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6.2: The Nature of Waves

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  • Introduction

    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.

    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 \(c\) be 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.

    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.

    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

    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.

    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:

    1. 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.
    2. 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!

    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

    It is notoriously difficult to develop a rational classification of water waves, basically because of the mathematical complexity mentioned above. One way to classify water waves I already mentioned: does the water move with the waves (translatory waves), or does the water merely oscillate as the wave passes, to return to its original position after the wave has passed (oscillatory waves)? But I also mentioned a more fundamental approach: waves for which the convective inertia terms can be neglected are called linear waves, and those for which the convective inertia terms are at least in part included in the analysis are called nonlinear waves.

    Yet another fundamental approach to classification is on the basis of the relative magnitudes of the three important length scales in the problem: wave height \(H\), wavelength \(L\), and water depth \(d\). Out of these three you can make three characteristic ratios: \(H/L\), \(H/d\), and \(L/d\). In deep water, \(H/d\) and \(L/d\) are both small, and the most important parameter is \(H/L\), called the wave steepness. In shallow water, on the other hand, neither \(H/d\) nor \(L/d\) is likely to be small, and the most important parameter is likely to be \(H/d\), called the relative height. In an intermediate range of water depths, the situation is more complicated.