-11-19T19:02:42+00:00","index":"","fulltext":""},{"type":"reviewerAgreed","content":"101951422488241184540199436954724664895","date":"2025-10-26T20:49:26+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-10-17T11:40:11+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"323223322778048844936069051370124802601","date":"2025-10-15T01:09:21+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-10-15T00:27:24+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2025-10-14T19:19:53+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-10-11T09:36:54+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-10-11T09:36:19+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-10-07T19:42:50+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"6e77b809-5c99-4125-9404-725adcef52a5","owner":[],"postedDate":"October 28th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":56895538,"name":"Biological sciences/Neuroscience"},{"id":56895539,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-03-30T16:26:39+00:00","versionOfRecord":{"articleIdentity":"rs-7802344","link":"https://doi.org/10.1038/s41598-026-36204-y","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2026-03-23 16:09:13","publishedOnDateReadable":"March 23rd, 2026"},"versionCreatedAt":"2025-10-28 16:43:34","video":"","vorDoi":"10.1038/s41598-026-36204-y","vorDoiUrl":"https://doi.org/10.1038/s41598-026-36204-y","workflowStages":[]},"version":"v1","identity":"rs-7802344","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7802344","identity":"rs-7802344","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}