GFD Lab Demo II Waves I. 15i2001

P.B. Rhines, E.G. Lindahl

Summary

piano

DR, modes, standing waves

1 ripples

2 small tank grav waves

hydrostatic pressure

particle paths

U-tube oscillator

boundary layer

3 8’ tank homog, 2L grav current

4 small tank 2L paddle waves

resonance; velocity, mixing

Renwick particle paths, y

5 16’ flume: solitons

6 racetrack flume: constriction, grav waves, turning point

sand, wave action

A plucked string has a dominant tone, set by the travel time for waves rebounding from end to end. Its timbre, or tonal character, depends on the mix of overtones—higher modes—and their decay times. Piano, harpsichord, harp are all different in this way. A guitarist makes muffled harmonics by supressing the fundamental, lowest tone.

We hear all this because the resonant oscillator radiates through the air: it is quite inefficiently coupled, allowing the note to resound for some time before dying. The coupling with air is greatly increased by an intermediary: the small volume of the string first excites a sound board, whose breadth makes for better radiation.

These waves ideally are non-dispersive, each different wavelength propagating with the same phase speed. In this way sound, light and electromagnetic waves, the most familiar to us, are atypical of the vast majority of waves in oceans and atmospheres.

Waves move energy and exert forces on their carrier medium, while only to a small extent, dependent on amplitude, moving the medium itself. In this lab we look at basic gravity waves on a water surface and internal gravity waves in a fluid made of layers differing in density.

(i) Ripples - 1

Apparatus

overhead projector

flat-bottomed glass dish

dye

This projected demonstration has many applications beyond waves, and it is worth building a glass tank with optically flat bottom: see hardware chapter.

Place about 1 cm. of water in the dish on the projector, and focus so that waves on the surface are vivid. Wavecrests act as lenses, giving a bright rendition of the shape of the surface. Make waves with a small piece of plastic, say a flexible ruler. It can be bent into a curve to make a focusing reflector. Draw it across the surface and observe leading ripples and trailing gravity waves. Now drop some dye in the fluid and make waves in one corner. As well as outward radiation from the corner, there is a sizable mean drift of the fluid, streaming out with the waves. This is a first example of the subtle interaction of waves and mean circulation in fluids, a dominant topic in advanced GFD. Shaking the basin laterally, and tipping it slightly, gives an artistic set of waves refracting and steepening on a miniature beach. Refraction is the bending of wave crests which extend across regions differing in propagation speed, and it is most easily seen at the beach, where crests line up with the shore as they come into shallow water.

(ii) Gravity waves in a narrow tank

Apparatus

narrow 1m-long plexiglass tank (chap xx)

dye

particles

Fill the tank to ¾ of its height. Holding in your hands rock it gently, experimenting with different periods, and resulting different wavelengths. Viewed from the side, the free surface makes intriguing, ribbon-like waves. With particles in the water, notice the movement of the water beneath the surface: forward, in the direction of propagation beneath a crest (high elevation) and backward beneath a trough (low elevation).

Use a small piece of plexiglass just narrower than the tank to make waves from one end. Observe again the movement of a cloud of dye, and particles in the midst of the fluid. This movement occurs in a region about ¼ wavelength deep, beneath the surface. In long (shallow-water) waves, with wavelength greater than about ¼ the mean fluid depth, H, fluid particles move almost in blocks or columns, with little difference in velocity, top to bottom. The force balance in such waves is a bit surprising: while gravity, a vertical force, is the origin of the waves, it is the horizontal pressure gradient (due to the slope of the free surface) that drives horizontal acceleration. The vertical velocity w is less than the horizontal, u, in a ratio H/L, and the vertical force balance is hydrostatic, as in a resting fluid (the pressure being approximately equal to the weight of water overhead). Here the wavelength, l, is 2p/k, where k is the wavenumber. Similarly the wave period, T, is 2p/w, where w is the frequency.

The free surface is a pressure gauge graphically showing us that most mysterious of fluid quantities. For short (deep-water) gravity waves, however, none of this is true. u and w have the same amplitude and the pressure is not hydrostatic (the clue being that the horizontal velocity and acceleration vary with depth). Qualitatively, however, pressure is still higher beneath a crest than beneath a trough.

When a short wave propagates down the tank, particles suspended in the water move in small circles. Dye will reveal a bit of creep in the direction of the wave motion (as in the ripple tank, (i)). When a standing wave has formed the particles move in small curved arcs, with no enclosed area. These differing orbits are symptomatic of the energy transport in the wave. In a single propagating wave the forward energy flux that creates new wave crests is equal to the product of horizontal velocity and pressure. The forward motion where the water is deeper (i.e., where pressure is greater) shows that there is a non-zero average product of the two. In a standing wave, by contrast, the horizontal velocity is greatest at a node, where the free surface does not move, and the horizontal velocity vanishes beneath both crest and trough. There the average product of pressure and horizontal velocity vanishes: the energy flux is zero, with equal left- and right-propagating components.

A wave-packet is a small group of short waves, and you can just make out that the packet moves more slowly than individual wavecrests (in a longer tank it becomes obvious). The speed of individual crests, or phase speed, is ÖgH for long waves and Ög/k for short waves.

With the paddle at one end search for a resonance. A friend of ours, Prof. Pierre Welander, told of standing in a swimming pool, rocking gently back and forth at the carefully calculated natural frequency for the pool. Over time a resonant wave built up, eventually sloshing up onto the well-dressed party goers. Resonance occurs at periods which are equal to the time taken by a wave to propagating down the channel and back. The gravest or lowest mode is the longest such wave, and the longest such period (lowest frequency), with ½ the wavelength equaling the tank length. Search for this, and for higher modes, by feel. The force you exert on the paddle will markedly decrease near resonance, as the wave ‘does work on you’. Amplitudes will grow, and your students may get wet.

Dispersion. The wave-form for short waves is close to a sinusoid, with some effects of nonlinear steepening. Long waves of sinusoidal shape are more difficult to produce. The reason is the non-dispersive property. Every quirk and irregularity in the forcing or the initial conditions is preserved in a non-dispersive wave, whereas short waves naturally sort out into sine-wave (Fourier-) components.

Linear and nonlinear. The linear nature of small-amplitude waves tells us that any two pure sine waves can be added together without any interaction…they simply ignore one another. With large amplitude however, the presence of one wave changes the ‘medium’ for another. With long waves, at a wavecrest the water is deeper than average, and so the wave speed is larger. In addition, there is horizontal velocity which further speeds up the propagation speed (since the wave is propagating relative to the movement of the water). Thus a crest catches up with a trough, and its forward face steepens, leading to breaking.

(iii) Solitons and undular bores

An Englishman named Scott Russell reported in 1844 riding on horseback along barge canals, observing trains waves that propagated more as individual humps than as continous trains of many wavecrests. This occurs as a second kind of nonlinear effect occurs, when the curvature of the free surface increases and the wave steepens. It is then no longer a perfectly ‘long’ wave and begins to show dispersion. A single hump of water, propagating as a long wave will thus tend to steepen at its forward face, but then dispersion will develop in the region of high surface curvature. Shorter gravity waves have smaller group velocity than long waves, and thus they fall behind the leading wave.

Apparatus long plexiglas tank

This is a visually remarkable experiment. Level the tank and fill it to about 5 cm depth. Then use a piece of scrap plexiglas to ‘toss’ some water down the channel. Watch its self-organization into a set of wave crests moving down the tank. Nonlinearity here is measured by h/H, so it is very likely that the waves are nonlinear. The dominant pulse forms, steepens, and radiates shorter waves. If the scale of the experiment and H are fairly large, these will be dominantly gravity waves. At small scale, and mean depths H ~ 2 cm or less, ripples, due to surface tension, will be prominent. As these have group velocity greater than their phase velocity, they race out ahead of the dominant pulse.

It is worth trying some creative lighting of this experiment. A slide projector with a ‘slit slide’ can be directed at a small glancing angle onto the surface, so that the crests light up in a darkened room.

Undular bores refer to a change in mean fluid height, accompanied by a set of solitary waves. We will in chap xx look at hydraulic jumps and other wave/mean stream phenomena, and see there that a change in free surface height can indeed advance for long distances, provided energy can be removed from it continually. Here, it is the wave pulses falling behind or ripples racing ahead that drain off the energy.

A solitary wave or soliton is a single wave crest that propagates without any such radiation or linear dispersion. It is a perfect balance between the nonlinear steepening of the profile and the linear dispersion which tries to blunt it. Solitons are found in many wave media, and have great practical value in long-distance light transmission along an optical fiber. By defeating dispersion, solitons carry information more cleanly, and are thus worth money.

Modes in a channel develop with non-uniform surface height across the channel (y-direction). These can be considered to be waves propagating at an angle to the down-channel (x- ) direction, which reflect back and forth across it. A standing wave forms in the across-channel direction. If you plot the dispersion relation (w(k,l)) and pick out a single cross-channel wavenumber (the ‘gravest mode’ would be l = y-wavenumber = l₀ = p/channel width), then the remaining dispersion relation, w(k; l=l₀) suddenly has become dispersive! That is, it is a curvy dependence with group velocity that is not a constant. Thus, a simple, small ‘pulse’ of surface height at time t=0 will disperse into sinusoidal waves, down-channel, if the ‘pulse’ has cross-channel structure like sin(l₀y).

(iv) Internal waves in a layered fluid.

Oceans and atmospheres are part of a great heat engine driven by sunshine. When a fluid is heated from below it develops convection currents, and its density is unstably distributed in the vertical. If however, the fluid is driven irregular distributions of heat and cold, it will tend to be stably stratified, with dense fluid sinking and spreading, buoyant fluid rising and spreading. This stable stratification determines much about the transport of trace chemicals, heat, and even biological communities. It is part of the important ‘geography’ of the biosphere.

As a first look at the effects of stable stratification, a two-layer fluid is constructed and wave modes excited.

Apparatus

small plexiglas tank (1m x 10 cm x 30 cm)xx

electric motor, paddle or robotic driver

d.c. power supply

dye

particles

7% salinity water, fresh water

The fluid has two wave modes, external and internal gravity waves. The motor driven paddle excites principally the internal mode at low frequency. Or, in the 2001 demonstration, we used a ‘Robby the Robot’ stepper motor drive to move the entire tank left and right on rollers. This is equivalent to a ‘tidal forcing’ in which the gravity vector swings left and right of vertical (think in terms of the potential function F). There is some turbulence and mixing near the paddle, but both progressive and standing waves are visible; search for a resonance where the paddle is at an anti-node of the horizontal velocity (node of the interface displacement). Note the roughly two-layer nature of the velocity, with strong shear at the interface. This shear will tear up the interface unless the density contrast is great enough.

Even in this simple experiment we are confronted with the mixing that occurs in real ocean/atmosphere fluids.

The low frequency forcing of the robotic arm can drive the ‘lowest’ or ‘gravest’ mode, that is a gravity wave of length twice the channel length, so that there is one hump on the interface, rising and falling. A hand-paddle can readily generate shorter, higher frequency waves; these may be ‘long’ or ‘short’ (hydrostatic or non-hydrostatic) depending on H/L.

Note how small the free surface movement is as a big internal wave moves by. The slower propagation of internal waves makes them more nonlinear than surface waves with the same

typical fluid velocities, say U. The measure of non-linearity of a wave is often

e = U/c

where c = w/k is the phase speed.

(v) Surface Waves, mean flows and wave action conservation

Waves propagate through a fluid. If the fluid moves as a whole, then the wave propagation is shifted in speed; in the extreme, yet common, case they can be made to stand still on a moving stream. Not only does a mean current translate the waves, but if the current varies in space it will squeeze or stretch the wave train. In doing so it can exchange energy and momentum can be deposited in the mean flow.

Apparatus

reentrant water channel (flume)

constriction (‘peninsula’)

In this experiment we use a reentrant water channel driven by propellers. The speed of the flow is about 30 cm sec^-1, and twice this in a narrow constriction formed by a semi-circular ‘peninsula’. First examine the mean flow itself. Using dye injected with a long-nosed dropper, notice the acceleration of the flow as it approaches the ‘narrows’. There is some turbulence in this approach, but the real region of unsteady flow develops just downstream of the narrowest point, where the rapid flow separates from the boundary. Strong turbulence and a permanent gyre of reversed mean flow (a ‘back eddy’) occur. The height of the water surface is proportional to the hydrostatic pressure in the water (Lab. 1), and sighting along the wall of the channel, one can see a slight dip in the free surface in the narrows. This is provides a pressure difference with respect to the upstream flow, which is needed to accelerate the horizontal flow.

Using a square piece of aluminum nearly as wide as the channel, send surface gravity waves upstream into the rapid flow in the narrow passage. In absence of mean flow the waves propagate through the narrows with some little attenuation. With a mean flow the waves are at first able to move upstream, but as they come into stronger adverse current, they are slowed. In addition the mean velocity gradient ‘accordians’ the waves, squeezing the crests together. These two factors together act to stop the waves. There is nothing very special about the point at which they are stopped (that is, where their forward group velocity relative to the water is just equal to the opposing mean current). At that point the compression of the waves continues, they become yet shorter and slower and are swept back downstream.

This is a classical turning point from wave theory. It occurs in this problem at the point where the water flow has a speed equal to ½ of the initial group velocity of the gravity waves. If a packet of just 3 or 4 waves is created downstream, you can watch it move (with crests moving at twice the group velocity yet disappearing) and amplify. It lingers at the turning point and then is swept back downstream.

There are some very general and useful principles which govern the amplitude increase of these waves. If wave energy were conserved, we might see a modest increase in amplitude as the wave packet is squeezed into a smaller space. But in fact there is more amplification than that, for it is the total wave action and not total energy that is conserved by the waves. Wave action, A, is wave energy divided by the frequency—measured in a frame of reference in which the mean current vanishes. This is the ‘intrinisic’ frequency of the waves measured moving with the mean flow.

Here, as the gravity waves approach the narrows, they become shorter and hence their intrinsic frequency rises. The total energy in the packet rises in proportion. The energy density rises even more, since the area is compressing.

This is a very useful lesson for small boat work in coastal waters. Tidal currents are often intensified by narrows, by the tip of an island, or by shoaling topography. Swell (that is, long-distance travelling waves from open ocean), if it propagates against these tidal convergences, will produce dangerously large waves. Just up-current of the waves, the water is often slick and level, a fact which has misled and injured many a kayaker.

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