INTRODUCTION THE NATURE OF WAVES Introduction The Equations of Motion Classification of Water Waves WATER MOTIONS DUE TO WAVES WAVE BOUNDARY LAYERSCHAPTER 6 OSCILLATORY FLOW INTRODUCTION 1 The first thing that comes to your mind when I mention water waves are probably the waves that appear on the water surface when the wind blows. These range in size from tiny ripples to giants up to a few tens of meters high and up to a few thousands of meters long. But many other kinds of waves make their appearance on water surfaces in nature. Here are the important kinds: • Flood waves in rivers.—Very long and very low, these waves propagate downstream at a speed that is different from the speed of the flowing water itself. It is important to try to predict both the speed and the maximum height of the flood wave. • Seiches in lakes and estuaries.—These are standing waves that are set up in an elongated basin by a sudden change in water-surface elevation in part of the basin, for example by a sudden drop in atmospheric pressure or by transport of surface water by a sudden strong wind. They may have just one node or more than one node. • Tidal bores in estuaries.—These are waves of translation, in which the water moves along with the wave. They have steep turbulent fronts, which can be hazardous to small boats. • Tsunamis (seismic sea waves) in the ocean.—These are extremely long and low-amplitude waves with high propagation speeds that are generated by sudden large-scale movements of the sea floor, usually by movement on faults but also by volcanic eruptions or submarine landslides. Their extreme destructiveness comes about because their amplitude increases spectacularly, sometimes to several tens of meters, when they shoal. • Internal waves in the atmosphere and the ocean.—When a fluid is stratified by density, waves can develop within the layer through which the density varies. This is easiest to appreciate when the density change is compressed to a jump discontinuity at a well-defined surface, but internal waves can exist also in layers with only gradual change in density. Internal waves care common in many settings in the ocean, and water velocities associated with the waves are in some cases strong enough to move bottom sediment. They are also common in the atmosphere. 2 Water waves of these kinds are called gravity waves, because, as you will see in a minute, gravity is the important force involved. But another important kind of waves, pressure waves, are present also in fluids, both air and water. 184THE NATURE OF WAVES Introduction 3 In a very fundamental sense, the waves that are of interest to us here can be viewed as a manifestation of unsteady free-surface flow subjected to gravitational forces. That is, any unsteady flow with a deformable free surface can be considered to be a kind of wave. 4 Do not let it bother you that real water waves involve changes in the water-surface geometry even when you follow along with the waves. You know from Physics I that a function of the form y = f(x-ct) represents a wave traveling with speed c in the positive x direction—and the shape of the wave does not change if you just travel along with the wave. And c could be a function of t, meaning that the speed of the wave changes everywhere with time but the shape of the wave train still stays the same. But now suppose that you took one additional step: let c a function of x rather than t. Then the shape of the wave changes as it moves: there is no speed at which you can travel, along with the wave, to keep the wave shape looking the same. The best way to think about this situation is that each point on the wave (you could call such points wavelets) has its own speed, so that, as all of them move, the overall shape of the wave changes with time. 5 In terms of the forces involved in wave motion, the motions of the water in the interior and the geometry of the free surface are an outcome of the interaction between pressure forces and gravity forces. Although it may or may not help you any, one way of thinking about waves is to consider that gravity tries to even out some initial nonplanarity of the water surface, and in doing so produces a usually complex unsteady flow in which the water-surface geometry changes as a function of time, but the characteristic amplitude of the water-surface disturbance has no way of actually decreasing unless viscous forces act also. 6 Real waves do decrease in amplitude, of course, because of the slight shear and therefore viscous friction in the interior of the water. But unless the waves produce water motions at the bottom, the rate of viscous dissipation of the wave motion is very slight. Mathematically, this means that the viscous term in the equation of motion can be ignored. Only when an oscillatory boundary develops at the bottom is the viscous dissipation substantial. The Equations of Motion 7 The equation of motion that describes water waves is just the Navier–Stokes equation without the viscous term but including a term for gravity. It turns out that this equation for inviscid flow affected by gravity can be put into the form of a wave equation, so you mathematically the existence of waves should not surprise you. 1858 If you had never fooled around with waves before, your natural inclination upon reading the foregoing paragraph would probably be to try to solve the equations to account for the observed behavior of water waves. And people have been doing this since the middle of the 1800s. But there are two serious impediments to simple solutions: • The equation is nonlinear, because of the presence of the convective acceleration term, which as you know from Chapter 3 involves products of velocities and spatial derivative of velocities. • An even more serious problem is that one of the boundary conditions—the geometry of the free surface—is itself one of the unknowns in the problem! 9 So it is unfortunately true that there is no general solution to the problem. People have therefore tried to make various simplifying assumptions that allow some mathematical progress in certain ranges of conditions for water waves. Much mathematical effort has gone into developing these partial approaches and establishing their limits of approximate validity. Classification of Water Waves 10 It is notoriously difficult to develop a rational classification of water waves,