Using schlieren optics as a tool to see the invisible, we describe a technique of visualizing traveling ultrasonic (28 kHz) sound waves in real time. Suitable for lecture demonstration purposes or as an instructional laboratory experiment, our setup can readily demonstrate the reflection of sound waves from surfaces, diffraction effects around objects, interference, and standing waves. Additionally, the incorporation of color filters provides information such as gradient directions and sound wave phase differences not obtainable with just a white light source. As an example, acoustic standing waves are analyzed.

Refraction of light due to small changes in the index of refraction n in air can be made visible with schlieren optics. There are many possible optical arrangements for schlieren systems. We shall only describe our arrangement in this apparatus note, which is illustrated in Fig. 1. Readers interested in the various techniques may consult a comprehensive book on the subject by Settles.1 

Fig. 1.

General layout of schlieren experiment (not to scale).

Fig. 1.

General layout of schlieren experiment (not to scale).

Close modal

A long focal-length mirror is used to focus a point source of light onto a thin wire, which acts as a light block. Positioned closely behind the wire is a camera (Fig. 2), which is focused on objects immediately in front of the mirror in the “test area.” If there is a change in the index of refraction in the air within the test area, the image of the point light source will be deflected. Deflections are small and typically several arcseconds.2 If it is deflected past the edge of the wire, the additional light that enters the camera will appear as streaks of light originating from the area where there was a change in the index of refraction; this is the so-called schlieren effect.

Fig. 2.

Wire light block positioned in front of camera.

Fig. 2.

Wire light block positioned in front of camera.

Close modal

It is the blocking of the direct light from the point light source that makes the schlieren effect dramatic; blocking the direct light produces a dark background in the mirror and only the light bent by inhomogeneous refractive regions in the air is rendered visible against this dark background. Practical considerations in optimizing the schlieren effect will be discussed in Sec. IV.

The brightness of the schlieren effect is proportional to the magnitude of the change in refractive index of the medium in which light propagates—the greater the change in refractive index, the greater the deflection of the point light image resulting in more light passing over the edge of the wire. Fundamentally, it is changes in the medium's density that are responsible for changes in the refractivity, (n–1). For air and other gases, there is a simple linear relationship between the refractive index and the gas density, ρ (Ref. 3)
n 1 = k ρ .
(1)
k is the refractivity constant. Also known as the Gladstone-Dale coefficient, k is nearly constant over most of the visible spectrum. Its value for air is approximately 2.3 × 10−4 m3/kg. The expression for the angular deflection (of the point light image) by a density gradient, say dρ/dx, is given by4 
δ = k L d ρ d x ,
(2)
where k is the Gladstone-Dale coefficient and L is the span of the disturbance in the direction of the optical axis. Changes in air density can be the result of changes in temperature, flow dynamics, differing gases, pressure changes, or any combination of these. One can see convection currents rising from a warm hand or, alternatively, cold air sinking from a glass of ice water. The hot air from an ordinary hair dryer has the appearance of exhaust from a jet engine. Gases other than air are rendered visible. For example, sulfur hexafluoride gas (six times denser than air) can be visibly “poured” from a bottle into a glass. The images are strikingly beautiful and never cease to amaze. Students need to be acquainted with Snell's law of refraction and the formation of real images by spherical mirrors to understand the schlieren technique. It has been a popular lecture demonstration in our classes for many decades.

Since sound waves are pressure waves, and pressure variations result in density gradients, one might ask, “Can the schlieren method be used to see audible sound waves?” Bershader et al. calculated that a schlieren system with 0.2 arcseconds sensitivity cannot render 1 kHz sound visible in air without exceeding sound intensities beyond the threshold of pain.5 This sensitivity is an order of magnitude greater than one can normally achieve, so the answer to the above question is “no” for practical purposes. We present evidence of both optical and acoustic limits to the practicality of visualizing sound as a lecture demonstration or as an undergraduate instructional lab.

In the past year we have combined our schlieren setup with an acoustic levitation apparatus to actually “see” a 28 kHz standing wave.6 The rate at which the density changes in a 28 kHz standing wave is expected to be at least 56 times faster than a 1 kHz traveling wave (since pressure antinodes occur every half-wavelength). Thus, with a modest schlieren sensitivity of 10 arcseconds, one should be able to see pressure gradients at this ultrasound frequency. Indeed, that is the case. Heartened by this success, we have enhanced the technique to be able to visualize traveling ultrasonic waves as well.

Clearly, sound waves travel at speeds too fast to see in real time. However, just as one can “freeze” the motion of a rapidly spinning object with a stroboscope, sound waves can also be rendered motionless with the stroboscopic effect. If one illuminates the waves with pulses of light at the same frequency as the waves, then the waves will appear to be standing still. Stroboscopic illumination in conjunction with schlieren optics has been used before in certain applications,7–9 but the optical and electronic instrumentation was quite elaborate and sophisticated and not at all appropriate for lecture demonstrations or instructional laboratories. A search of the literature indicates that our experiment is the first to show traveling sound waves in air using readily accessible equipment. Apparatus and equipment details are presented in Sec. IV.

An ultrasonic transducer is set up in front of the mirror, generating 28 kHz sound waves. These are rendered visible by pulsing the point light source at the same frequency. The ultrasound wavefronts appear as bands of light across the width of the mirror. Frozen in space, one can hold a scale in front of the mirror to measure the wavelength.

Since we are dealing with traveling waves, it is more satisfying to see the waves progress forward rather than remain motionless. Setting the strobe frequency a few Hz less than the ultrasound frequency gives the illusion that the waves are progressing forward; the greater the frequency difference, the faster the progression rate. For example, if the strobe frequency is 3 Hz less, the slightly longer period between strobe flashes allows the wavefronts to progress about 4 cm further each second—a visually appealing progression. One can also set the strobe frequency to be a few Hz greater than the ultrasound frequency. The wavefronts then appear to be moving backwards—instructive and amusing, but not appropriate for the following demonstrations to be performed.

The next step is to show that these sound waves behave just like water waves that students may have already seen in ripple tank demonstrations. For example, the waves can be reflected in any direction by simply holding a flat piece of plastic (or metal or glass) in front of the oncoming waves. Diffraction of sound waves around the edge and into the shadow of the reflecting plate is also clearly visible. Images of these phenomena are shown in Figs. 3 and 4.

Fig. 3.

Reflection of sound waves.

Fig. 3.

Reflection of sound waves.

Close modal
Fig. 4.

Diffraction of sound waves.

Fig. 4.

Diffraction of sound waves.

Close modal

The creation of standing waves by the superposition of two traveling waves is easily shown by reflecting the waves back on themselves. By moving the reflecting plate just partially into the oncoming waves, students can see waves traveling in both directions. The plate can then be moved in all the way to completely superimpose the incident and reflected waves. This usually produces a somewhat crude-looking standing wave in the sense that one sees hints of it happening. By carefully adjusting the distance between the reflecting plate and transducer to be an integral number of half-wavelengths, the standing wave will become strong and stable with just one problem: the distance between maxima is a full wavelength rather than half. Half of the expected maxima are missing! The reason for this is that we are strobing at the same frequency as the waves with a very short pulse of light. This is a sampling issue. Strobing at 56 kHz reveals the entire standing wave.10 The scale with wavelength markings can be placed next to the standing wave to show that the bands of light occur every half-wavelength. Lastly, the fact that there is a bright band of light right next to the reflecting plate is consistent with the premise that the bands of light correspond to changes in air density. The air near the surface of the plate undergoes maximal density changes: condensation when the pressure is above atmosphere and rarefraction when below atmosphere. The bands of light occur at pressure antinodes.

The white bands of light tell us where there is a density gradient in the air, and the brightness of those bands corresponds to the magnitude of the gradients, but they do not tell us the direction of the gradients. Is the density increasing or decreasing? In other words, the information we obtain from the brightness of the bands of light is simply a scalar quantity. One needs additional information to know the direction of the gradients. The incorporation of color can give us that additional information and turn the information gleaned from the image into a vector quantity. The next step in the demonstration is to show how.

We use two narrow strips of color filters,11 butted together and placed behind the wire block, see Fig. 5. The red filter is above the green (the relative orientation is arbitrary, but must stay fixed throughout the experiment). Now, if there is a pocket of air in the test area in front of the mirror that is less dense than the surrounding air, that pocket will behave like a weak diverging lens (recall that light rays are always bent toward the region of higher refractive index, and away from regions of lower refractive index). Light rays that were once focused onto the wire block will thus converge slightly behind it, passing through the color filters along the way (see Fig. 6). If, on the other hand, there is a pocket of air that is denser than the surrounding air, it acts like a weak converging lens. Light rays will be focused slightly in front of the wire and then diverge after the focal point, subsequently passing through the color filters—however, in this case the colors are reversed. For example, if the bottom of a hot object appears green, the bottom of a cold object will appear red.

Fig. 5.

Color filters attached to wire light block.

Fig. 5.

Color filters attached to wire light block.

Close modal
Fig. 6.

Illustration of light rays refracted by pockets of different air densities.

Fig. 6.

Illustration of light rays refracted by pockets of different air densities.

Close modal

The vector aspect of the color schlieren can be beautifully demonstrated by holding a hot object (soldering iron) and cold object (ice cube tray), side by side, in front of the mirror (see Fig. 7). Notice the reversal of colors around the two objects. As one approaches the hot soldering iron from below, the air next to the iron is less dense and diverges the light rays down through the green filter. Thus, green corresponds to a decrease in air density. Above the iron, the air becomes more dense as we move away from the iron and light rays passing through that area diverge up through the red filter: red corresponds to an increase in air density. The same holds for the cold ice cube tray; approaching the tray from below, one encounters denser air and one sees the color red. Above the tray, the density of the air decreases as we move away and one sees the color green.

Fig. 7.

Color schlieren image of hot and cold objects.

Fig. 7.

Color schlieren image of hot and cold objects.

Close modal

Note that the opposite colors represent these density gradients if one approaches the objects from above, rather than below. The spatial gradient is then the negative of that encountered from below.12 The color representations of the gradients are defined by the orientation of the color filters. Thus, provided the orientation of the filters is not changed in an experiment, we now can define gradient directions by colors. Whereas before we could only judge the magnitude of the density gradient by the brightness (a scalar quantity), we can now also deduce whether it is a positive or negative gradient (vector quantity). Again, the colors that represent these gradients are arbitrary and must first be defined by known density gradients (such as the air around a hot object). Only then can one analyze gradients in unknown circumstances. We next analyze an acoustic standing wave with color information.

With color filters in place as described above, and the strobe frequency set to twice the ultrasound frequency, the previous bands of white light now appear as alternating bands of green and red (see Fig. 8). The green and red bands are separated by a half-wavelength. Recall that a green light represents a decrease in air density. That means the air in that region is changing from a pressure above atmosphere to a pressure below atmosphere. The red band represents the opposite: a pressure change from below atmosphere to a pressure above.

Fig. 8.

Color schlieren image of standing wave.

Fig. 8.

Color schlieren image of standing wave.

Close modal

Note that since the strobe is synchronized with the transducer generating the waves, we always see the same phase of the standing wave. That is to say, even though the density alternately decreases and increases during one oscillation cycle of the standing wave, we only see the first half of the cycle. If we were to see the whole cycle, the band would be a mixture of red and green, becoming an indistinct yellowish color. Instead, we can adjust the synchronization by adding a 180° phase shift to observe the second half of the cycle. When that is done, the red bands become green and the green become red!

Lastly, insert small Styrofoam balls into the standing wave. They settle into both the red as well as green bands of light, as shown in Fig. 9.

Fig. 9.

Color schlieren image of levitating Styrofoam balls.

Fig. 9.

Color schlieren image of levitating Styrofoam balls.

Close modal
Schlieren sensitivity is conventionally defined as the proportional change in illumination relative to the undisturbed background illumination, as seen by the camera. The background illumination in the mirror is mostly due to the light coming from the point light source image that is not obstructed by the light block. The point light source is a 0.5 mm diameter circular disk, and so is its image. The unobstructed light that passes the light block is thus a segment of a circle and the background illumination will be proportional to the area of that segment. An approximate formula for the area, A, is given by A = (2/3)ch, where c is the chord length and h is the height (sagitta) of the segment.13 When the circular image is deflected, the change in illumination will be proportional to the change in area, ΔA. Since the deflection, d, is small, the chord length will not change significantly and the increase in area can be approximated by a rectangular area, ΔA ≈ cd. Given these parameters, we can approximate the proportional change in the illumination, I, and thus the schlieren sensitivity, by
Δ I I = Δ A A = d ( 2 / 3 ) h = 3 ( δ 2 f ) 2 h = 3 δ f h ,
(3)
where δ is the deflection angle, f is the focal length of the mirror, 2f is the radius of curvature of the mirror (see Fig. 1), and h is the height of the unobstructed light source image.

An accurate determination of the sensitivity can be quite involved as it depends on some parameters and factors in a nonlinear fashion. Equation (3) is sufficiently accurate for our purposes. The three most important factors are mirror focal length, the size and shape of the point light source and structure of the light block, and the unobstructed height of the source image. Equation (3) suggests infinite sensitivity as h → 0. In practice, one cannot eliminate all stray light from reaching the mirror and achieve a completely dark background. Furthermore, diffraction of light around the light block ultimately limits the sensitivity.

From Eq. (3), we see that changes in illumination depend on how much of the point light source image is deflected past the light block into the camera. Here, we will simply characterize the sensitivity of our apparatus by the distance the light is deflected away from the light block due to refraction in the test area—the greater the deflection, the greater the change in illumination. The deflection, d, at the location of the light block is d = δ2f. The longer the focal length of the mirror, the greater the deflection and change in illumination—the mirror acts like an optical lever and increases the sensitivity proportional to its focal length. Note that the deflection is quite small—deflection angles are typically several arcseconds. Using a 2-meter focal length mirror, for example, d is approximately 0.1 mm.

Our setup is a single-mirror system.14 Spherical mirrors are ideal for this purpose for the following reason. Suppose one positions a point light source on the mirror's optical axis at its center of curvature (a distance equal to twice its focal length). The diverging illuminator light that bathes the mirror is reflected straight back onto itself (since the radius is perpendicular to the spherical surface) and forms a real image superimposed on the point light source. We shall denote the light reflected from the mirror as the analyzer beam. If the point light source is displaced slightly off the optical axis, its image will similarly be displaced on the opposite side of the axis. The schlieren setup is configured with this small off-axis separation between the illuminator and analyzer light beams, providing room for a video camera to be positioned behind the real image of the point light source. Note that the off-axis distance is very small compared to the radius of curvature of the mirror (R = 2f) so that, for all practical purposes, the configuration can be thought of as a coincident schlieren system as opposed to an off-axis system.15 

The area of interest where we wish to see schlieren effects lies directly in front of the mirror. The diverging light from the point light source and the converging analyzer light reflected from the mirror pass through this area. Since the test area is traversed twice by the same light ray, the ray will undergo two deflections. This gives an advantage of twice the sensitivity of a single-pass schlieren method and (theoretically) an additional factor of 2 in Eq. (3).16 On the other hand, since one beam is diverging and the other converging, the two paths through the test area are not identical. Consequently, there is a slight blurring of the image, which may reduce the resolving power. In practice, this has not been a problem.

To achieve high sensitivity, the point light source and its corresponding image must be small. The reason is simply that the deflection of the image is only a fraction of a millimeter. If, for example, the light source image is 1.0 mm in diameter and the image is deflected 0.1 mm, then only 10% of the available light from the image enters the camera resulting in a relatively weak illumination change. On the other hand, if the point light source image is 0.1 mm in diameter and the maximum image deflection is 0.1 mm, then 100% of the light from the image passes the light block and one has effectively maximized the dynamic range of light intensities entering the camera. An increase in dynamic range gives rise to greater contrast in the schlieren image, giving rise to higher sensitivity. Of course, as the point light source decreases in size, the brightness of the image also decreases resulting in poor contrast and low sensitivity. We have settled on the following compromise to minimize image size and maximize image brightness.

The point light source is a 10-watt white LED,17 providing approximately 700 lumens. To secure even illumination, Vellum translucent film is used to diffuse the light, which then passes through a 0.5-mm diameter pinhole. The entire assembly is housed inside a 1-in. diameter aluminum tube. For continuous operation, a power supply capable of delivering 14 V at 0.7 A is required. For pulsed operation, it is necessary to combine a function generator18 with an audio power amplifier19 to drive the LED.

For typical schlieren demonstrations, the image of the point light source is focused onto a knife-edge (razor blade). This blocks the analyzer beam from entering the camera. If there is a change in the index of refraction in the test area that causes a vertical deflection of the image past the knife-edge, the light that passes over the edge is seen by the camera as coming from the area where there is a change in the index of refraction. A light block oriented horizontally detects only vertical refractive gradients; horizontal changes in the index of refraction go undetected, and vice versa, for a vertically oriented light block. To be able to observe waves traveling in any direction, one could use a light block oriented 45° with respect to the horizontal. However, only index gradients that refract light past the edge of the light block would be detected. The opposite gradient that refracts light further onto the light block goes undetected. To circumvent that problem, we have chosen a thin wire to act as a light block, as shown in Fig. 2 and described below.

The image of the point light source is focused onto the wire, oriented at 45° with respect to the horizontal. The point light source image size dictates the wire diameter. To secure a dark background in the mirror, one would guess that the wire diameter should be the same as the image size. We have found that schlieren sensitivity seems to be optimum when 90% to 95% of the image is blocked. The image size is 0.5 mm (=0.0197 in.) and thus we have chosen size 7 (0.018 in. diameter) piano wire. With this geometry, refractions in any direction are detected (refractions parallel to the wire being the exception). The ability to detect positive and negative index gradients improves the overall sensitivity of the schlieren system.

For maximum schlieren sensitivity, it is important to accurately focus the point light source onto the wire. This can be executed visually using the following procedure. Move the wire into place from the side and observe the image in the mirror. If the wire is properly positioned in the image focus plane, the background illumination in the mirror will uniformly dim as the wire is moved in (from the side) to block the light source image. If the wire is slightly in front of the focus plane (closer to the mirror), a shadow of the wire will be seen moving in the opposite direction that the wire is moving. If the wire is behind the focus plane (closer to the camera), the shadow will move in the same direction as the wire. Fine-tune the position of the wire along the optical axis so that no shadow appears and the background illumination dims uniformly.

The specific camera and lens model/manufacturer is not important, but note that, for tight cropping of the schlieren image, the video camera needs to be equipped with a long focal-length zoom lens (in the 150–300 mm range). We use a Blackmagic Design Cinema Camera (MFT mount). The active region of the sensor is 15.81 × 8.88 mm, giving it a crop factor of about 2.3. For tight shots we use an Asahi Takumar 300 mm f/6.3 lens with a Zhongyi Lens Turbo 2 speed booster.

The ultrasonic waves are generated by a 28 kHz transducer designed for ultrasonic cleaning baths.20 These are typically driven by a push-pull oscillator specially designed for this application. Many are controlled by an Arduino microcontroller. Ours is driven harmonically, powered by a Samson Servo 120 audio power amplifier. The input signal comes from a sine wave function generator.21 A homemade impedance matching transformer22 couples the 8 -Ω output of the audio power amplifier to the ultrasonic transducer.

This experiment is rich in content. First and foremost, it is the first time that traveling sound waves have been visualized for use in lecture demonstrations in real time. Previous experiments involved elaborate and sophisticated electronic instrumentation. Excepting the mirror, the experiment can be set up with equipment found in most instructional laboratories. It is an engaging application of geometric optics, wave phenomena, and acoustic levitation. As an instructional laboratory experiment, the schlieren system is rich in possibilities and could be explored further as a Fourier optical processor.23 

With the addition of color as a tool, we have shown how the schlieren system can be used in a lecture demonstration to analyze the dynamics of a standing wave in acoustic levitation. The phases of the standing wave are made evident with the addition of color information. One would not expect to see either color since the density gradient continuously oscillates from positive to negative. However, the illumination pulse is synced to the transducer and is so short (10 μs) that only half of the pressure oscillation is captured. The alternating red and green bands of light in the schlieren image are striking. When the illuminating pulse is delayed by 180°, the reversal of colors is an extra bonus.

We would like to acknowledge and thank our anonymous reviewers for their helpful comments and suggestions for improving the manuscript. A special thank you to David Jackson (Dickinson College) for numerous thought provoking discussions.

There is very little in the literature concerning safety guidelines for ultrasound. One should not assume that it is harmless simply because one cannot hear it. With safety in mind, we drive the ultrasonic transducer with the minimum power required to achieve levitation of the small Styrofoam spheres—approximately 5–8 W.

Without recourse to measuring the sound level with a dB meter designed for ultrasonic frequencies, the following measurements were made to determine the ultrasound intensity. An Earthworks M-30 microphone was used to measure the 28 kHz sound level at a distance of 40 cm from the transducer (the distance one's ears might be during the experiment). According to the spec sheet, the M-30 microphone frequency response is flat (±2 dB) up to 30 kHz. The gain of the microphone preamp was adjusted so that the output was 1 Vp-p, as measured on an oscilloscope. A small loudspeaker was then placed next to the ultrasonic transducer and driven by an audio generator/amplifier at 1 kHz. A calibrated General Radio model 1551-B Sound Level Meter was positioned right next to the Earthworks microphone. The volume of the 1 kHz signal was then adjusted so that the signal from the Earthworks microphone/preamp was again 1 Vp-p, as measured on the oscilloscope. At that volume, the GR Sound Level Meter registered 78 dB—a level considered to be safe. Nevertheless, we use hearing protection whenever the transducer is on—safety first!

1.
G. S.
Settles
,
Schlieren and Shadowgraph Techniques—Visualizing Phenomena in Transparent Media
(
Springer-Verlag
,
Berlin, Heidelberg, New York
,
2001
). The author is one of the leading authorities on the subject. This is an excellent book that covers the various techniques in detail as well as providing a historical background. There is a wealth of information here with 1020 references.
2.

Reference 1, p. 51. Those well-trained in the schlieren arts recognize the sensitivity level of an image by the type of disturbances it reveals. This is seldom done in terms of refractive-index gradients, but rather ranked in terms of the deflection angle, δ, in arcseconds. If one can see the warm air generated by rubbing your hands together, this corresponds to δ being about 5–10 arcseconds.

3.

Reference 1, p. 26. For gases other than air, k may vary roughly from 1 to 15 (×10-4 m3/kg).

4.
D.
Bershader
,
S. G.
Prakash
, and
G.
Huhn
, “
Improved flow visualization by use of resonant refractivity
,”
AIAA Paper No. 76–71 of the 14th Aerospace Sciences Meeting
(
1976
), p.
4
.
5.

Reference 4, pp. 3–4. Schlieren sensitivity is directly proportional to the refractivity constant (Gladstone-Dale coefficient) of the gas. By seeding the air with 0.1 mol % of non-resonant sodium vapor (kNa/ kair ≈ 106), the authors improved the sensitivity of their apparatus by a factor of 1000. However, it appears that they erred in their calculation of sound overpressure in air: an incorrect value for L made their result too low by a factor of 100. This makes the prospect of detecting 1 kHz sound waves in air even worse.

6.
The video “
Ultrasonic Levitation
” <http://www.youtube.com/watch?v=XpNbyfxxkWE> uses Schlieren imaging to show Styrofoam balls suspended of an ultrasonic standing wave.
7.
L. F.
Lawrence
,
S. F.
Schmidt
, and
F. W.
Looschen
, “
A self-synchronizing stroboscopic schlieren system for the study of unsteady air flows
,” National Advisory Committee for Aeronautics Technical Note 2509 (1951). Developed for the study of air flows about aerodynamic bodies in wind tunnels, the authors describe a schlieren system having two light paths, one of which serves as a reference path with a phototube that triggers a stroboscopic light source for the other path.
8.
R. A.
Kadlec
and
S. S.
Davis
, “
Visualization of quasiperiodic flows
,”
AIAA J.
17
,
1164
1169
(
1979
). Kadlec and Davis improved on the Lawrence design (see previous reference) with extra sensors and modern flash lamp to freeze wave phenomena and analyze their phase relationships.
9.
D. R.
Andrews
, “
Study of wavefronts in acoustic diffraction patterns using a stroboscopic schlieren technique
,”
Proc. SPIE
348
,
565
570
(
1983
). Andrews uses the technique to study ultrasonic waves in water (wavelength = 2 mm) at repetition rates in the 200 to 2 kHz range.
10.

To prevent drifting, it is useful to synchronize the light pulse generator with the transducer's sine wave generator. The Pasco model PI 8587C has a TTL output that can be used as a trigger for the light pulse generator.

11.

Kodak Wratten 2 color filter #29 (red) and #61 (green). These are available from Kodak Cinema & Television (800) 621-3456. The website is motion.kodak.com.

12.

Let the positive y-direction be upward. Then, moving upward, a negative refractive-index gradient would be dn/dy < 0. Moving downward (in the negative y-direction), dy < 0, and thus a negative refractive-index gradient becomes 0. Hence the different colors above and below the object.

13.
J. W.
Harris
and
H.
Stocker
, “
Segment of a circle
,” in
Handbook of Mathematics and Computational Science
(
Springer-Verlag
,
New York
,
1998
), p. 92, Sec. 3.8.6,. The error is < 0.8% for 0 < θ ≤ 45° and < 3.3% for 45° < θ ≤ 90.°
14.

12.5″ diameter, 3.12 m focal length, f/10, protected aluminum mirror (originally purchased from Edmund Scientific for $600). The company no longer exists and the closest equivalent is a Techspec Precision Parabolic Mirror available from Edmund Optics, Part No. 32-277-522. When set-up space is limited, we also have an 18″ diameter, 2 m focal length, f/4.3 mirror salvaged from a spectrometer. The longer focal length mirror provides greater schlieren sensitivity, but the latter is a higher quality mirror producing better images. Note that at f/10 or higher, the difference between a spherical and parabolic mirror is insignificant for this application and a smaller diameter mirror can cost an order of magnitude less than a 12″ diameter mirror (currently $2500). Many schlieren videos on youtube.com use 6″ mirrors with excellent images.

15.

Reference 1, pp. 46–48, for a full explanation of coincident and off-axis geometries.

16.

Reference 1, pp. 42–46.

17.

LED Engin LZ4-00CW08 cool white, 1 channel, Standard Star MCPCB. Forward voltage = 14 V and current = 0.7 A.

18.

Pasco model PI 8587C digital function generator and Tektronix Arbitrary Function Generator model AFG 1022 are two options we have used. To secure color schlieren images of the standing wave, the function generator must have an external trigger option. The Tektronix generator also has a phase shift adjustment, an added plus.

19.

Most audio power amplifiers have a flat frequency response (within a decibel or so) up to 100 kHz. The power amp should be able to drive a 20-Ω load with a compliance of ¾ A. We use a McIntosh 30-watt audio amplifier, model MC-30.

20.
American Piezo
(www.americanpiezo.com) 28 kHz Cleaning Transducer model #90-4040. It is a 50 W transducer but, for safety reasons, we operate it at a minimum power of around 8 W.
21.

Pasco model 8587C digital function generator.

22.

The core of the transformer consists of two C-shaped pieces of ferrite which, when put together make a square. The primary is 10 turns of #18 wire and the secondary is 100 turns of #22 wire. The inductance of the primary is 230 μH w/ secondary open and 16 μH w/ secondary shorted. The operational inductance is such that its impedance is well matched to the 8 Ω output of the amp. The inductance of the secondary is 18.8 mH w/ primary open and 1.8 mH w/ primary shorted. It's operational inductance is around 10 mH. The static capacitance of the transducer is 3550 pF. To resonate at 28 kHz, we want an inductance of 9.1 mH. The inductance of the secondary is a close match for that. The output of the transformer can be as much as 400 Vp-p, but we operate it at approximately 50 Vp-p (the minimum power to secure levitation).

23.
See, for example,
E.
Hecht
, “
The spatial distribution of optical information
,”
Optics
, 2nd ed. (
Addison-Wesley
,
Reading MA
,
1987
), Chap. 14 or Reference 1, pp.
341
352
.