Here's the general theory of small-amplitude surface or gravity waves on deep water
The surface of the ocean is in constant agitation. When a part of the surface rises above the general level, the pressure increases below it. When a part sinks below, the pressure is less, simply because of the weight of the water. The pressure difference causes water to flow to reduce the heights and fill in the sags. The moving water has inertia, so it overshoots, and what was once an elevation becomes a depression, and vice versa. This is the recipe for oscillations or wave motion, and we propose to study the motion mathematically.
We postulate that the water (or other liquid) is incompressible and nonviscous, and that no external forces act on it except gravity. Therefore, we assume that the motion will be irrotational, describable by a velocity potential φ. The equation of continuity for an incompressible fluid then demands that φ satisfy Laplace's equation. Our wave functions will be harmonics, a great aid to the analysis. For oscillatory solutions, we will assume a time dependence eiωt, and for plane wave solutions, ei(ωt - kx), where x is the coordinate in the direction of propagation. Note that all we have used at this stage is the existence of φ and the continuity equation (conservation of mass), no dynamics.
It is the boundary conditions that will introduce the dynamics and determine the nature of the solutions. Over any rigid, fixed surface bounding the fluid the normal velocity must vanish. This means that the normal derivative of the velocity potential must be zero over such surfaces--a necessary, but not the critical condition. The most important boundary condition will be the one at the free surface. In fact, this is where the dynamics will enter, and we must go to the equation of motion to establish it. In the case of potential flow, a first integral of the equations of motion can be made. We substitute v = grad φ in Euler's equation, express the result as a gradient, and integrate with respect to the space coordinates. The result is shown in the Figure. There is also an additive arbitrary function of time, but since this can have no space dependence, it can be neglected. In steady flow, this expression reduces to Bernoulli's Equation. Here, we retain the time derivative of the velocity potential, which is the important thing. If the wave amplitude is small, we can neglect the square of the velocity compared to the other terms. Then, the time derivative of the velocity potential is equal to the potential of the external forces, here gravitation, when evaluated at the free surface, where the pressure is constant regardless of displacement, and may be taken as zero. We make little error by evaluating the derivative at y = 0 instead, which is essential to our argument since we do not actually know the shape of the surface. In addition to this, the y-component of the velocity determined from the velocity potential must equal the velocity of the free surface. This gives a requirement on the y-derivative of the velocity potential, and again we evaluate it at y = 0. When we eliminate the yet unknown wave displacement between the two conditions, we find the boundary condition, shown in the Figure, that must be satisfied by the velocity potential.
Let us first look at plane waves travelling in the x-direction. We can solve Laplace's equation by separation of variables, writing φ as a product of functions of x, y and z. A suitable solution of this type is φ = cosh k(y + h)ei(ωt - kx), which already satisfies the boundary condtion on the bottom, y = -h. We assume that the body of fluid is of indefinite size in the x and z direction. The boundary condition at the free surface gives gk sinh kh = ω2 cosh kh, or ω2 = gk tanh kh. If the depth is much greater than the wavelength, tanh kh = 1, and ω2 = gk, from which c2 = g/k = (g/2π)λ. The phase velocity c depends strongly on the wavlength, so the propagation is dispersive. We can still superimpose solutions to get new solutions, but wave shapes are not preserved. This is the important limit of waves in deep water.
The particle velocities may now be found from the velocity potential by differentiation. The results are u = -ik cosh k(y + h) and v = k sinh k(y + h), for the vertical and horizontal components of the velocity, times the exponential factor and a constant. To find the amplitudes, we simply divide by iω, to find x = cosh k(y + h), y = i sinh k(y + h), where we have again suppressed common factors. Normalizing to unit x-amplitude at y = 0, we have x = cosh k(y + h) / cosh kh, and y = i sinh k(y + h) / cosh kh. Now, if kh is large, the x and y amplitudes are equal at y = 0, and 90° out of phase, so the movement is circular. With increasing depth, the amplitudes decrease, the y amplitude decreasing more rapidly. Near the bottom, the motion is predominantly in the x-direction, but is quite small. If the depth is equal to the wavelength, kh = 2π and the x amplitude on the bottom is only about 1/268 of its value at the surface. This means that a wave cannot 'feel' the bottom until its depth is rather less than a wavelength. The phases of the x and y motions show that the particle velocity is in the direction of the wave propagation at the top of the orbit. Near the surface, the fluid moves in circles at constant angular velocity, so the square of the velocity is constant. This shows that it can be neglected in forming the boundary condition.
In the Figure above, the wave moves to the right if the radii rotate anticlockwise, and to the left if the radii rotate clockwise. The profile of the wave is not sinusoidal (the troughs are steeper than the crests). We could not have successfully assumed a sinusoidal wave, as we do in other cases of wave motion. This wave is indeed sinusoidal in a sense (φ varies sinusoidally with x and t), but the wave shape is not sinusoidal. The wave shape we have found is valid only for waves of very small amplitude. For waves of finite amplitude, different shapes arise. In fact, the crests are steeper than the troughs, and this becomes accentuated as the amplitude increases. There is a great deal of interesting lore on wave types. Scott Russell discovered a 'solitary' wave that moves without change of shape, the first of the solitons.
The other limit is shallow water, or kh small. Then, tanh kh is approximately kh, and we find ω2 = (gk)(kh), or c2 = gh. The phase velocity is now constant, independent of the wavelength, so the propagation is nondispersive. At the surface, the x amplitude is 1, and the y amplitude is ikh, very much smaller. The movement is predominantly to-and-fro, with no appreciable y component, and about the same at the surface as at the bottom. Of course, the surface will rise where fluid collects, and fall where it is removed, but this motion does not enter into the dynamics. It is the custom to call such waves tidal waves, since the tides are an example of them, with wavelengths much greater than the depth of the oceans. Such waves will exert viscous forces on the bottom that are not present with surface waves.
If you drop a pebble into a pool of water, you will observe a ring-shaped wave disturbance moving outwards from the point of impact. If you watch it closely, you will see waves forming at the rear of the group whose amplitude passes through a maximum as they move faster than the group, and finally disappear in front of the group. The group of waves moves on steadily while this happens. The energy in the wave is moving at the speed of the group, not at the phase velocity of the waves. In this case, the group moves at half the phase velocity. This is a general result: energy moves at the group velocity when wave propagation is dispersive. Only in a few extraordinary cases is there a separate energy velocity different from the group velocity. We know that ω/k is the phase velocity. It can be shown that the group velocity U is dω/dk for waves of a small range of k values. For surface waves in deep water, 2ω(dω/dk) = g, so U = g/2ω = ω/2k = c/2, as observed. For tidal waves, ω = ck, where c is a constant, so U = c. Nondispersive waves are so common that we come to forget the difference between U and c. It is not surprising that group velocity was recognized first with surface waves on water.
There is a useful formula for the group velocity in terms of the wavelength dependence of the phase velocity. Lamb gives a derivation of it that is simple, but not easy to comprehend unless the matter is well understood. Since it is a rewarding exercise, we give it here. In the Figure, we have plotted the straight lines representing the propagation of groups of different wavelengths with their group velocities U, originating from a disturbance at x = 0, t = 0. Think of these as contour lines on a surface λ(x,t), which gives the dominant wavelength in group passing point x at time t. Then, U can be defined by the top equation, which says that λ is constant on the line x = U(λ)t. It gives the change in λ in time δt, which is zero. There is also a function c(x,t) giving the phase velocity, whose lines λ = constant are steeper than those of U. The change in λ along such a line in time δt is not zero, but some increment δλ. This increment is the change in λ seen by a moving wavfront, which is the increase in c per wavlelength, times δt, the rate of separation of groups differing by δλ (it is difficult to see this). Transposing this term, we find an equation for no change in wavelength, which defines the group velocity. If we substitute ω/k for c, we find U = dω/dk. In terms of the index of refraction n = c/v (v is now the phase velocity) for light, the group velocity U = (c/n)[1 + (1/n)dn/d&lambda)]. Since n generally decreases with increasing λ for optically transparent materials, U < c. Speed of light measurements measure group velocity.
There are several graphical representations of group velocity. If the phase velocity is plotted as a function of wavelength, a tangent from any point intercepts the velocity axis at the group velocity U, as shown on the left in the figure. This comes directly from the equation we derived above. A plot of ω as a function of k is called the dispersion curve or relation. The slope of a line from any point to the origin is the phase velocity, while the slope of the tangent at the point is the group velocity. A dispersion curve that is a straight line from the origin represents nondispersive propagation.
H. Lamb, Hydrodynamics, 6th ed. (Cambridge: Cambridge Univ. Press, 1953), Chapter IX.
Composed by J. B. Calvert
Created 13 July 2000