Nonlinear Waves

Introduction

We derived the properties of an ocean surface wave assuming waves were infinitely small [math]\displaystyle{ ka = O }[/math] (0). If the waves are small [math]\displaystyle{ ka \lt \lt 1 }[/math] but not infinitely small, the wave properties can be expanded in a power series of [math]\displaystyle{ ka }[/math] (Stokes, 1847). He calculated the properties of a wave of finite amplitude and found:

[math]\displaystyle{ \zeta = a\cos(kx-\omega t) + \frac{1}{2}k a^2\cos2(kx - \omega t)+\frac{3}{8}k^2 a^3\cos3(kx - \omega t)+... \,\! }[/math]

The phases of the components for the Fourier series expansion of [math]\displaystyle{ \zeta }[/math] in (16.14) are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is [math]\displaystyle{ a_{max} = 0.07L ( ka = 0.44 ) }[/math]. Such steep waves in deep water are called Stokes waves (See also Lamb, 1945, [math]\displaystyle{ \S \; }[/math]250).

Knowledge of non-linear waves came slowly until Hasselmann (1961, 1963a, 1963b, 1966), using the tools of high-energy particle physics, worked out to 6th order the interactions of three or more waves on the sea surface. He, Phillips (1960), and Longuet-Higgins and Phillips (1962) showed that [math]\displaystyle{ n }[/math] free waves on the sea surface can interact to produce another free wave only if the frequencies and wave numbers of the interacting waves sum to zero:

[math]\displaystyle{ \omega_1 \pm \omega_2 \pm \omega_3 \pm ... \omega_n = 0 \,\! }[/math]

[math]\displaystyle{ k_1 \pm k_2 \pm k_3 \pm ... k_n = 0 \,\! }[/math]

[math]\displaystyle{ \omega_i^2 = g k_i \,\! }[/math]

where we allow waves to travel in any direction, and [math]\displaystyle{ k_i }[/math] is the vector wave number giving wave-length and direction. (16.15a, b) are general requirements for any interacting waves. The fewest number of waves that meet the conditions of (16.15) are three waves which interact to produce a fourth. The interaction is weak; waves must interact for hundreds of wave-lengths and periods to produce a fourth wave with amplitude comparable to the interacting waves. The Stokes wave does not meet the criteria of (16.15) and the wave components are not free waves; the higher harmonics are bound to the primary wave.

Wave Momentum

The concept of wave momentum has caused considerable confusion (McIntyre, 1981). In general, waves do not have momentum, a mass flux, but they do have a momentum flux. This is true for waves on the sea surface. Ursell (1950) showed that ocean swell on a rotating Earth has no mass transport. His proof seems to contradict the usual textbook discussions of steep, non-linear waves such as Stokes waves. Water particles in a Stokes wave move along paths that are nearly circular, but the paths fail to close, and the particles move slowly in the direction of wave propagation. This is a mass transport, and the phenomena is called Stokes drift. But the forward transport near the surface is balanced by an equal transport in the opposite direction at depth, and there is no net mass flux.

Solitary Waves

Solitary waves are another class of non-linear waves, and they have very interesting properties. They propagate without change of shape, and two solitons can cross without interaction. The first soliton was discovered by John Scott Russell (1808-1882), who followed a solitary wave generated by a boat in Edinburgh's Union Canal in 1834.

Scott witnessed such a wave while watching a boat being drawn along the Union Canal by a pair of horses. When the boat stopped, he noticed that water around the vessel surged ahead in the form of a single wave, whose height and speed remained virtually unchanged. Russell pursued the wave on horseback for more than a mile before returning home to reconstruct the event in an experimental tank in his garden. - Nature 376, 3 August 1995: 373.

The properties of a solitary waves result from an exact balance between dispersion which tends to spread the solitary wave into a train of waves, and non-linear effects which tend to shorten and steepen the wave. The type of solitary wave in shallow water seen by Russell, has the form:

[math]\displaystyle{ \zeta = a\;sech^2\left[\left(\frac{3a}{4d^3}\right)^{1/2}(x-ct)\right] }[/math]

which propagates at a speed:

[math]\displaystyle{ c = c_0\left(1+\frac{a}{2d}\right) }[/math]

You might think that all shallow-water waves are solitons because they are non-dispersive, and hence they ought to propagate without change in shape. Unfortunately, this is not true if the waves have finite amplitude. The velocity of the wave depends on depth. If the wave consists of a single hump, then the water at the crest travels faster than water in the trough, and the wave steepens as it moves forward. Eventually, the wave becomes very steep and breaks. At this point it is called a bore. In some river mouths, the incoming tide is so high and the estuary so long and shallow that the tidal wave entering the estuary eventually steepens and breaks producing a bore that runs up the river. This happens in the Amazon in South America, the Severn in Europe, and the Tsientang in China (Pugh, 1987: 249).