By Hilary Ockendon

Applied arithmetic is the paintings of making mathematical versions of saw phenomena in order that either qualitative and quantitative effects might be anticipated by way of analytical and numerical tools. Theoretical Mechanics is anxious with the learn of these phenomena which might be ob­ served in way of life within the actual global round us. it is usually characterized via the macroscopic process which permits the concept that of a component or particle of fabric, small in comparison to the scale of the phenomena being modelled, but huge in comparison to the molecular dimension of the cloth. Then atomic and molecular phenomena seem purely as amounts averaged over many molecules. it really is hence average that the mathemati­ cal types derived are when it comes to services that are non-stop and good behaved, and that the analytical and numerical tools required for his or her improvement are strongly depending on the speculation of partial and traditional differential equations. a lot natural learn in arithmetic has been stimu­ lated by means of the necessity to enhance types of actual occasions, and experimental observations have frequently ended in very important conjectures and theorems in research. it truly is hence very important to provide a cautious account of either the actual or experimental observations and the mathematical research used. The authors think that Fluid Mechanics deals a wealthy box for il­ lustrating the artwork of mathematical modelling, the ability of mathematical research and the stimulus of functions to simply saw phenomena.

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At time t = 0, f~ A(k)eikxdk -~ n0 (x) so that n0 is the Fourier transform of A(k) A(k) 1 [ -~ = 2W e -ikx n0 (x)dx. 14) The integral will converge if noCx) tends to zero for large enough lxl. 9). They give the profile shape in the form of a double integral which cannot in general be explicitly integrated. We can however·evaluate the integral asymptotically for large time, and to do this we use the method of stationary phase. First we observe that, since each wave travels with 30 II • FREE BOUNDARY PROBLEMS finite speed, if t is large then so is x.

Thus for large values of x and t the wave +This method is a special case of the method of steepest descents for complex integrals. 3. 15) has a physical interpretation and is called the group velocity. If an observer moves with the group velocity relative to the waves he will see a profile which is dominated after a long time by waves whose wave number is k0 • Thus, after a long time, waves of any given wave number k will travel with their wave speed c(k), called their phase speed, only if no other harmonics are present in the initial disturbance to interact with them and cause them to disperse.

The convergence of this integral is assured if n0 (r) tends to zero fast enough as r + •. 24) even 36 II • FREE BOUNDARY PROBLEMS asymptotically for large t and we shall not do so here. The WKB expansion method may also be used in a three dimensional situation to study waves a long time after their initiation. As in the last section we introduce a large parameter A and put x = AX, s = As, - - iAecx s t) t = At and ~ = A(x,y,s,t)e ' ' • By writing w = -et' i = e- and X m = es, we obtain the dispersion relation w = w(t,m).

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