Pulse propagation in the quiescent environment during direct numerical simulation of Rayleigh-Taylor instability: Solution by Bromwich contour integral method

In: "Three-dimensional direct numerical simulation (DNS) of Rayleigh-Taylor instability (RTI) trigerred by acoustic excitation -- Sengupta et al. { 34, 054108 (2022) "} the receptivity of RTI to pressure pulses have been established. It has also been shown that at the onset of RTI these pulses are one-dimensional and the dissipation of the pressure pulses are governed by a dissipative wave equation. The propagation of these infrasonic to ultrasonic pressure pulses have been studied theoretically and numerically by a high fidelity numerical procedure in the physical plane. The numerical results are consistent with the theoretical analysis and the DNS of RTI noted above. The properties of pulse propagation in a quiescent dissipative ambience have been theoretically obtained from the linearized compressible Navier-Stokes equation, without Stokes' hypothesis. This analysis is extended here for a special class of excitation, with combination of wavenumbers and circular frequencies for which the phase shift results in an imposed time period is integral multiple of, and the signal amplification is by a real factor. Here, the governing partial differential equation (PDE) for the free-field propagation of pulses is solved by the Bromwich contour integral method in the spectral plane. This method, for an input Gaussian pulse excited at a fixed frequency, is the so-called signal problem. Responses for the specific phase shifts integral multiple of can reinforce each other due to the phase coherence. It is shown that these combinations occur at a fixed wavenumber, with higher frequencies attenuated more in such a sequence.