ON THE DISPERSION-RELATION FOR TRAPPED INTERNAL WAVES

An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/c(p) = 1/c(p0) + k/omega(max) + epsilon(k) where c(p) is the wave phase speed for a particular mode, c(p0) is the phase speed at k = 0, omega(max) is the maximum possible wave angular frequency and omega(max) less-than-or-equal-to N(max) where N(max) is the maximum buoyancy frequency. Also, epsilon(0) = 0, epsilon(k) = o(k) for k large, and is bounded for finite k. In particular, when epsilon(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/c(p) almost-equal-to (integral(infinity)0 N2 (y)y dy)-1/2 + k/omega(max). The eigenvalue problem is analysed for a class of buoyancy frequency squared functions N2(x) which is taken to be a class of real-valued functions of a real variable x where 0 less-than-or-equal-to x less-than-or-equal-to infinity such that N2(x) = O(e(-betax)) as x --> infinity and 1/beta is an arbitrary length scale. It is demonstrated that N2(x) can be represented by a power series in e(betax). The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that epsilon(k) can be neglected and that, in addition, (omega(max) = N(max). More generally, it is demonstrated that when k --> infinity it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that if N2(x) is O(e-(betax)) as x --> infinity and has a maximum value N(max) then a sufficient condition for 1/c(p) approximately k/N(max) to hold for large k for the lowest mode is that N2(t)/t is convex for 0 less-than-or-equal-to t less-than-or-equal-to 1 where t = e(-betax).