Oceanography

Non-linear waves

One can derive the properties of an ocean surface wave assuming waves were infinitely small Ka=O (0). If the waves are small ka << 1 but not infinitely small, the wave properties can be expanded in a power series of Ka (Stokes, 1847). He calculated the properties of a wave of finite amplitude and found:

The phase of the components for the Fourier series expansion of ? in above equation

ς = a cos (k x – ω t) +  ^½

 in are such that non-linear waves have sharpened crests and flattened troughs. The maximum amplitude of the Stokes wave is a _max= 0.07L (ka=044). Such steep waves in deep water are called stokes waves(See also lamb, 1945, §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 6^th order the interaction of three or more waves on the sea surface. He, Phillips (1960), and longuet-Higgins and Phillips (1962) showed that n 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:

w₁ ± w₂ ± w₃ ±….. w_n =0   --  2

k₁ ± k₂ ± k₃ ±…. K_n=0        -- 3

w_i ² = g k_i

Where we allow waves to travel in any direction, as above equation 2 & 3 and k_i is the vector wave number giving wave-length and direction are general requirements for any interacting waves. The fewest number of waves that meet the conditions of 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 and the wave components are not free waves; the higher harmonics are bound to the primary wave.