The Dispersion of a Flood Wave

The inversion of this transform gives a somewhat cumbersome integral, of which the physical meaning is far from obvious, and Lighthill Whitham naturally prefer to elucidate this form the asymptotic behaviour of the transform, by the method of steepest descents. The method presented here also uses the transform without the need for inversion and obtains a description of the wave in terms of its moments. 

If the disturbance /(f) is hump-like, falling to zero either at some finite / or sufficiently fast as t — °°, it will possess a set of moments 

If the disturbance is observed at a station downstream the corresponding moments 

Under conditions which will be discussed in the next paragraph these moments are given by 

Equation (11) simply means that the total disturbance ffi v(x, t) dr is the same at any point x as it is at x = 0. We could without loss of generality put a0 = fio = 1. Equation (12) shows that the mean time of the disturbance at x differs from that at x = 0 by a quantity strictly proportional to x moreover, if the wave velocity is judged by the progress of this mean time the wave moves with a constant velocity 3vq/2, the kinematic wave velocity. Equation (13) shows that the increase of the variance is also proportional to x and the constant of proportionality is a measure of the rate of dispersion. 

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