Energy flow and radiation of linearized acoustic–gravity waves and propagation of boundary waves in a gravitationally stratified isothermal compressible inviscid semi-infinite fluid from a time-varying bottom boundary are investigated in the frequency–wavenumber domain. Impedance Z, the ratio of the bottom vertical displacement to the fluid pressure above it, is a function of the frequency and horizontal wavenumber (ω, k) of the bottom boundary undulation. The amplitude and phase of Z at the bottom boundary divide the (ω, k) coordinates into wave-type regimes. In contrast to the pure acoustic or gravity wave case, the phase of Z is continuous but changes quickly across the regime boundaries between the propagating waves and trapped waves at the bottom, except for the Lamb wave branch along which the amplitude is infinite and across which the phase jumps by π. The phase of Z determines the efficiency of the work against the fluid by the deforming bottom boundary, showing reduced upward wave-energy flow from the bottom near the regime boundaries in which the phase of Z approaches ±π/2. For precise modelling of pressure waves and the energy flow of acoustic and gravity waves in the fluid originating from a time-dependent bottom-surface deformation with an apparent phase velocity comparable to the speed of sound in the fluid, it is necessary to include the dependency on (ω, k) of impedance Z.
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