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.