Deep-Water Waves

Water waves are fluid edge waves; they also are called surface waves or gravity waves . These waves arise from a much different mechanism than do edge waves in elastic bodies. This is evident from the fact that water waves travel much more slowly than do sound waves. However, they do have the same basic structure as elastic edge waves: harmonic variation in the horizontal direction and exponential decay with depth. Water waves occur on and near the free surface; they cannot occur in a closed container. These waves occur because gravity acts to pull down the water that stands higher than average and push aside water lying beneath, as illustrated in Figure 11.1. This figure is a “snapshot”; the wave might be progressive (moving either to the left or right) or standing (with fixed nodes) or some combination of these two limiting cases. The adiabatic incompressibility and kinematic viscosity of water are K ≈2×109 Pa and ν ≈ 10 -6 m 2 ·s -1 . The pressure at the bottom of a typical ocean, which is 4200 m deep, is about 4 ×10 7 Pa. This compresses water by about 2%. Since we are interested in surface waves, it is safe to treat water as completely incompressible. Viscous effects are important only near solid boundaries (and then only within thin boundary layers) or if the water is turbulent (as it is in breakers and whitecaps). So it is safe to treat water as inviscid in the study of non-breaking waves. Waves on open water are almost always progressive (i.e., moving in a certain direction). Standing waves occur when two progressive waves travel past each other in opposite directions, such as commonly occurs near a vertical breakwater. We will consider progressive waves in this chapter, with standing waves considered in § 12.3. We will find in § 11.1 that the equation governing motions within water is not the wave equation, but Laplace's equation.1 The wave character of the solution comes from the boundary conditions at the surface. These conditions are developed in § 11.2. Water Wave Equation As with sound and seismic waves, let's focus our attention on plane waves traveling in the x direction, with motion and variation in both the horizontal ( x ) and vertical ( z ) directions.