Guided mode disappearance during upslope propagation in variable depth shallow water overlying a fluid bottom

The adiabatic mode theory for upslope propagation in shallow water breaks down when the depth H(x) decreases to a critical value Hc at which the mode’s phase velocity equals sound speed c2 in the underlying fluid bottom (density ρ2). Matched asymptotic expansion techniques yield the acoustic pressure in the transition region (of lateral extent c2/ωε2) in terms of Airy functions. The expansion parameter ε whose smallness justifies the required approximation is the cube root of ( ρ2/ρ1)[(c2/c1)2−1]‖H′‖. During the early approach to the critical range (taken as x = 0), the amplitude decreases within the water layer as (−x)1/2; beyond x = 0, asymptotically as x−5/2. A cylindrically spreading (as r−1/2) wave of narrow beamwidth 0.94 ε originates from the transition region (near x = 0) and propagates at angle ε with the interface obliquely downward into the bottom fluid. The derived characteristic critical-depth transition function is discussed in detail. The results are in accord with computations reported by Jensen and Kuperman [J. Acoust. Soc. Am. 67, 1564–1566 (1980)].