Gravity Waves

Abstract

Waves may occur at fluid interfaces when gravity or surface tension provides a restoring force that pushes a deformed surface back toward its equilibrium position. The general formulation of the surface wave problem is nonlinear, even when the flow is inviscid. An appropriate linearization for small surface slope leads to traveling-wave solutions that are dispersive in deep water and nondispersive in shallow water. In particular, the phase and group speeds of deep-water gravity waves are different and both depend on wavelength with longer waves traveling faster. In shallow water, gravity-wave phase and group speeds are equal and depend on the depth of the water in which they travel. The complexity of the situation increases when the waves are nonlinear, when they occur between fluid layers of differing density, and when they occur on density gradients within a stratified fluid. In the latter case, the phase and group velocities are not even in the same direction.

Keywords

Capillary waves; Dispersion; Energy flux; Gravity waves; Group velocity; Interface waves; Internal waves; Linear waves; Nonlinear waves; Phase velocity; Standing waves; Surface waves; Water waves

Chapter Objectives

• To develop the equations and boundary conditions for surface, interface, and internal waves

• To derive relationships for linear capillary-gravity wave propagation speed(s), pressure fluctuations, dispersion, particle motion, and energy flux for surface waves on a liquid layer of arbitrary but constant depth

• To describe and highlight wave refraction and nonlinear gravity wave results in shallow and deep water

• To determine linear density-interface wave characteristics with and without an additional free surface

• To present the characteristics of gravity waves on a density gradient with constant buoyancy frequency

8.1. Introduction

There are three types of waves commonly considered in the study of fluid mechanics: interface waves, internal waves, and compression and expansion waves. In all cases, the waves are traveling fluid oscillations, impulses, or pressure changes sustained by the interplay of fluid inertia and a restoring force or a pressure imbalance. For interface waves the restoring forces are gravity and surface tension. For internal waves, the restoring force is gravity. For expansion and compression waves, the restoring force comes directly from the compressibility of the fluid. The basic elements of linear and nonlinear compression and expansion waves are presented in Chapter 15, which covers compressible fluid dynamics. This chapter covers interface and internal waves with an emphasis on gravity as the restoring force. The approach and results from the prior chapter will be exploited here since the wave physics and wave phenomena presented in this chapter primarily involve irrotational flow.

Perhaps the simplest and most readily observed fluid waves are those that form and travel on the density discontinuity provided by an air-water interface. Such surface capillary-gravity waves, sometimes simply called water waves, involve fluid particle motions parallel and perpendicular to the direction of wave propagation. Thus, the waves are neither longitudinal nor transverse. When generalized to internal waves that propagate in a fluid medium having a continuous density gradient, the situation may be even more complicated. This chapter presents some basic features of wave motion and illustrates them with water waves because water wave phenomena are readily observed and this aids comprehension. Throughout this chapter, the wave frequency will be assumed much higher than the Coriolis frequency so the wave motion is unaffected by the earth’s rotation. Waves affected by planetary rotation are considered in Chapter 13. And, unless specified otherwise, wave amplitudes are assumed small enough so that the governing equations and boundary conditions are linear.

For such linear waves, Fourier superposition of sinusoidal waves allows arbitrary wave-forms to be constructed and sinusoidal waveforms arise naturally from the linearized equations for water waves (see Exercise 8.3). Consequently, a simple sinusoidal traveling wave of the form:

η(x,t)=acos[2πλ(x−ct)]

(8.1)

is a foundational element for what follows. In Cartesian coordinates with x horizontal and z vertical, z = η(x,t) specifies the waveform or surface shape where a is the wave amplitude, λ is the wavelength, c is the phase speed, and 2π(x − ct)/λ is the phase. In addition, the spatial frequency k ≡ 2π/λ, with units of rad./m, is known as the wave number. If (8.1) describes the vertical deflection of an air-water interface, then the height of wave crests is +a and the depth of the wave troughs is –a compared to the undisturbed water-surface location z = 0. At any instant in time, the distance between successive wave crests is λ. At any fixed x-location, the time between passage of successive wave crests is the period, T = 2π/kc = λ/c. Thus, the wave’s cyclic frequency is ν = 1/T with units of Hz, and its radian frequency is ω = 2πν with units of rad./s. In terms of k and ω, (8.1) can be written:

η(x,t)=acos[kx−ωt].

(8.2)

The wave propagation speed is readily deduced from (8.1) or (8.2) by determining the travel speed of wave crests. This means setting the phase in (8.1) or (8.2) so that the cosine function is unity and η = +a. This occurs when the phase is 2nπ where n is an integer:

2πλ(xcrest−ct)=2nπ=kxcrest−ωt,

(8.3)

and x_crest is the time-dependent location where η = +a. Solving for the crest location produces:

xcrest=(ω/k)t+2nπ/k.

Therefore, in a time increment Δt, a wave crest moves a distance Δx_crest = (ω/k)Δt, so:

c=ω/k=λv

(8.4)

is known as the phase speed because it specifies the travel speed of constant-phase wave features, like wave crests or troughs.

Although instructive, (8.1) and (8.2) are limited to propagation in the positive-x direction only. In general, waves may propagate in any direction. A useful three-dimensional generalization of (8.2) is:

η=acos(kx+ly+mz−ωt)=acos(K·x−ωt),

(8.5)

where K = (k, l, m) is a vector, called the wave number vector, whose magnitude K is given by:

K2=k2+l2+m2.

(8.6)

The wavelength derived from (8.5) is:

λ=2π/K,

(8.7)

which is illustrated in Figure 8.1 in two dimensions. The magnitude of the phase velocity is c = ω/K, and the direction of propagation is parallel to K, so the phase velocity vector is:

c=(ω/K)eK,

(8.8)

where e_K = K/K.

From Figure 8.1, it is also clear that c_x = ω/k, c_y = ω/l, and c_z = ω/m are each larger than the resultant c = ω/K, because k, l, and m are individually smaller than K when all three are non-zero, as required by (8.6). Thus, c_x, c_y, and c_z are not vector components of the phase velocity in the usual sense, but they do reflect the fact that constant-phase surfaces appear to travel faster along directions not coinciding with the direction of propagation, the x and y directions in Figure 8.1 for example. Any of the three axis-specific phase speeds, is sometimes called the trace velocity along its associated axis.

If sinusoidal waves exist in a fluid moving with uniform speed U, then the observed phase speed is c₀ = c + U. Forming a dot product of c₀ with K, and using (8.8), produces:

ω0=ω+U·K,

(8.9)

where ω₀ is the observed frequency at a fixed point, and ω is the intrinsic frequency measured by an observer moving with the flow. It is apparent that the frequency of a wave is Doppler shifted by an amount U·K

in non-zero flow. Equation (8.9) may be understood by considering a situation in which the intrinsic frequency ω is zero, but the flow pattern has a periodicity in the x direction of wavelength 2π/k. If this sinusoidal pattern is translated in the x direction at speed U, then the observed frequency at a fixed point is ω₀ = Uk. The effects of uniform flow on frequency will not be considered further, and all frequencies in the remainder of this chapter should be interpreted as intrinsic frequencies.

Example 8.1

If a surface wave described by (8.2) has small surface slope, |∂η/∂x| ≪ 1, what does this imply about the vertical velocity of the surface, ∂η/∂t?

Solution

Differentiate (8.2) with respect to x and t to find:

surfaceslope=∂η∂x=−kasin[kx−ωt]andsurfacevelocity=∂η∂t=+ωasin[kx−ωt].

Thus, the surface slope has amplitude ka. These relationships and k = ω/c imply:

1c∂η∂t=−∂η∂x.

So when ka = 2πa/λ (the amplitude of ∂η/∂x) is much less than unity, the vertical velocity of the surface (∂η/∂t) is much less than the wave speed, c.