Demonstration and study of the dispersion of water waves with a computer-controlled ripple tank ͒ Bernhard Ströbela Fachbereich Mathematik und Naturwissenschaften, Hochschule Darmstadt–University of Applied Sciences, Haardtring 100, 64295 Darmstadt, Germany ͑Received 31 August 2010; accepted 16 January 2011͒ The design of a ripple tank built in an undergraduate student project is described. Water waves are excited acoustically using computer programmable wave shapes. The projected wave patterns are recorded with a video camera and analyzed quantitatively. From the propagation of wave packets in distilled water at three different depths, the phase and group velocities are measured in the frequency range from 2 to 50 Hz. Good agreement with theory is found. The propagation of wave trains of different shapes is recorded and explained on the basis of the stationary phase approximation. Various types of precursors are detected. For a depth slightly above the critical depth and thus nearly dispersion-free, the solitary-like propagation of a single pulse is observed. In shallow water, the compression of a chirped pulse is demonstrated. Circular waves produced by falling water drops are recorded and analyzed. © 2011 American Association of Physics Teachers. ͓DOI: 10.1119/1.3556140͔

I. INTRODUCTION enon that is important in fundamental physics ͑matter waves͒ and in technology ͑propagation of light pulses in optical The ripple tank is a standard part of physics lecture dem- fibers͒. onstrations and undergraduate laboratories. Following the 1,2 tradition of Young’s classic experiments, demonstrations A. Dispersion relation of water waves with ripple tanks concentrate on the observation of interfer- ence, diffraction, and reflection. By moving a point source If we neglect viscosity and friction with a rough bottom, over the water surface, the Doppler effect and the Mach cone the dispersion relation for a linear wave on the surface of a also can be demonstrated.3 Transparent insets can be used to liquid of depth h, with density ␳, acceleration of gravity g, vary the water depth, which allows the simulation of differ- and surface tension ␴ is9–11 ent media and the demonstration of refraction and internal ␴k3 reflection. Quantitative measurements are usually not made ␻ = ͱͩgk + ͪtanh kh, ͑1͒ with educational ripple tanks, mainly due to the delicate ␳ character of many experiments and the lack of a means of ␻ ␲/␭ recording. where is the angular frequency and k=2 is the wave number. The phase velocity vp and the group velocity vg In this paper, we show that a ripple tank is also well-suited ͑ ͒ to demonstrate and investigate various effects associated follow from Eq. 1 as with the dispersion of water waves. It can be used to illus- ␻ g ␴k ͱͩ ͪ ͑ ͒ trate the propagation of wave packets in dispersive media, vp = = + tanh kh, 2 k k ␳ including a visualization of the group velocity. Previous pub- lications on ripple tank measurements mainly focused on and nonlinear effects.4,5 Crawford described the generation of ␴k2 3␴k2 wave packets and the observation of the group velocity in a khͩg + ͪsech2 kh + ͩg + ͪtanh kh 25-ft-long ornamental water basin.6 Ripper considered tran- d␻ ␳ ␳ vg = = . sient phenomena connected with the propagation of water dk ␴k3 7 2ͱͩgk + ͪtanh kh wave pulses. The present paper includes the observation of ␳ these effects. ͑3͒ Figure 1 shows the phase and group velocities of waves on II. BACKGROUND distilled water of different depths. First, consider the deep ӷ As noted by Feynman, water waves, which are easily seen water limit kh 1. When a wave is present on the water and are frequently used as an example of waves in elemen- surface, gravity and surface tension tend to flatten the crests tary courses, are the worst possible example because “they and troughs and restore equilibrium. The latter increases with have all the complications that waves can have.”8 In contrast surface curvature, and therefore dominates at short wave- to electromagnetic and sound waves in most media, water lengths or high frequencies. This type of wave is called a waves are strongly dispersive and depend on the surface ten- capillary wave. In the capillary wave limit, sion and water depth. The dispersion leads to remarkable 3v ␴k2/␳g ӷ 1, v Ϸ ͱ␴k/␳, v Ϸ p . ͑4͒ effects, especially in the propagation of a single pulse with a p g 2 broad frequency spectrum. However, water waves allow the direct observation of wave packets of a given frequency with At long wavelengths or low frequencies, gravity dominates a group velocity different from the phase velocity, a phenom- as the restoring force. In the gravitational wave limit,

581 Am. J. Phys. 79 ͑6͒, June 2011 http://aapt.org/ajp © 2011 American Association of Physics Teachers 581 ͑ ͒ ͑ ͒ ͑␳ / 3 ␴ / ͒ Fig. 1. Phase velocity vp solid line and group velocity vg dashed line of waves in distilled water =1000 kg m and =0.072 N mat25°C for different depths h, as calculated from Eqs. ͑2͒ and ͑3͒.

v ͱ ͑ ͒ ␴ 2/␳ Ӷ Ϸ ͱ / Ϸ p ͑ ͒ v0 = gh. 6 k g 1, vp g k, vg . 5 2 With decreasing h, the normal dispersion range becomes less distinct and eventually vanishes below the critical water depth,7 At ␴k2 /␳g=1 or k=ͱ␳g/␴, both restoring forces contribute ͱ ␴/␳ ͑ ͒ equally, and the phase velocity reaches a minimum. Because hc = 3 g, 7 dispersion is zero at this value of k, the group velocity equals Ͻ which is 4.7 mm for distilled water. For h hc, both vp and the phase velocity. For distilled water, this case occurs for vg increase monotonically with increasing frequency, and ␭ / Ͼ ͑ ͒ =1.70 cm, vp =vg =23.0 cm s, and f =13.6 Hz. At higher vg vp anomalous dispersion . This behavior is apparent in frequencies, surface tension dominates. The phase velocity the first three plots of Fig. 1. At a depth of about 5 mm, water vp increases with increasing frequency, and the group veloc- waves are nearly dispersion-free at low frequencies. This ity exceeds the phase velocity. In analogy to optics, this be- depth has been recommended for ripple tank experiments if havior is called anomalous dispersion. Below a frequency of dispersion effects are to be avoided.12 13.6 Hz, gravity dominates as the restoring force. The phase velocity vp decreases with increasing frequency, and vp ex- ceeds vg. This behavior is normal dispersion. Pedagogically, it is a virtue of water waves that they show both normal and B. Wave propagation anomalous dispersion behavior in a convenient frequency Typically, the water in a ripple tank is excited by a con- and wavelength range. tinuous, approximately sinusoidal disturbance, resulting in ӷ In finite water depth when kh 1 is no longer valid, the an almost monochromatic wave.13 For the propagation of vertical motion of the water particles is encumbered, reach- such a wave, only the phase velocity vp matters, which de- ing zero at the bottom. This restriction is reflected in the scribes the motion of the crests and troughs. tanh kh term in the dispersion relation, which reduces both ␻ If instead a sine function of frequency 0 with an approxi- the wavelength and vp at a given frequency. In the shallow mate Gaussian-shaped envelope and a width of at least sev- water limit ͑khӶ1͒, which is approached at any nonzero eral periods is used as the initial disturbance, a wave packet → depth as f 0, dispersion vanishes, and both vp and vg is generated, which travels over the water surface. Although approach the value crests and troughs continue to move with the phase velocity

582 Am. J. Phys., Vol. 79, No. 6, June 2011 Bernhard Ströbel 582 C. Precursors

Transient wave phenomena associated with the propaga- tion of broadband signals in dispersive media are known as precursors. The broadband signal might be a short single pulse or the abrupt switching-on or switching-off of a sinu- soid. The name precursor refers to the Fourier components which travel faster than the main signal. Precursors, how- ever, can also follow the signal. Precursors were predicted by Sommerfeld and Brillouin17 as early as 1914 for electromagnetic waves in conjunction with a discussion of the nonviolation of causality for super- luminal group velocities near resonant absorption frequen- cies. Precursors of electromagnetic waves are difficult to ob- serve due to their short duration and strong absorption in regions of prominent dispersion. Fig. 2. Group velocities of water waves as a function of water depth under In contrast, precursors of liquid surface waves are readily ͑ ͒ the same conditions as in Fig. 1. Limiting velocities broken lines are v0 for 7,14 ␭→ϱ ͑ ͒ observable because these waves have low frequencies and shallow water limit and vmin at the group velocity minimum for strong dispersion over the entire spectrum. Three types of depths above the critical depth hc =4.7 mm. No group velocity exists in the shaded region. Water depths below 2 mm are not shown because they are precursors can be observed in water waves above the critical impractical due to the limited wetting ability of distilled water. The depth depth: the high frequency Sommerfeld precursor ͑period in- values used for the experiments are marked with arrows. creases with time; from the capillary branch of the group velocity curve͒, the low frequency Sommerfeld precursor ͑period decreases; from the gravity branch͒, and the Brillouin precursor ͑constant period; from the group velocity mini- mum͒. vp, the envelope of the packet moves with the group velocity vg. Inclusion of the quadratic term leads to additional spread- ing and chirping of the packet. A short single pulse, resulting, for example, from an im- pact on the water surface, has a broad spectrum centered D. Circular waves about zero frequency. Such a pulse does not propagate as a wave packet, but instead splits into its different frequency The impact of an object on a liquid surface is another 18 components and corresponding wavelengths. Nevertheless, example of a broadband initial disturbance. Le Méhauté developed a theory for the circular waves generated by drops vg retains a meaning in this case. With the method of station- ary phase,14,15 it can be shown that in the asymptotic limit, hitting a water surface. In contrast to earlier analyses pio- neered by the classical work of Cauchy and Poisson nearly that is, for both large distances x and times t from the origi- 19 nal disturbance ͑kxӷ1 and ␻tӷ1͒, the different frequency 200 years ago, which were limited to gravity waves, Le components of the original pulse propagate with their respec- Méhauté showed that for small bodies such as water drops, tive group velocities. Therefore, for a given distance x and capillary and viscous effects have to be taken into account. time t from the impact, we will find oscillations of such a The resulting circular wave pattern is almost independent of wavelength with a group velocity v ͑␭͒=x/t. The crests and details such as size and velocity of the impinging small body, g and is dominated by wavelengths near the group velocity troughs of these waves move with their appropriate phase minimum ͑4.33 cm in deep water͒. This fact has remarkable velocities v . p consequences for the backscattering of radar waves from Figure 2 depicts the group velocity for different wave- 20 21 rain-roughened seas. Experiments with a rain generator lengths as a function of water depth. It is seen that for any have shown a wave number spectrum of the agitated water depth, there exists a minimum group velocity. Below the surface with a distinct maximum, corresponding to a wave- critical depth h , the minimum group velocity is the shallow c ͱ length of 5.3 cm. water limit velocity v0 = gh, which is approached for kh There is almost no wave motion within an area limited by Ӷ1, and thus ␭→ϱ and f →0. For hϾh , the minimum c the radius rmin=vmint. Le Méhauté called the slowest but group velocity vmin detaches from v0, now decreasing with dominant wave group the “trailing wave,” which is equiva- increasing depth, and eventually reaches its deep water limit lent to the Brillouin precursor. Zhu et al.22 observed the trail- Ͼ of 17.7 cm/s, with a wavelength of 4.33 cm. For h hc, there ing wave over a distance of 1.4 m from the impact and de- is a range of group velocities between v0 and vmin where two scribed it as a dispersive wave packet. At small distances wavelengths ␭ coexist, which are consistent with the from the impact, the trailing wave is preceded by waves of ͑␭͒ / asymptotic limit condition vg =x t. Whether one of these the capillary branch whose wavelength decreases with in- wavelengths dominates in a specific situation depends on the creasing radius. These high frequency waves are subject to details of the pulse spectrum and wave attenuation. strong damping. Therefore, with increasing time and distance From Fig. 2, it can be seen that near the critical depth hc, from the impact, the low frequency waves of the gravita- the group velocities for long wavelengths ͑say ␭Ն2cm͒ tional branch may emerge whose wavelengths now increase nearly coincide. Under such conditions, single pulses of ap- with increasing radius. These high frequency and low fre- propriate shape and sufficient length propagate nearly undis- quency waves are equivalent to the high frequency and low torted, like solitons.16 frequency Sommerfeld precursors, respectively.

583 Am. J. Phys., Vol. 79, No. 6, June 2011 Bernhard Ströbel 583 Fig. 3. Computer-controlled ripple tank.

III. EXPERIMENTAL SETUP and a central bushing connected with the hose. The second audio out channel of the sound card can be used to operate As an undergraduate student project, a computer- the LED in a strobe mode. This mode is not used for the controlled ripple tank apparatus ͑Fig. 3͒ was designed and 23 experiments described. built based on an aluminum extrusion construction kit. The The waves projected onto the screen were recorded with a tank includes a transparent 4 mm Plexiglas bottom plate with 60 frame/s progressive mode camera.30 With this frame rate an affixed PVC frame shaped as a 1:4 sloped beach to reduce and an exposure time of ͑1/256͒ s, images and videos of reflections. The tank can be filled with water up to a height waves up to about 50 Hz frequency could be recorded. The of 20 mm.24 The waves above a window of 40 cm images were captured using the MATLAB image acquisition ϫ30 cm are fully projected by a high-power white LED source25 via a 45° inclined deflecting mirror onto a toolbox. vertical screen made from Plexiglas RP, a special material As a familiar interpretation, the bright lines observed on designed for rear projection.26 The overall dimensions of the the screen of a projection ripple tank represent the wave apparatus containing the ripple tank is 752 mm ͑width͒ crests, which act as cylindrical collecting lenses. In reality, ϫ630 mm ͑depth͒ϫ1450 mm ͑height͒. the situation is more complicated. Llowarch noted that “the ripples have a complicated lens action on the light, so that Both the window size and the maximum filling height of the tank exceed the dimensions of commercial educational the relation between the patterns of ripples on the water sur- 27 face and the pattern of light and shade on the screen is not ripple tank systems, which was one reason to implement 31 our own design. Common educational ripple tanks can be necessarily a simple one.” This statement is especially true equipped with a computer interface, so that many of the ex- for strongly curved surfaces, that is, for short wavelengths periments described in this article can also be performed and large amplitudes. The bright pattern is better described in with these systems. terms of three-dimensional caustics intersected by the two- dimensional projection plane, which are blurred by the ex- The shape of the initial disturbance to excite the waves is 32,33 programmed in MATLAB and sent using the sound command tension of the light source and/or chromatic effects. It to the computer sound card,28 essentially a digital to analog should not be assumed that the brightness on the projection converter. The output voltage is applied, after a variable am- screen reproduces the exact wave shape, that is, the elevation plification, to a loudspeaker that is tightly mounted against a above the average surface level. metal sheet with a central connection tube. A plastic hose In addition, the initial disturbance produced by the nozzle conveys the alternating pressure from the loudspeaker to a is not proportional to the calculated shape. The sound card nozzle-type wave generator attached to the water surface.29 has a strong high-pass character at low frequencies.34 Nev- For the generation of a plane wave, this nozzle is a slightly ertheless, the significant quantities of the experiments ͑wave- inclined PVC bar aligned parallel to the narrow side of the lengths, frequencies, and propagation velocities͒ are not bottom window with a blind slot milled into its underside, affected by these limitations. Some of the experiments de-

584 Am. J. Phys., Vol. 79, No. 6, June 2011 Bernhard Ströbel 584 Fig. 4. ͑a͒ Shape and ͑b͒ amplitude spectrum of a squared cosine pulse of duration ␶. scribed in Sec. IV were performed with different excitation the standard pulse. The bandwidth of the signal is ␶−1 amplitudes, and similar results were found, which shows that =0.2T−1. As an example, for a frequency of 5 Hz, a squared nonlinear effects also do not play an important part in these cosine pulse of 1 s duration with five complete periods of experiments. 200 ms is applied, with an effective frequency band of ͑5Ϯ1͒ Hz. A. Pulse shapes Although Gaussian pulses have perfect spectral properties B. Camera calibration without any side lobes, such pulses are not practical due to their infinite length. The experiments described in this paper To measure wavelengths and velocities, the digital camera use squared cosine shaped pulses, which combine a limited image must be calibrated by converting its pixel scale to temporal width with good spectral behavior. For a pulse meters. To do so, a stripe of transparent foil with two at- length ␶, the squared cosine time function is tached round labels is placed on the bottom window of the empty tank. A program was written to recognize automati- cos2͑␲t/␶͒ for − ␶/2 Ͻ t Ͻ ␶/2 ͑ ͒ ͭ ͮ ͑ ͒ cally the images of the labels and to determine their center s t = 8 ͑ ͒ ͑ ͒ 0 otherwise, positions to subpixel accuracy. With m1 ,n1 and m2 ,n2 denoting the pixel coordinates of these positions, and d the with the amplitude spectrum center to center distance of the labels in meters, a scaling S͑f͒ = ͑␶/4͓͒sinc͑␶f −1͒ + 2 sinc͑␶f͒ + sinc͑␶f +1͔͒, ͑9͒ factor F with units of pixels per meter can be calculated, ͱ͑ ͒2 ͑ ͒2/ ͑ ͒ where f is the ordinary frequency and sinc͑x͒ F = m1 − m2 + n1 − n2 d. 10 ϵ ͑␲ ͒/͑␲ ͒ ͑ ͒ sin x x . The Fourier transform S f can easily be Foil stripes with different label distances d were used in calculated by writing s͑t͒ as the product of a squared cosine different places and orientations on the bottom window, and and a rectangular function and applying the convolution the value of F was reproducible and constant to within 1% theorem. Figure 4 shows such a pulse shape together with its accuracy over the whole field so that optical effects such as amplitude spectrum. It can be seen that the first zero of this distortion can be neglected. spectrum occurs at a frequency of 2␶−1, with only weak side When water is filled into the tank, the value of the scaling lobes following. At f =␶−1, the amplitude is half of its maxi- factor has to be corrected slightly. The LED, which is essen- mum value, and the power is just one quarter. Therefore, an tially a point source, projects the labels on the dry tank bot- effective bandwidth of ␶−1 may be assumed for a squared tom onto the screen with the magnification M =q/p, where p cosine pulse of duration ␶. is the distance from the LED to the surface of the bottom For the generation of narrow-band wave packets, the window and q is the distance from the LED via the mirror to squared cosine pulse shape is used to modulate a cosine sig- the screen ͑see Fig. 6͒. With water in the tank and the waves nal of period T. For a better comparison of the experiments on the water surface projected, p is reduced by the water at different frequencies, a pulse with five complete periods depth h. The distance q is reduced by the displacement a of ͑␶=5T͒ is used in all cases. Figure 5 shows the shape and the virtual image of a point source seen through a parallel amplitude spectrum of this pulse, which we will refer to as plate of thickness h and index of refraction n,

Fig. 5. ͑a͒ Shape and ͑b͒ amplitude spectrum of a squared cosine pulse of duration ␶ with five complete periods ͑T=0.2␶͒. This wave shape is referred to as a “standard pulse.”

585 Am. J. Phys., Vol. 79, No. 6, June 2011 Bernhard Ströbel 585 Fig. 6. ͑a͒ An object placed on the surface of the tank bottom window is projected onto the screen with the magnification M =q/p. ͑b͒ Waves on the surface of a water layer of depth h are projected with the magnification MЈ=͑q−0.25h͒/͑p−h͒.

n −1 a = h · = 0.25h for water with n = 1,333. ͑11͒ n Therefore, the magnification MЈ of the water wave projec- tions on the screen becomes q − 0.25h MЈ = . ͑12͒ Fig. 8. Sequence of images of a wave packet with f =7 Hz in 16 mm deep p − h water, traveling in the upward direction. The crest marked by an arrow For a correct calibration, the measured scaling factor F has to travels faster than the packet and eventually disappears at its front, which indicates normal dispersion ͑v Ͼv ͒. be modified by the ratio of MЈ/M, which amounts to an p g effect of about 1.7% for the dimensions of our ripple tank and h=16 mm. By combining Eqs. ͑10͒ and ͑12͒, we find that the correct scaling factor for the calculation of distances N = 128͓C͑W/R −1͒ +1͔. ͑14͒ and velocities is Here, C is a factor Ն1, which can be chosen to optimize the MЈ ͱ͑m − m ͒2 + ͑n − n ͒2 p͑q − 0.25h͒ FЈ = F · = 1 2 1 2 · . display of faint wave patterns at the expense of a certain M d q͑p − h͒ amount of clipping of the stronger patterns. ͑ ͒ ͑13͒ Figure 7 c shows a typical normalized record of a plane wave packet after a 90° rotation ͑waves travel in the upward direction in this representation͒. The signal/noise ratio can be C. Image processing further enhanced by averaging some or all of the image col- umns. The averaged columns of successive frames are put Although not seen by the human eye, the camera reveals together to create distance-time diagrams that give a compre- ͓ that the projection screen is unequally illuminated see Fig. hensive impression of the propagation of the waves under ͑ ͔͒ 7 a . To avoid this problem with the recorded wave patterns, dispersion. These distance-time diagrams are discussed in the a reference image without any waves excited is recorded following as images with the physically correct labeling of before each series of waves. The camera stop should be ad- the x and t axes. justed to prevent saturation of this image. From the reference image R and the wave images W, normalized 8 bit wave images N are calculated according to the heuristic formula IV. RESULTS AND DISCUSSION

Three different water depths were chosen for the experi- ments: “deep” water, h=16 mm ͑the tank is nearly filled͒; near the critical depth, h=5.5 mm ͑chosen for an extended low-dispersion frequency range͒; and shallow water, h =2.5 mm ͑chosen for a complete and persistent wetting of the tank bottom͒.

A. Phase and group velocities To measure the phase and group velocities, standard pulses ͑see Fig. 5͒ of different frequencies were used to ex- cite wave packets. Figure 8 shows six snapshots of a wave Fig. 7. ͑a͒ Reference image without waves excited. ͑b͒ Raw wave image. ͑c͒ Normalized wave image with C=5 ͓see Eq. ͑14͔͒. The arrows indicate ad- packet of 7 Hz traveling in the upward direction. It is appar- ditional circular waves originating from the edges of the straight wave gen- ent that the phase velocity exceeds the group velocity in 35 erator at the lower end of the image. this case, which is typical for normal dispersion. The

586 Am. J. Phys., Vol. 79, No. 6, June 2011 Bernhard Ströbel 586 Fig. 10. Distance-time diagram of a wave packet in shallow water, h =2.5 mm, with f =7 Hz,asinFig.9͑a͒, but with anomalous dispersion. The visually estimated center of the packet is marked by a line. A Sommerfeld precursor ͑s͒ precedes the packet.

Fig. 9. Distance-time diagrams of wave packets in water with h=16 mm. ͑a͒ f =7 Hz, normal dispersion. The center of the packet, marked by a line, moves with the group velocity, which is less than the phase velocity ͑motion of the crests and troughs͒. At the end of the tank, the packet is reflected ͑upper right͒. ͑b͒ f =14 Hz, nearly dispersion-free. Note the widening of the packet as it travels through the medium, and the downward chirping ͑fre- quency decreases with time͒ which becomes most apparent after reflection. Both effects can be understood from the increase of the group velocity with increasing frequency. ͑c͒ f =28 Hz, anomalous dispersion. The center of the packet moves faster than the crests and troughs. The packet is followed by a faint Brillouin precursor ͑white arrows͒. distance-time diagrams shown in Figs. 9 and 10 consist of similar time-slices ͑averaged wave profiles͒ taken every ͑1/ 60͒ s and placed side by side along the time axis, giving a complete view of the propagation of the wave packets. These distance-time images should not be understood as two- dimensional images of waves traveling in the tank. Distance-time diagrams were taken for many frequencies, and the phase and group velocities determined by visually fitting the slopes. The results for the three water depths are shown in Fig. 11, together with the theoretical curves from ͑ ͒ ͑ ͒ Eqs. 2 and 3 . The measured values agree with theory to Fig. 11. Measured and theoretical values of the phase velocity ͑filled circles within a few percentage points except for the highest and and solid line͒ and the group velocity ͑open circles and dashed line͒ of lowest frequencies examined. At high frequencies, the group waves on distilled water at three values of h.

587 Am. J. Phys., Vol. 79, No. 6, June 2011 Bernhard Ströbel 587 Fig. 12. Unfolded distance-time diagram of the wave train developing from Fig. 14. Unfolded distance-time diagram of the wave train developing from a short single pulse in water with h=16 mm. A high frequency Sommerfeld a short single pulse in water with h=5.5 mm. Because the medium is nearly precursor ͑HS͒, a low frequency Sommerfeld precursor ͑LS͒, and a Brillouin dispersion-free for the lower frequency range, the pulse propagates like a precursor ͑B͒ are observed. The origin is at the beginning of the initial pulse. solitary wave. The higher frequency components lead the wave train as a The pulse is not visible because the water meniscus M distorts the projection Sommerfeld precursor S. in the vicinity of the wave generator. The pattern marked with R results from a faint second wave emitted backward from the wave generator and reflected by the near end of the tank. At the point P ͑t=2.75 s, x=0.69 m͒, a period T=0.25 s and a wavelength ␭=0.085 m are observed, thus f =1/T=4 Hz, ␭ / / / vp = f =0.34 m s, and vg =x t=0.25 m s, assuming the asymptotic limit. ͑ ͒ ͑ ͒ ␭ / / Equations 1 – 3 give =0.0844 m, vp =0.338 m s, and vg =0.257 m s Figure 13 shows the propagation of a longer sinusoidal for h=16 mm and f =4 Hz. pulse with a rectangular envelope. The front and rear ends of the pulse move with the group velocity. As expected, the discontinuities at the beginning and the end of this pulse velocity is difficult to determine because the waves decay induce precursors. rapidly. This effect may in part account for the apparent sys- For h somewhat greater than hc, there is an extended fre- tematic error of the calculated group velocities from the the- quency range which is almost dispersion-free. At this depth, oretical values. a short single pulse propagates like a solitary wave, as can be Ͻ seen from Fig. 14. In shallow water with h hc, the group velocity increases monotonically with frequency over the en- B. Pulse propagation tire frequency range. This property is demonstrated in Fig. To excite a broad spectrum of frequencies, single pulses of 15. A short single pulse in shallow water produces a wave pattern with dispersion, which is simpler than that found for 20 ms duration were applied. Figure 12 shows the propaga- ͑ ͒ tion of this initial disturbance in water with h=16 mm. To deep water see Fig. 12 . allow a longer traveling distance, the wave was reflected by Dispersion does not necessarily spread pulses. If an appro- ͑ Ͼ ͒ priate chirped pulse ͑that is, a pulse in which the frequency a barrier. As expected for “deep” water h hc , three kinds ͒ of precursors are visible. The validity of the stationary phase increases or decreases with time is applied as the initial approximation is demonstrated in Fig. 12 for one exemplary disturbance, the dispersive medium is able to compress such point P. a pulse considerably. Chirped pulse compression is used in radar technology36 as well as for the generation of ultrashort

Fig. 13. Distance-time diagram of the wave train developing from a rectan- gular pulse of1sduration and f =20 Hz for h=16 mm. The front and rear / ͑ ͒ ends of the pulse move with the group velocity vg =0.28 m s white lines , / which is different from the phase velocity vp =0.24 m s. In front of the Fig. 15. Unfolded distance-time diagram of the wave train developing from pulse, a faint high frequency Sommerfeld precursor ͑HS͒ is visible. The a short single pulse in shallow water ͑h=2.5 mm͒. The group velocity pulse is followed by a pronounced Brillouin precursor ͑B͒ originating from monotonically increases with frequency. The higher frequencies lead the the signal’s rear. Other precursors are superimposed by the signal itself. wave train as a Sommerfeld precursor S.

588 Am. J. Phys., Vol. 79, No. 6, June 2011 Bernhard Ströbel 588 impact ͑say for xϽ20 cm͒, at least if the surface is hit by a small object. For small wavelengths, tanh khϷ1, and the depth effect becomes negligible. The most prominent feature of the circular waves, particularly at later times, is their pro- nounced inner edge which delimits the trailing wave. Within this inner edge, which moves with the minimum group ve- locity ͑between 15 and 22 cm/s, depending on water depth, see Fig. 2͒, the water is almost quiescent.

V. CONCLUDING REMARKS It was demonstrated that a number of quantitative experi- ments on water wave dispersion can be performed with straightforward and inexpensive equipment. The interfacing Fig. 16. Distance-time diagram of the compression of a chirped pulse ͑see of a ripple tank to a computer opens up new possibilities for the inset for the shape͒ for h=2.5 mm. After the compression, the chirp is a classical field of research. inverted. Shallow water was chosen for this experiment for its monotonic The study of the dispersion of water waves is stimulating, and nearly linear frequency dependence of the group velocity. because it combines everyday observations ͑for example, cir- cular waves formed by rain drops in a puddle and the wake of a boat͒ with fundamental concepts ͑phase and group ve- laser pulses.37 Figure 16 demonstrates this effect for water locities and precursors͒, physics history ͑the Cauchy–Poisson waves. The expected intensification of the compressed pulse problem͒, and current technological aspects ͑propagation of could not be observed. a signal and chirped pulse compression͒. The experiments described in this article do not exhaust C. Circular waves the potential of a computer-controlled ripple tank. Further investigations as a part of student projects could include a The ripple tank was employed to study circular waves more thorough study of circular waves and transient effects produced by drops falling into water. Droplets of about 3 mm ͑precursors͒, the effect of reduced surface tension by the ad- 38 diameter were released 2 cm above the water surface. Fig- dition of surfactants, the change of surface tension as a func- ure 17 shows parts of the circular wave patterns taken 0.5 s tion of surface age,39 the propagation of water waves for a after the impact. The patterns recorded for the three different continuously varying depth, nonlinear effects due to the in- water depths are combined into one image to allow for visual teraction of waves, and a quantitative analysis of common comparison. The additional graph in the upper right quadrant ripple tank experiments ͑diffraction, reflection, and refrac- shows the wavelengths expected from the stationary phase tion͒. condition. That is, over the distance x, the wavelengths asso- /͑ ͒ ciated with vg =x 0.5 s are depicted for the three cases, as ACKNOWLEDGMENTS calculated from Eq. ͑3͒. Theory and experiment are in good agreement. The author is indebted to Kenneth Justice who designed It might be surprising that there is little visible difference the electronics and guided the students in the mechanical and between the circular wave patterns at the different water electronic workshops. Markus Appel is acknowledged for his depths. The similarity can be explained by the predominance support of the plastic machining. This work benefited from of short wavelength components in the neighborhood of the the commitment and the ideas of the many students involved within the past three years including Drago Babic, Thomas Degenhard, Amir Djamali, Youssef Elabridi, Stefan Groß, Christian Iser, Nico Kappenstein, Emanuel Kurek, David Lang, Martin Lawrenz, Benjamin Lochocki, Matthäus Panc- zyk, John Rieker, Hans Scheibner, Xavier Uwurukundo, and Michael Wormer. The author thanks Johannes Ströbel for critical reading and valuable suggestions.

͒ a Electronic mail: [email protected] 1 Thomas Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts ͑J. Johnson, London, 1807͒, Vol. 1, Plate XX, Figs. 265 and 266. 2 E. S. Barr, “Men and milestones in optics II: Thomas Young,” Appl. Opt. 2, 639–647 ͑1963͒. 3 W. Klein and G. Nagel, “Continuous ripple-tank demonstration of Dop- pler effect,” Am. J. Phys. 48, 498–499 ͑1980͒. 4 J. Wu and I. Rudnick, “An upper division student laboratory experiment which measures the velocity dispersion and nonlinear properties of gravi- tational surface waves in water,” Am. J. Phys. 52, 1008–1010 ͑1984͒. 5 G. Kuwabara, T. Hasegawa, and K. Kono, “Water waves in a ripple Fig. 17. Montage of images of circular waves produced by falling drops in tank,” Am. J. Phys. 54, 1002–1007 ͑1986͒. distilled water of three different depths. All images were taken at 0.5 s after 6 F. S. Crawford, “Water wave machine for demonstrating group velocity,” the impact. The graph in the upper right quadrant shows the wavelengths as Am. J. Phys. 41, 1203–1204 ͑1973͒. a function of distance at t=0.5 s, as expected from the theory. 7 T. Ripper, “A simple method for observing precursors in water waves,”

589 Am. J. Phys., Vol. 79, No. 6, June 2011 Bernhard Ströbel 589 Am. J. Phys. 78, 134–138 ͑2010͒. generated by water drop,” Chin. Sci. Bull. 53, 1634–1638 ͑2008͒. 8 R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on 23 Extrusion Base 30 by KANYA Ltd., ͗www.kanya.com͘. Physics ͑Addison-Wesley, Reading, MA, 1963͒, Vol. I, Chaps. 51–54. 24 A desired water depth can most easily be attained by filling in the corre- 9 Th. Dorfmüller, K. Stierstadt, and W. T.Hering, Bergmann/Schaefer: Le- sponding measured volume. Careful leveling of the tank bottom plate is hrbuch der Experimentalphysik, Bd. 1 Mechanik, Akustik, Wärme, 11th important, especially for experiments with shallow water. ed. ͑de Gruyter, Berlin, NY, 1998͒, pp. 709–714. 25 OSRAM OSTAR ͑white, 420 lm, viewing angle of 120°, light-emitting 10 L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. ͑Butterworth- surface of 3.2 mmϫ2.1 mm͒. To minimize blurring of short wavelength Heinemann, Oxford, 1987͒,p.247. patterns, the rectangular light-emitting surface of the LED was assembled 11 G. D. Crapper, Introduction to Water Waves ͑Ellis Horwood, Chichester, with its short side parallel to the wave propagation direction. 1984͒,p.31. 26 Manufacturer of Plexiglas RP: Evonik Industries AG, ͗corporate.evoni- 12 PSSC, Physical Science Study Committee, Laboratory Guide for Physics k.com͘. ͑Heath, Boston, 1960͒. 27 Ripple tanks can be purchased from CENCO Physics ͑Sargent Welch͒, 13 The higher harmonics from the intrinsic nonlinearity of water wave mo- Conatex Didactic, PHYWE, and Leybold ͑LD Didactic͒. tion are neglected in this context, because the resulting changes in wave 28 The sound card was acquired from Realtek High Definition Audio, ͗ww- profiles are beyond the scope of this paper. w.realtek.com͘. 14 T. H. Havelock, The Propagation of Disturbances in Dispersive Media 29 Best results are achieved when the nozzle is inclined and placed slightly ͑Cambridge University Press, Cambridge, 1914͒. above the water level so that the meniscus hangs down from its bottom 15 E. Falcon, C. Laroche, and S. Fauve, “Observation of Sommerfeld pre- end. cursors on a fluid surface,” Phys. Rev. Lett. 91, 064502 ͑2003͒. 30 Black-and-white FireWire camera, TheImagingSource, DMK 21BF04, 16 Genuine solitary pulses or solitons can only be expected if nonlinear 640ϫ480 pixels. effects contribute, which are neglected in this paper. 31 W. Llowarch, “Application of the Schlieren technique to the study of 17 A. Sommerfeld and L. Brillouin, “Über die Fortpflanzung des Lichts in ripples,” Nature ͑London͒ 178, 587–588 ͑1956͒; W. Llowarch, Ripple dispergierenden Medien,” Ann. Phys. 349, 177–202 ͑1914͒; English Tank Studies of Wave Motion ͑Oxford University Press, Oxford, 1961͒. translation: L. Brillouin, Wave Propagation and Group Velocity ͑Aca- 32 J. V. Hajnal, R. H. Templer, and C. Upstill, “A ripple tank for studying demic, New York, 1960͒. optical caustics and diffraction,” Eur. J. Phys. 5, 81–87 ͑1984͒. 18 B. Le Méhauté, “Gravity-capillary rings generated by water drops,” J. 33 Caustics from surface waves can often be observed on the bottom of a Fluid Mech. 197, 415–427 ͑1988͒. water-filled pool in bright sunlight. 19 The classical Cauchy–Poisson problem of the fluid response to a local- 34 The sound card is usually connected to a speaker via capacitive coupling ized initial disturbance of the free surface was the subject of a contest of to protect the latter from direct voltage. the Paris Academy of 1813, which was won by Cauchy in 1816. Poisson, 35 The wake of a boat, which is essentially a wave packet in the normal who as a member of the jury was not allowed to participate, refined dispersion regime, behaves in the same way. Cauchy’s ideas in a later publication. Details can be found in A. D. D. 36 J. R. Klauder, “The design of radar signals having both high range reso- Craik, “The origins of water wave theory,” Annu. Rev. Fluid Mech. 36, lution and high velocity resolution,” Bell Syst. Tech. J. 34, 809–820 1–28 ͑2004͒. ͑1960͒. 20 M. J. Smith, E. M. Poulter, and J. A. McGregor, “Doppler radar back- 37 D. Strickland and G. Mourou, “Compression of amplified chirped optical scatter from ring waves,” Int. J. Remote Sens. 19, 295–305 ͑1998͒. pulses,” Opt. Commun. 56, 219–221 ͑1985͒. 21 L. F. Bliven, P. W. Sobieski, and C. Craeye, “Rain generated ring-waves: 38 The small drop height was chosen to prevent additional rings produced by Measurements and modelling for remote sensing,” Int. J. Remote Sens. splashes. 18, 221–228 ͑1997͒. 39 A notable decrease of the surface tension of open distilled water was 22 G. Z. Zhu, Z. H. Li, and D. Y. Fu, “Experiments on ring wave packet observed over several days.

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