Imaging Sound

 

1. Introduction

Waves

Examples of waves are all around us. Perhaps the most familiar type is a water wave that can be directly seen and felt. We can record a video of water waves and observe how the ripples behave. If we capture an image or record a video, we can easily measure the wavelength (distance between peaks) or speed of the waves as they travel on the surface. 

Sound also behaves as a wave. Sound is a series of compressions and expansions (rarefactions) of whatever medium that propagates it. In air, the wavelengths can be a few tenths of a millimeter to hundreds of meters, and audible sound travels at 343 meters/second (767 mi/hr). 

Our ears generally respond to the time average (loudness) and spectrum (pitch) of these waves, especially at higher frequencies, but we cannot directly sense the peaks and valleys of the waves. There are methods to visualize sound. One of the more dramatic visualizations is the Rubens Tube, where sound reverberates in a tube of flammable gas that supports standing waves, creating regions of high and low pressure (time averaged) visualized as high and low flames (shown below: Moises Alves demonstrates sound waves using a metal tube, a trumpet, and 100 tiny flames. Picture: Moises Alves). 

An optical technique for seeing sound uses Schlieren Imaging to detect faint deflections of light by the pressure waves of sound. There are links for this method at the end of the post show very striking images.

Light also acts as a wave, traveling almost 1 million times faster than sound as oscillating electric and magnetic fields, with a much wider range of wavelengths. The portion of the spectrum we can see has wavelengths less than a thousandth of a millimeter. As with our hearing, our eyes detect just the amplitude (brightness) and spectrum (color) of a light wave because the wave frequency and speed is far too high to perceive (even electronically).

 

Imaging Water Waves: The Ripple Tank

A common demonstration of wave effects uses a device called a Ripple Tank. A motor drives an object up and down in a transparent tray of water, creating ripples. The shadow of the waves are projected onto a surface below the tray. Flashing the illumination at a rate near the oscillation frequency of the waves can appear to freeze the wave motion. This is Stroboscopic Imaging.

For a point-like object, the waves are circular:

A Sound Ripple Tank

This project describes a type of ripple tank for sound, diagrammed below. A single tone is played into a row of speakers, and the sound travels through air in a chamber to a row of microphones. The mics are scanned along the propagation direction x. A computer triggers the recording of the mic signals in sync with the speaker signal, freezing the motion of the sound wave by stroboscopic imaging. If the images are captured at an increasing delay time, a succession of frames are recorded that can be played back as a movie of the sound wave, similar to what is seen in a water ripple tank.

 

The photos below show the actual apparatus that is described in detail in a later section. The sound wave is confined to travel in a slab of air between an aluminum floor and a clear plastic ceiling (the top lid in the photo). This forms a waveguide for the sound, making it act as if it were two dimensional. The electronics are along the left in the photo. The project required about 10 months of design and development. When it all finally worked, the first movie was of a stunningly clear circular wave resembling ripples in a pond.

2. Results

The sound movies can be viewed by clicking the usual play button (these are ironically silent movies). The small red arrows indicate the position of small speakers that generated the waves. Each movie will play for about 1 minute. If you want continuous play, try a right click on the video and choose the Loop option. You can also choose to view full screen (recommended).

Centered Single Source

This movie is of sound waves at 9000 Hz emitted by a single speaker (located at the red arrow position). The image on the left uses grey scale and the one on the right is the same data using a Look Up Table (LUT) for color enhancement. The movies have 16 frames, played back here at 10 frames/second (fps).

Analysis

• Only the center speaker in a row of 32 is active, 9.7 cm from the right edge.
• The clear set of circular waves centered on the speaker verifies:
• Synchronous capture of sound pressure at all points and time.
• Timing resolution better than \(\Delta t = 7.4 \mu s\).
• Successful strobe imaging over the sound period T of 111 \(\mu s\).
• Looping of frames results in a smooth wave motion, verifying precise repetition over T.
• Some image noise is due to interference with edge reflections.
• The microphones show reasonable equalization.
• In this movie, there are approximately 17 peaks along x, over a distance of 624 mm, giving a wavelength \(\lambda\) = 624mm/17 = 36.7mm. At f = 9000 Hz, the speed of sound can be estimated as \(c = \lambda*f = 330  m/s\), which is within 1% of the typical value of 334 m/s.
• At 1 frame/ \(\Delta t\), the system is strobe-filming at an effective frame rate of 135,000 fps. Playing back at 10 fps effectively slows down the sound by a factor of 13,500:1.

Two Sources vs. Separation

When two speakers are active, separated by a distance s (in units of speaker pitch \(p_s\) = 13.8 mm), their waves add leading to constructive and destructive interference.

• The sound waves from two sources show classic interference effects, including radial lines of destructive interference, indicated by the dashed lines. The same patterns occur when a laser shines through two slits. 
• Lines of destructive interference become more frequent and smaller in angle as the source separation increases.
• For the smallest spacing s = 1 the two sources act nearly as a single source.
• For s = 29 a dense grid-like interference pattern emerges.

A composite movie (2 fps) of the first frame for each of the speaker separations is shown below as another way to see the trend as s increases:

What We Hear: Amplitude

Because the time between the peaks of sound waves (period T) are usually very short, our ears do not normally react to the individual pressure oscillations. Instead, we sense the time average amplitude of sound (A). An Amplitude image can be calculated from the movie data by averaging or calculating the standard deviation of the frames over a full period, and gives an idea of what it would sound like if we could walk around inside the waveguide chamber (assuming you could hear 9000 Hz). 

For the case of two active speakers separated by \(6p_s\), the movie data and the Amplitude image (color enhanced) are shown below (the amplitude is color enhanced using a "Fire" LUT):

Dark regions of the Amplitude image indicate areas of destructive interference (low energy). A walk through the sound field would be loudest in the bright regions and quiet in the dark regions. The movie and Amplitude show a fan out of constructive and destructive regions. There is additional structure that results from interference due to reflections from the sides. The fine vertical and horizontal bars represent the system pixels (in this case, 49 x 32). 

For a single source (see below) crossing bands are evident in the Amplitude image from interference between the side bands and the direct speaker wave. Without side reflections, we would expect just a bright region near the source that fades with distance. 

Amplitude images for two speakers separated by various distances s (units of \(p_s = 13.8mm\)) are shown below. The dark lines of destructive interference are are easier to see than in the movies.

Simulation Method

The movies and Amplitudes seem to agree with what is expected from point sources of sound. To get more quantitative, and to better understand the images, it is very useful to use a model simulation for comparison with the data, as described below.

• The waveguide chamber is considered 2D (x, y) with uniform pressure along z.
• Each speaker is treated as a point source, generating an expanding cylindrical wave.
• All speakers are separated by a spacing \(p_s\) with coordinates \(x_s = 0\), \(y_{sj} = jp_s\). The speakers are all in phase at the frequency f, emitting an equal pressure amplitude A. Their outputs are A or 0, depending on the programmed choice.
• The microphones are in a row parallel to the speakers, separated by a spacing \(p_m\), and move together along x at the same x increment \(p_m\). Each mic therefore has coordinates \(x_m = x_0 + i_x p_m\) and \(y_m = mp_m\).
• Using Huygens method, a 2D pressure wavelet from a point source can be represented as:

\(S(x_m, y_m, \tau) = \frac{Ae^{i(kr_m-\omega\tau)}}{\sqrt{r_m}}\)

\(k = \frac{\omega}{c},  \omega = 2\pi f\)

\(r_m = \sqrt{x^2_m + (y_m - y_{sj})^2}\)

where c = speed of sound, \(\tau\) = frame delay (0 to T), T = wave period = 1/f, and \(r_m\) = distance from the mic at (x,y) to the speaker at (0, \(y_{sj}\)). The pressure variation at (x,y) is the Real component of the sum of S over all the active speakers. Explicitly, for each frame at time \(\tau\) for each speaker:

\(S(x_m , y_m , \tau) = \frac{A}{\sqrt{r_m}}[cos(kr_m)cos(\omega \tau) + sin(kr_m)sin(\omega \tau)]\)

Single Speaker Case:

Using this simulation, we see that the single speaker calculation is strongly correlated to the data. The Amplitude simulation shows only localized energy that falls off with distance, while the Amplitude image data shows interference lines. This is a strong indication that these interference features are from wall reflections excluded from the simulation. 

Two-Speaker Case:

(Frame 1 images)

(Amplitude Images)

The key features of the Amplitude data are nicely represented by the simulations, despite the small side wall noise.

Young's Experiment:

Interference of sound from two speakers is analogous to light transmitted through two narrow slits, known as Young's experiment. The same simulation method can be used to derive an equation for the angle \(\theta\) of the destructive (dark) interference lines seen in these sound Amplitude images:

\(tan\theta = \frac{m\lambda}{2s}\)

where \(\lambda\) is the wavelength, s is the separation of the speakers, and m is an odd integer. Dashed lines at these calculated angles are super-imposed on the Amplitude images below, corresponding nicely with the observed dark regions:

Detecting Source Locations

• How do creatures so quickly and accurately discern the direction of a sound source using two ears? 
• How do modern acoustic locators work?

Using 2 Detectors (Ears):

If we activate the system speakers in sequence and look just at the first movie frame, the sound waves look like:

If we are standing at far left end of the chamber, facing the speakers with our ears labeled A and B (separated by 20 cm), notice the different behavior of the waves at each ear for speakers at two positions:

 

A plot of the mic signals at A and B are shown below for both speaker positions: 

 

There is a time delay between the signal peaks that varies with the speaker position. The signal peaks are simultaneous if the distance from the mics to the source is equal. By sensing the coincidence in the signals from our ears we can determine when we are facing a sound source. This is known as Interaural Time Difference or ITD. ITD allows us to detect astonishingly small delays of about 10 micro-seconds, providing sound direction to within 1 degree.

In this experiment at 9000 Hz, there are 2 additional speaker positions that show peak overlap, shown below, where the peaks are one period apart, so they would also seem coincident. This could lead to confusion in direction using ITD, were it not for other audio means that humans use like intensity difference. The movies show the speaker numbers (out of 32) where the A and B mics signals are in sync.

Sound Location Using 32 Detectors

Using one speaker and the signals from all 32 microphones (not just 2), the movie data can be analyzed to determine the source location. The analysis method is to simulate the emission from a “target” point source that best correlates to the movie data. Mathematically, the simulated point source output from a target position is multiplied by the data, and the product is added up to give a correlation coefficient C.

For example, with just speaker #6 on, sound waves are generated as shown in the image above. If a simulated target source is placed at each of the speaker positions, the correlation C with the data will maximize when the target originates at the #6 position. 

Plots of C vs. target position shows the peak at the same location as the source, showing that speaker position can be readily detected by the mic array. In the above plot, C is also shown as a function of the x data range used to calculate C. As more of the data is included, the correlation peak is reduced at some point, indicating that the data behaves less like a simple point source when closer to the speakers. As few as 3 x positions (corresponding to x data range = 26mm) can correctly locate the speaker position.

The process of correlating the data pattern with those from target sources is one way of locating a source. Another means is to apply a prescribed phase delay to the mic acquisitions that turns it into a collective lens focused on the target point. For example, if only the speaker at the center of the speaker row is emitting, the circular wave front will be detected first by the closest mic at the center and later by the side mics farthest away. If the data acquisition was delayed in time or position for each mic so that the wave front signal is acquired by all the mics at the same time (in phase), then the total signal sum would be maximized. This set of delays imposed on the mics acts as a phased lens with a focal point at the target. If there is no source at the target, the same phased lens will collect a weaker total. It is mathematically equivalent to the correlation method discussed above. This approach is used to focus detector arrays in ultrasonic imaging, radio astronomy and telemetry.

Detecting Multiple Sources

Using two speakers and the signals from all 32 microphones, the movie data can be analyzed in the same way to determine both source locations. For example, with speakers #6 and #25 on (shown above), if a simulated source (target) is placed at each speaker position and its emission is multiplied by the data and summed over a range of x, the C value maximizes when the target is placed at either source position. The plot below shows peaks in C are at positions #6 and #25 to #27, which is within an accuracy of 1 position.

Sound Imaging Resolution

The plots of C vs. target speaker position shows peaks very close to the correct location of the sources, until the separation is below 5 \(p_s\) = 7 cm or 1.8 wavelengths. This is the spatial resolution limit of this sound imaging system. 

By analogy, optical imaging has a similar resolution limit. The Rayleigh limit results from the overlap of diffracted light from light sources separated by a distance s' when the light is sent through a lens of diameter D with focal length f. For a 2D system, the minimum resolvable source separation is \(s' \approx (f/D) \lambda\). The correlation method for the sound imager is equivalent to using a lens in optics, where D = chamber width and f is an effective focal length. The above result suggests an effective f/D = 1.8. For a mic array width D = 441mm, this implies an effective f ~ 800 mm for sound location.

Creating Plane Waves

Increasing the number of centered adjacent speakers evolves the waveform from a point like source, to a fan and ultimately forms a plane wave. his is illustrated in the movies shown below. 

Here is a composite movie of the first frame to better see how the plane wave evolves as the number of sources increases:

Amplitude Images

While the wave movies take on a plane wave form with increasing number of sources, the amplitude close to the speakers shows complex interference fringes at the edges of the source cluster and in-between since the speakers spaced at \(p_s \approx \lambda/3\). Larger wavelengths (lower frequencies) or smaller spacing would improve the amplitude uniformity. These patterns are fundamental to the speaker arrangements as seen in the simulated waveforms and amplitudes.

Simulations

The simulations can be extended beyond the x limit of the chamber showing the Amplitude pattern close to the speakers (near field) includes crisscrossing interference patterns, but eventually settles into a steady beam farther away from the sources (far field). This is shown by simulation for the case of 32 sources extended to x ~ 3.9m:

Single Slit Diffraction

When a laser beam travels through a narrow slit, instead of just projecting the image of a slit with sharp edges, a central bright spot emerges with weaker side lobes. Each peak diverges at a characteristic angle determined by the ratio of the wavelength \(\lambda\) to the slit width D. Narrowing the slit or increasing the wavelength increases the angle of the lobes. 

In this system, a virtual slit can be made by enabling n adjacent central speakers, making \(D = n p_s\). Diffraction theory predicts the lobe peaks occur at an angle \(\theta\) given by \(\frac{D}{\lambda}sin\theta = b\), where b = 1.43 and 2.46 for the first and second lobes, respectively. In the previous example of n = 12 active speakers D = 165 mm, \(\lambda\) =  38.1 mm, corresponding to \(\theta_1\) = 19.2° and \(\theta_2\) = 34.5°. Lines at these two angles are superimposed on the sound Amplitude data (below left), showing reasonable agreement with the theoretical result. The simulation and a profile plot also show the diffraction peaks, within the resolution of the image.

There is a region near the speakers where the data and simulation show two strong lobes and a center dark region. As shown earlier, if the number of speakers increases (larger D), the sound Amplitude image shows multiple interference fringes. If D is held constant but more speakers are used at a smaller spacing, the simulation result is unchanged. This indicates that the near field features are not the result of the speaker spacing but are a consequence of summing wavelets very close to the sources.

Two Source Groups vs. Separation

We have seen what happens when waves from single speakers interfere and how the pattern evolves with speaker separation. A similar effect is seen when groups of speakers interfere as the group size n and their separation s change. This is analogous to transmitting light through two slits of varying width and separation. Borrowing from the optics equations, at a large distance from the sources, the Amplitude images should behave according to:

\(A = \frac{sin^2\beta}{\beta^2} cos^2\alpha\)

\(\beta = \frac{\pi n p_s}{\lambda} sin\theta\)

\(\alpha = \frac{\pi s p_s}{\lambda}\)

This is essentially the expression for the single speaker case, where the \(\alpha\) factor depends on separation, multiplied by a factor for a single group case \(\beta\) that depends on the group size n or slit width. In these examples for n = 2, the results are visually the same as n = 1. As the group size increases, the higher angle bands diminish in Amplitude relative to n = 1. In the waveform movies below, note that s is measured between the center of each speaker group.

Waveform Data

Amplitude Images

Notice the near field (closest to the speakers), that larger n creates a dark region between the groups. This is a result of the waves from the outer groups travel as forward beam in a more plane-wave fashion as n increases (discussed further in the next section). The simulations using point sources are in good agreement with the system data.

Simulation

The Poisson Spot

A well-known demonstration of wave behavior of light, is to shine a coherent beam on a small opaque disk to create a shadow with a surprising tiny bright spot in the center (attributed to Poisson and Arago). A similar experiment can be created in this system by just disabling the center speakers. This results in  a weak line after the dark region due to constructive interference of waves diffracting from the edge of the speaker groups. This is equivalent to diffracting around a barrier. 

A Speaker Grating

In optics, a grating can be a series of evenly spaced slits, prisms or mirrors, often used for splitting light into its component colors for spectroscopy or special effects. In a transmission grating, some light travels straight through and the rest is deflected into multiple diffraction angles that depend on the grating spacing and light wavelength. 

Beyond the near field, the transmitted light intensity pattern \(A^2\) can be calculated using the same point source sum method where n grating lines are spaced at \(s p_s\):

\(A^2 = \frac{sin(n\alpha)}{sin(\alpha)}\)

\(\alpha = \frac{\pi s p_s sin\theta}{\lambda}\)

This is analogous to n speakers spaced s speakers apart. The following waveform images, Amplitude images and simulations illustrate various grating configurations. In some cases, the near field interference completely fills the image space for this system. 

Waveform Data:

Amplitude

Simulations:

Far Field Effects from a Speaker Grating

The Amplitude images can be compared with the grating equation only in the far field (\(x \gg \lambda\)). A few examples are shown below. These compare 1) previous simulations of wave Amplitude within the chamber area (near field),  2) the same simulation over a much larger area (far field), color enhanced, 3) a cross-section plot across y, and 4) a plot of the grating equation as a function of angle. The angles and amplitudes of the beams agree with the diffraction equation. At a wavelength of 38 mm, a larger chamber ~ 2 x 2.5 meters is required to see the far field grating beams.

For the case of n = 3 speakers, s = 9 speaker widths apart, a simulation of the waveform over the extended path shows the beams at an instant in time:

Phase Grating vs. Amplitude Grating

Using alternating groups of speakers is an example of an Amplitude Grating, resulting in sound beams traveling at specific diffraction angles in the far field. Another way to get similar outputs is to activate all the speakers, but delay or advance the timing of the wave by some fraction of the period T. This is often referred to as a Phase Grating. Delays can be created electronically or by sending the speaker output through a long channel. To give an example, consider a case where a group of 4 speakers is electronically delayed by T/2 or travels an extra distance of \(\Delta L = \lambda /2\), compared to their neighbors. This results in a phase shift of \(\pi\). If we consider 3 such groups with a phase delay of \(\pi\), separated by s = 9 speakers without phase delay, the phase profile would be as shown in the plot below. The resulting simulated output amplitude is shown on the right, including the far field (red arrows indicate 0 phase delay and bule is \(\pi\) radians). The result is very similar to the Amplitude Grating discussed previously for n = 3, s = 9, where the first order diffraction beam is evident at the same angles. The suppression of the center beam is interesting. If an amplitude grating is made with the same profile, the central beam is still present. It’s absence appears to be a consequence of using the phase grating approach with all of the speakers active.

Regarding Near Field Effects

A close examination of the data and simulated Amplitude images shows the near field pattern can change significantly with s spacing. For example, there are bright spots (hot spots) at the y center of the images that moves to lower x values as the s decreases, as do the other features of the image. This is shown in these simulated Amplitude images for f = 9000 Hz, tracking two hot spots along the x axis, indicated by green and blue arrows. 

These near field features are known in optics as part of the Talbot effect where the waveform tends to replicate the grating pattern at successive locations along x. The characteristic Talbot length \(L_T = (s p_s)^2/\lambda\). In the above example, this gives \(L_T\) = 125mm, measured from the speakers. Since these images begin at x = 97mm, the columns of bright regions indeed correspond to some of the \(L_T\) positions. The central hot spot is then one of the replicas of the central speaker.

The position of the approximate maximum of these spots \(x_p\) is plotted below as a function of \(L_T\) showing good linearity. The plot indicates \(x_p\)(green arrow) = (1/2) \(L_T\), and \(x_p\)(blue arrow) ~ (5/3) \(L_T\). From the case of n=5, s=7, both spots are visible and appear to be part of source images 3 replicas apart. For a given s, higher n tends to brighten and shorten the spot.

These behaviors indicate that all speakers add constructively at the hot spot location. The phase of waves from a speaker at an angle \(\alpha\) to the spot compared to \(\alpha\) = 0 is approximately \(\Delta\phi \approx x_p \frac{2\pi}{\lambda}(\frac{1}{cos\alpha} – 1)\). Any \(\Delta\phi\) that is a multiple of \(2\pi\) will add constructively at \(x_p\) and brighten the hot spot. 

Since the near field pattern location along x has a factor of  \(1/\lambda\) in \(L_T\), we can imagine using this property as an acoustic spectrometer. A single mic could scan along x at the center of n speakers each emitting the same sound. Each frequency component of the sound would peak at a unique x value, providing a measure of the sound spectrum vs. x. Higher n values would lead to narrower spots and higher resolution. For best accuracy in determining the frequency from the near field images, one could correlate the entire data image to the simulations at target frequencies to maximize their correlation. The simulations and below show the peak spot location as a function of the sound wavelength, and are in agreement.

Scaling this experiment to audio frequencies such as 50 – 1000 Hz, using 7 speakers at a spacing of 30 cm, moving an ear or mic out from the middle of the array should allow one to hear or record the audio spectrum moving from x = 2cm (50 Hz) to 44 cm (1000 Hz). Far field diffraction orders could also be demonstrated for other combinations of a series of speakers for interesting demonstrations.

There is also the potential for optical spectroscopy, in principle. If light is transmitted through a grating or divided by a fiber beam splitter, a probe fiber could travel along x in the near field to extract a spectrum in a very compact assembly (for visible light \(L_T \approx\) 1 micron). However, many other means are employed today for small spectrometers using other types of interference effects.

Near field optics is of particular interest in modern lithography used to pattern layers in integrated circuitry. Talbot features and lithographic masks to correct for near field effects are the rule rather than the exception.

Wave Reflection

Flat Reflector

So far, the experiments have examined different configurations of speakers to demonstrate wave behavior with sound. Reflection is another common behavior for waves. In this system, objects placed in the wave path must fit into the waveguide gap and the mic array scan can begin only after the object's end point. 

To demonstrate a plane wave reflection from a flat mirror, a single wooden bar, sealed to the waveguide plates with foam, was placed at an angle of 13° to the x axis. The incident sound wave was generated by enabling 5 adjacent speakers (#3 - #7), and the scan started after the bar. The resulting wave pattern shows diffraction off the edge of the bar and a clear beam traveling 26° - 28° to x, which is the reflection angle expected for plane waves. Without the bar, the same speaker output produces near circular waves diffracting around the edge of the speaker cluster. A photo of the system is shown below, turned on its side to correspond with the diagrams and sound movies.

Parabolic Reflector: Collimating Sound

A parabolic mirror can be used to create a collimated beam from a point source like a small speaker.
Two halves of a parabola were cut from MDF and placed in the center of the waveguide, sealed on the top and bottom faces with foam. The shape was defined by \(y^2 = 4fx\), where f = focal length measured from the bottom of the parabola along x. A single speaker was enabled at the focal point of the parabola pieces.

The wave movie is shown below, starting after the parabola, and shows a plane wave beam at the exit, validating the design.

Parabolic Reflector: Focusing Sound

In the reverse orientation, the same parabolic reflector can focus a plane wave to a point at f. A collimated beam is approximated by a row of 15 adjacent speakers at the wide portion of the parabola.

The wave movie shows a concentration of the sound at the exit of the parabolic reflector. Further refinement of the shape would be required to sharpen the focus.

Phase Profile using Tilted Channels

Creating a Phase Delay

A sound wave can be delayed by increasing its path length using a channel. A series of channels could then be used to create a phase profile for the speakers, as a means to direct and shape waves. To check the feasibility of this method, two parallel wooden bars were separated by ~15mm and placed at an angle \(\theta\) to the x axis. The bars were sealed to the waveguide surfaces using foam. 

In order to measure a delay, two waves were generated. In region A, 6 adjacent speakers were enabled to create a narrow plane wave beam, and in region B a single speaker was enabled, feeding into the channel of length L. The mics were scanned from the end of the channel at \(x_0 = L cos\theta\) to the end of the chamber. An example movie is shown below:  

The waves in region A are mostly planar while at the end of the channel in B the waves are circular until they begin interfering with the A waves. The first 4 -5 peaks from each region in the first frame of the movie were used to measure the lag \(\Delta x\) of B behind A. A plot of \(\Delta x\) vs. the channel tilt angle is shown below the images. 

The propagation length in the channel of region B is longer than that of the plane wave in region A by \(\Delta x = L(1-cos\theta)\). Using L = 330 mm, the plot of the measured and calculated lag shows a somewhat higher value than expected from the path length difference, possibly from interference between the two wave patterns. The agreement is sufficient to validate the method of tailoring the phase lag using tilted channels or tubes.

Creating an Angled Plane Wave

If N tubes of length \(L_i\) are arranged to give a linear increase to the delay time (phase \(\phi\)) for each channel, the sum of point sources then produces a plane wave that is tilted by an angle \(\beta\) from the y axis. For each speaker i, at position \(y_i\), a desired phase delay \(\phi_i\) is related to the path difference \(\Delta L_i\) by:

\(\Delta \phi_i = \frac{2\pi}{\lambda} \Delta L_i\)

\(\Delta L_i = L_i - x_0\)

\(L_i = \frac{x_0}{cos\theta_i}\)

The plane wave tilt \(\beta\) is related to the position of the end of each tube \(y'_i\) by 

\(y'_i = y_i + x_0 tan\theta_i\)

\(tan\beta \approx \frac{\Delta L_N - \Delta L_0}{y'_N - y'_0}\)

A plot of an example linear phase shift design and the corresponding tube angles is shown below for \(x_0\) = 220 mm. 

The output of each speaker was directed by a 1x1cm square aluminum tube of length L tilted by angle \(\theta\) to the x axis. The tubes all ended at a distance \(x_0\) from the speakers to establish a row of point sources with specified phase lags. A picture of the apparatus with the tubes is shown below.

The wave movie confirms a tilted plane wave with a tilt angle very close to that from the designed phase profile.

Acoustic Prism:

From the relation between the phase shift and the tube length, we also see that the phase depends on the wavelength. As a result, we can expect that different wavelengths will be tilted to different angles, as the simulated images illustrate below:

 

In this case of frequencies from 8 kHz to 10 kHz, the angle change is calculated to be about \(3^\circ\). Also note the visibly larger wavelength for 8 kHz compared to 9 or 10 kHz. 

This method uses a linear phase profile to angularly separate sound wavelengths, which resembles the operation of an optical prism. However, the phase shift approach utilizes optical interference, while a glass prism uses refraction. Materials like glass refract (bend) light at the air/glass interface due to their material refractive index. Also, the higher acoustic frequencies are tilted less by the phase shift method, which is the opposite behavior for light and glass prisms.

Acoustic Lenses

The method of creating a phase delays using tubes of different lengths and tilt angles provides an opportunity to explore other phase profiles. Two examples are shown that focus and defocus sound.

Focusing Lens

If a phase profile is convex shaped, wavefronts in the center will be delayed more than at the edges, creating a wavefront shape that converges to a point, much like a positive lens or concave mirror. Such a profile is shown below. The same relations are used to determine the tube lengths and angles to achieve the profile design.

The tube configuration used two groups of  9 speakers and tubes, shown below, along with the wave movie, Amplitude image, and simulated amplitude image. The wave fronts do indeed converge from the ends of the tubes toward an elongated focal region in the center of the image. 

The use of a phase profile to focus sound can be utilized to localize audio signals to single listeners without the use of headphones. It is used to target locations for sonar applications and in ultra-sound imaging to provide a known source for echo detection. Focused ultra-sound can be used to ablate unwanted tissue in medical procedures. In telemetry, phased arrays are used to focus radio frequency transmission from multiple antennas for communication with a distant probes or orbiting satellites. 

Defocusing Lens

Inverting the shape of the focusing phase profile creates a larger delay at the edges causing the center to advance sooner. This produces a diverging wavefront, making a plane wave act like a distant point source.  The design phase profile and photo of the apparatus using 18 adjacent speakers and tubes are shown below.

The wave movie shows the expected curved wavefront toward the end of the image. In the near field the Talbot images complicate the pattern close to the end of the speaker tubes.

A diverging sound pattern from an array of speakers can be used to better disperse sound into an audience. The phase profile achieves this electronically without physically directing the speakers.

3. Operating Principles

In the following sections provides details of the sound imaging system, beginning with the basic strobe operation.

A Micro-Controller Unit (MCU) provides a square wave with a period T to a speaker driver that feeds into a speaker. The speakers rapidly compress and decompress the air to produce a fairly sinusoidal pressure wave. The microphones at some position (x,y) relative to the speakers detect the pressure wave and convert it to an electrical signal of the same period. There is a delay in the mic signal from the start of the speaker signal due to the distance traveled by the sound.

The mic signals are amplified and fed to a Sample and Hold circuit triggered to capture each mic voltage at a prescribed delay time \(\tau_i\) from the start of the speaker signal. The delay times span the period of oscillation T in time increments \(\Delta\). This can be repeated at the same delay time for any cycle of the vibration, so \(\tau_i = i\Delta + mT\), where \(\Delta = T/(N-1)\) (an integral fraction of T), i = 0 to N-1, and N is the number of movie frames corresponding to delay time \(\tau_i\). The time it takes to electronically acquire a mic voltage reading must be \(<\Delta\). Each reading is repeated at some multiple of the period (mT) and summed to get an average signal \(S_{avg}(x, y, \tau)\). This is stroboscopic imaging. The mics then advance along x and the acquisition process is repeated. The data is then arranged into an image file \(S_{avg}(x, y)\) for each delay \(\tau_i\) to create N frames for a movie covering a full wave period.

System Schematic

The sequence of events in more detail:

1. The user provides parameter values to a control program on the laptop to configure each experiment and uploads to the MCU using a Command Prompt.
2. The MCU creates a digital signal to all speakers and an enable mask based on the user configuration.
3. Signals are amplified to drive a row of 32 speakers.
4. Sound waves propagate between upper and lower plates separated by ~1/3 wavelength in a sealed chamber.
5. Sound waves are detected by a row of 32 mics. The mics can be moved as a unit along x, controlled by a motor and the MCU.
6. The mic signals are amplified and low-pass filtered in parallel.
7. The amplified mic signals are captured in parallel by Sample/ Hold (S/H) units at a time \(\tau\) after the start of the speaker signal. The captured levels are then digitized by an Analog to Digital Converter (ADC). This is repeated at the same \(\tau\) delay at subsequent periods for averaging.
8. The averaged digitized mic data at each x, y position is output to the MCU for each time delay \(\tau\) over the full period of the sound wave.
9. The MCU outputs the averaged mic data at each \(\tau\) to the laptop log file and advances the row of mics to the next x position until a specified limit is reached.

The Acoustic Chamber

In large environments a microphone picks up the sum of the direct sound wave and its many reflections, which can obscure the direct wave. Suppressing these reflections is a complex task usually requiring an anechoic chamber. The sound pressure waves at the mic decrease as 1/r from the source, limiting the distance of wave travel.

Confining the sound to a chamber with height less than the wavelength creates a 2-Dimensional waveguide, resembling ripples on the surface of water. Absorbing sound along the edges to reduce distortions from reflection is much simpler than sound absorption in a large chamber. The sound pressure waves decreases as \(\frac{1}{r^{1/2}}\) for a 2D wave, extending the travel distance compared to 3D.

Wavelength and Chamber Dimensions

To create a wave with a smooth wavefront using multiple speakers, a speaker spacing (pitch) \(p_s < \lambda/2\) is required. The speaker outer diameters are 1.3 cm so \(\lambda \geq 3 - 4\) cm  is necessary (comparable to the wavelength of water waves). From \(f = c/\lambda\), where c is the speed of sound in air (343 m/s), gives f = 8.5 – 11 kHz. Based on speaker availability, f = 9000 Hz, \(\lambda\) = 3.81 cm was chosen. A side benefit is that this pitch is high enough to be out of many people's hearing range, making it a quiet or silent experiment, especially with a closed chamber.

A reasonable chamber width is about 10 \(\lambda\). Allowing for a few cm on either side for sound absorbers to minimize reflections led to a chamber width of ~18". Setting the number of speakers equal to a power of 2 for digital control, led to a choice of 32 speakers. The length of wave propagation was chosen to be longer than the width, similar to most ripple tanks, ~ 30”. The gap of the chamber must be sub-wavelength to ensure 2D behavior and was set to 13mm. Finally, a surrounding enclosure wall with a top lid was used for sound isolation to form a waveguide chamber while allowing easy access.

System Parameters:

• f = 9000 Hz, T = 111\(\mu s\), \(\lambda\) = 38.1mm.
• At each (x, y) position and time delay \(\tau_i\) for each frame, \(N_{avg}\) = 64 readings are averaged.
• Each of 16 frames corresponds to a time delay \(\tau_i = i \Delta\tau\) for each strobe capture, where i = 0 to 15, \(\Delta \tau\) = T/15 = \(7.4 \mu s\). 
• The mics move along x in steps of 13 mm. At mic position ix = 0 the mics are 9.7 cm from the speaker row due to mechanical constraints. The maximum ix is 48 steps covering 624 mm.
• The 32 mics are spaced 13 mm apart on the mic bar, giving a y range of 403mm.
• The 32 mini speakers are at a separation of 13.8mm at x = 0. They are piezo-electric transducers (PZT) with best operation from 8000 - 10000 Hz.

Mechanics

Extruded aluminum struts (2020, 2040) form a frame and rails for 3 gantry stages joined by a top cross bar. The frame is bolted to a 3/4" thick wood base 24” x  41”. The central gantry is attached to a toothed-belt wrapped between a pulley on a NEMA 17 stepper motor and a pulley at the far end of the chamber. The motor moves the cross bar and 3 gantries along x. 

A 1.25" thick table-top, consisting of two 1/8” thick aluminum plates adhered (marine epoxy) to opposite sides of a 1” thick foam insulation, is bolted to the frame. The perimeter of the foam was coated with poly-urethane to minimize acoustic resonant cavities. The top plate of aluminum forms the bottom surface of the waveguide chamber. The cross bar is free to move under the table-top. Side struts are attached to the protruding ends of the cross bar that extend vertically over the table-top. A stainless steel 5 mil x ½” band is bolted to the side struts over the table-top to act as a platform for the mic array.

A perimeter wall made from 4” high x ¼” thick aluminum is bolted to the wood base using angle brackets to enclose the waveguide chamber. Holes for cabling were pre-drilled in the left perimeter wall. A 1” high x ¼” aluminum frame was constructed and placed on the perimeter wall edge, attached by a piano hinge to the back wall. A 2 mm silicone gasket seals between the lid and wall. A 3/8” thick polycarbonate (PC) window was bolted to the top of the frame to complete the lid. The bottom surface of the PC forms the top of the waveguide chamber. The waveguide gap between the table-top and lid bottom is 1.2 -1.3 cm. Lid clamps were positioned, 3 along the front wall and one on each side.

Drag chains were installed along the back and front long gap between the table and wall, each housing 16 mic cables. They are connected at one end to the mic side-struts and at the other to a milled hole in the left perimeter wall.

The steel mic band was covered in Kapton tape for electrical insulation. Twisted wire pairs (30 AWG) were soldered to the 32 electret mics and epoxied to the Kapton. The mics were aligned using a jig to define even spacing and alignment during epoxy cure. They face left toward the speakers at a spacing of 13 mm. The twisted pairs were soldered to the shielded cables inside the drag chains on either side.

32 PZT speakers were inset into a row of holes in a ¼” x 1” x 18" aluminum bar. The bar was bolted along the bottom edge to another ¼”x 1" x 18" aluminum bar to form a long right-angle bracket. The bracket was supported by two vertical rods in post-holders for height adjustment to place the PZT exit holes at the center of the waveguide gap. PZT driver boards were mounted on the horizontal portion of the bracket.

Melamine foam (1/2” thick x 2” wide) was adhered to the far end of the table-top to reduce back reflections. Silicone foam, shaped to a wedge cross-section by a sanding process, was adhered to the side edges of the table-top and the PC lid, above and below the mic bar. The wedge creates a diminishing gap between the top and bottom foam to reduce side edge reflections (see inset diagram)

The mic bar is wrapped in PTFE tape between the foam wedges to reduce friction when moving.

Electronics

Speakers

Given the need for many speakers at a spacing less than the acoustic wavelength, the choices became small voice coils or piezo-electric transducers (PZT). 

Voice coils use a coil attached to a diaphragm near a fixed permanent magnet. Current creates a magnetic force between the coil and magnet, moving the diaphragm back and forth with the applied signal. In a small speaker (8 ohm, 1W), the coil wire is very thin and the diaphragm is often thin soft plastic. This led to excessive heating of the coil that softened the diaphragm and often melted the wire.

PZT speakers utilize a ceramic disc that bends as a voltage is applied. They tend to have specific resonant frequencies due to their shape and tend to be far more robust for small sizes than voice-coils. This meant operating over a narrow frequency range. Initial experiments with bender components and DIY Helmholtz resonators led to high variability in response, leading to commercially packaged PZT. The source was Mouser for model Adafruit 1740 with following characteristics: 8 kHz resonance external drive, 9 volt typical, 13mm diameter, 2.5 mm thick, C ~ 15 nF, good response over 8 – 9.1 kHz. NOTE: The lead colors (red/ black) do not correspond to a consistent polarity. The user must drive the PZT and observe signal from a mic on an oscilloscope to verify the phase relative to the PZT drive signal. Roughly ½ of speakers were 180° out of phase, so their leads had to be reversed. This is crucial to obtain a phase equalized array.

Speaker Amplifier/ Driver

A digital output from a micro-controller (MCU) is not generally capable of driving a PZT speaker. After experimentation with several drive schemes, an efficient and cost-effective solution was a dual full H-bridge circuit DRV8833 in a readily available breakout board (1.5A, 18mm x 16mm). Note: This board requires either a pull up on NSLEEP (EEP) pin 1 or a jumper across J2 that connects EEP to Vcc through an on-board 47 Kohm resistor.

Speaker Drive Scheme

PC boards were designed and purchased containing two DRV8833 breakout boards to operate a total of 4 PZT speakers. 32 speakers required 8 driver boards and 16 DRV8833. All DRV8833 are powered by an external power supply (Vdrive/ GND1) consisting of an MP1584 converter with a 12v input. Each half of DRV8833 requires two complementary digital inputs PWM and nPWM into IN1/ IN2 or IN3/ IN4 to generate the PZT outputs O1/ O2 or O3/ O4. The inputs come from AND gates (74AHC08) that transmit PWM and nPWM or 0 depending on their Enable inputs. The PWM and nPWM signals and ENi enables come from the MCU board. In this way all of the speakers are selected by their ENi inputs and all operate with the same signals at the frequency f = 9 kHz, in phase. Trim pots (50 Kohm) are used between the outputs and the PZT to equalize the speakers. 

Microphones and ADC

Miniature electret mics were chosen (uxcell 6050 6mm x 5mm) for compactness and good response near 9000 Hz. The leads were soldered to twisted 30 AWG wire, which was fed along the mic bar and soldered to leads in shielded cable (Mogami W2697).  The cables were fed though two drag chains (R18 10x15mm) on either side of the waveguide, fed through the perimeter wall into a shielded box containing the preamps and ADC. A preamp MAX4466 was chosen for excellent response (no AGC) and compactness. Its native microphone was removed and the leads from the ends of the shielded cables were soldered to the mic inputs. The outputs of the preamp were sent through a simple low-pass filter into one of 8 channels of an Analog AD7606 module (the system required 4 AD7606 for 32 mics). A separate 3.3 V supply was used for the pre-amps.

Motor Control and Hall Sensor

The row of 32 mics are moved by a single stepper motor (NEMA 17) under the waveguide table. The motor controller (TMC2208) provides a sinusoidal signal to the motor coils to minimize motor acoustic noise. The MCU provides an enable signal (disabled before and after a run), a step signal, and direction input. A 12V supply is used for the coils, and a separate 3.3V supply for the controller. The origin point for the mics is determined by a small magnet that moves with the mic bar gantry and a  digital Hall sensor (KY-003 breakout using 3144 sensor) at the far end of the chamber. Note: the Hall sensor only responds to the magnet’s North pole pointing to the flat face of the 3144. 

Micro Controller Unit (MCU)

The MCU must:

• Generate a square wave at the desired frequency
• Control which speakers are enabled
• Control the mic motor
• Trigger the capture of the mic signal at a precise delay from the speaker start point
• Extract the data from the AD7606 S/H
• Average the data and print to a data log. 

The timing precision (\(\approx 1 \mu s\))  is within the capability of an ESP32 S3 with built in waveform generation (MCPWM), timers (GPTimer) and motor control signals (RMT).

MCU Controller Board

A PC Board was designed and purchased to contain the following:

• MCU (ESP32 S3)
• 4 shift registers (74AHC595) to convert serial data (GPIO 10) to a 32 bit speaker Enable pattern
• Schmidt trigger (74AHC14) to create a complementary nPWM signal for DRV8833 speaker drivers from the MCU PWM output (GPIO 21)
• Level shifter (74LVC245) to convert 5V signals from AD7606 to logic 3.3V input to the MCU
• Connection headers to all speaker drivers, motor controller and all AD7606
• The ESP32 S3 is powered by a USB cable from a laptop to provide 3.3V for the MCU, upload parameters and receive the data log.

Power Supplies and Regulators

Electronics Layout

Software

The C code for the ESP-32 S3 was co-authored with ChatGPT. The program has over 500 lines of code and evolved as the project matured. There are other routines written in VBA in Excel for simulation and conversion of the MCU log file into text images. Much of the image processing uses scripts to create movie gif files and calculate standard deviation images using ImageJ. 

The basic process flow to obtain each movie is:

A block diagram of the Main.c program for the ESP-32 S3 is:

Speaker and Microphone Equalization

• During operation, the same electrical signals are sent to each enabled PZT speaker, but their sound level output can differ by a factor of two, mostly due to mechanical differences of the disks and housings. 
• Similarly, the same sound presented to the mics can produce a wide range of electrical signals. 
• Each of these devices is equipped with a trim-pot in their driver or amplifier to achieve equal output and response.

Attempts at Sound Refraction

One of the most well known optical effects is the refraction (bending) of light as it crosses a boundary between two different materials. Each medium has its own wave velocity c/n where c is the speed of light in a vacuum and n is the refractive index. This leads to a change in the direction of light propagation. With water waves in the shallow tray of a ripple tank, the surface wave speed decreases with the depth of the water, so refraction can be easily demonstrated just by placing a slab below the water surface to reduce the water depth. A triangular slab in the water then acts as a prism for water waves. 

In acoustics, the speed of sound in an ideal gas \(c = \sqrt{\gamma \frac{RT}{M}}\), where \(\gamma\) is a material property (ratio of specific heats), R is the ideal gas constant, T is temperature (Kelvin), and M is the gas molecular weight. In air at room temperature, c = 343 m/s. In \(CO_2\), the speed of sound is 267 m/s due to its larger M value. This has been used in demonstrations by filling a large balloon with \(CO_2\) to create a lens to focus sound. An attempt was made to try create a 2D lens in this system by filling an oblong balloon with \(CO_2\), flattened between the lid and floor of the waveguide, but its transmission of sound was quite poor, preventing any imaging attempts. In another attempt, a thin plastic membrane was adhered around the perimeter of two parallel triangular plates separated by a 1 cm spacer. This formed a small prism chamber that could be filled with \(CO_2\) and sealed. While this did transmit a plane wave, no wave deviation was observed, possibly due to gas leakage.

The temperature dependence of c was also explored to observe sound refraction. A thin heat pad 10 cm x 25 cm was laid along the sound propagation direction on a thin insulator. An attempt to observe sound waves after they passed over the heated region, relative to the cooler adjoining region, did not show any detectable phase shift. Calculations showed that the pad temperature might need to be so high that it could damage the system.

Potential System Enhancements

• Create a new table with an acoustically transparent top, perhaps using slats along x with an adhered membrane. Place MEMS microphones under the surface guided in the slots. This would allow imaging of the sound at all points before, under, and beyond objects.
• Shift from an MCU to FPGA circuit control to enable speaker phase and frequency variation, allowing electronic phase variation instead of using fixed guide tubes.
• Extend the mic bar closer to the speaker row for further near field exploration, with possible gain control on the mics or speakers to avoid mic saturation.
• Double the length of the system to observe both far field and near field wave phenomena.

Acknowledgements

Collaborators

Many thanks to John F., Paul B. and James W. for their guidance on the electronics and acoustics, Keith K. for machining, materials advice and shop technique, and my wife Barb W. for her endless patience and encouragement.

Role of ChatGPT

Many of the approaches used in this project were the result of interacting with the LLM ChatGPT. In particular:

• After presenting the basic concept of capturing a movie of sound waves using strobe acquisition with an Arduino, ChatGPT provided an initial recommendation for multi-MCU (4x ESP32 S3). It then provided multiple code attempts that ultimately proved that multiple MCUs could not properly synchronize 32 speakers and the stroboscopic mic acquisition. Chat GPT (and colleagues) then suggested gated speaker operation, and switching to a single MCU plus 4x Sample/ Hold circuits (AD7606) that enabled synchronized data capture.
• After sharing the complex sound patterns observed in an open room, ChatGPT suggested using a waveguide type chamber with a narrow gap, which was ultimately successful.
• ChatGPT helped source many of the electronic components and provided tutorials on drivers, pre-amps, power supplies, and power layout.
• ChatGPT provided details on the use of KiCAD to create schematics and PCB designs (with occasional corrections required due to version differences).
• Chat GPT5 provided all of the ESR32 S3 code based on my requested operation. This experience evolved from being highly error prone to nearly flawless code in the transition from V3 to V5.
• Exchanges with ChatGPT regarding acoustics were also useful for experimental design and analysis.

References

Videos Related to Imaging Sound

Schlieren Imaging

What Does Sound Look Like? | SKUNK BEAR

https://www.youtube.com/watch?v=px3oVGXr4mo&list=PLJatR-aiW35xmyFX-FsUWE5Yc-LlabcAc

Looking at Levitating Sound 1080p, SuperLaser123

https://www.youtube.com/watch?v=XzcfZxf1__E

Schlieren imaging Part 2 : Standing waves of sound

https://www.youtube.com/watch?v=tjOJMJ4wNt8

Schlieren imaging Part3 : Viewing continuous sound waves

https://www.youtube.com/watch?v=VOMYPu5DhMk

Schlieren imaging Part 4: Sound wave diffraction

https://www.youtube.com/watch?v=0axFaSNyEVI

Microphone Imaging

A site that shows transducers at 40 KHz (1x8) and a scanning mic to detect the output field: https://www.youtube.com/watch?v=z4uxC7ISd-c

\(CO_2\) lens and a scanning microphone: D. C. Thomas, Am. J. Phys. 77 (3), March 2009 p. 197