Hydro-acoustic frequencies of the weakly compressible mild-slope equation

Abstract

This draft was prepared using the LaTeX style file belonging to the Journal We present a novel analytical solution for hydro-acoustic waves in a weakly compressible fluid over a slowly varying bottom. Application of a multiple-scale perturbation technique and matched asymptotic analysis leads to a uniform analytical solution of the depthaveraged governing equations in three dimensions. We show that the slow depth variation plays a leading-order effect on the evolution of the normal mode amplitude and direction. This dynamics is much richer than the two-dimensional limit analysed in previous studies.For tsunamigenic disturbances, we show that the hydro-acoustic wave field is made up by longshore trapped and offshore propagating components, which were not explicated in previous work. For a plane beach, we find an exact analytical solution of the model equation in terms of integrals of Bessel functions. Our model offers a physical insight into the evolution of hydro-acoustic waves of interest for the design of tsunami early warning systems.

Acoustic–gravity waves from multi-fault rupture

“…More generally, acoustic-gravity waves propagating into shallow sea depth experience frequency filtering by the water layer (Abdolali et al 2015a;Cecioni et al 2015). Low-order modes are associated with smaller critical depths and are therefore able to propagate further onshore (Abdolali et al 2014;Renzi 2017). These results confirm that changing sea depth cannot be ignored when making these calculations, since it affects the timing and scale of the signals measurable at any particular point.…”

“…In figure 9, we can see the superposition of pressure signals emanating from multiple slender faults with differing orientations, resulting in areas of high pressure, and areas where the signal is weaker. The pressure contours of column three in figure 8 and the third column of figure 9 highlight the missing processes of refraction, diffraction and interference induced by the variable sea depth and areas of localised elevation (red coloured areas figure 6), with refraction dominating all modes in deep water (Renzi 2017). In figure 6, there is a transect with a seamount located approximately one third of the way along AB.…”

“…Low-order modes are associated with smaller critical depths and are therefore able to propagate further onshore (Abdolali et al. 2014; Renzi 2017). These results confirm that changing sea depth cannot be ignored when making these calculations, since it affects the timing and scale of the signals measurable at any particular point.…”

Effects of the sound speed vertical profile on the evolution of hydroacoustic waves

“…Now consider the physical scales of the problem. The typical maximum amplitude of tsunamigenic HA waves is A ∼ 10 −2 m (Kadri & Stiassnie 2012), whereas the wavelength is λ ∼ 10 4 m and the angular frequency is ω ∼ 1 rad s −1 (see Stiassnie 2010;Cecioni et al 2015;Renzi 2017). Furthermore, the sound speed parameter in (2.6) is ∼ 10 −1 (see Jensen et al 2011), thus O(G) = O( 2 ), while the nonlinearity parameter δ ∼ 10 −6 .…”

“…Further developments included the effects of bottom elasticity (Eyov et al 2013) and two-dimensional depth variations (Kadri 2015). Analytical and numerical models were also devised to understand the motion of HA waves in more complex, three-dimensional scenarios (see Sammarco et al 2013;Abdolali et al 2015a;Abdolali, Kirby & Bellotti 2015b;Cecioni et al 2015;Renzi 2017;Mei & Kadri 2018).…”