×

Approximate solution of nonlinear triad interactions of acoustic-gravity waves in cylindrical coordinates. (English) Zbl 1452.76210

Summary: The three-dimensional radial propagation of wave disturbances over a slightly compressible fluid of constant depth is discussed. We focus on resonant triads comprising two gravity modes and one acoustic mode. The derivation of the evolution equations in a non-integral form is made possible by approximating the radial solution by cosine functions in two regions, inner and outer, that are matched at a location where all relevant derivatives are in agreement with the exact Bessel solution. When the interaction takes place in the inner region of all modes, the amplitude evolution equations are found to be similar to the two-dimensional case. However, focusing of one gravity mode and de-focusing of the other is observed when the interaction involves an inner region of the acoustic mode, and outer regions of the gravity modes.

MSC:

76N30 Waves in compressible fluids
76Q05 Hydro- and aero-acoustics
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Craik, A. D.D., Wave interactions and fluid flows, p. 322 (1984), Cambridge University Press
[2] R6 · Zbl 1294.76222
[3] Kadri, U., Wave motion in a heavy compressible fluid: Revisited, Eur J Mech B Fluids, 49, 50-57 (2015) · Zbl 1408.76468
[4] R1 · Zbl 1381.76042
[5] Kadri U. Faraday waves by resonant triad interactions of surface-compression waves. 2017. Axive.org/pdf/1701.00667.pdf.
[6] Kadri, U., Multiple-location matched approximation for bessel function j_0 and its derivatives, Commun Nonlinear Sci, 72, 59-63 (2019) · Zbl 1464.33003
[7] Kadri, U., Time-reversal analogy by nonlinear acoustic-gravity wave triad resonance, Fluids, 4, 2, 91 (2019)
[8] Longuet-Higgins, M. S., A theory of the origin of microseisms, Phil Trans R Soc Lond, 243, 1-35 (1950) · Zbl 0041.14003
[9] Miles, J. W., Internally resonant surface waves in a circular cylinder, J Fluid Mech, 149, 1-14 (1984) · Zbl 0581.76027
[10] Phillips, O. M., Wave interactions – the evolution of an IDE, J Fluid Mech, 106, 215-227 (1981) · Zbl 0467.76020
[11] Pidduck, F. B., On the propagation of a disturbance in a fluid under gravity, Proc R Soc A, 83, 347-356 (1910) · JFM 41.0842.02
[12] Pidduck, F. B., The wave-problem of cauchy and poisson for finite depth and slightly compressible fluid, Proc R Soc A, 86, 396-405 (1912) · JFM 43.0850.01
[13] Whipple, F. J.W.; Lee, A. W., Notes on the theory of microseisms, Mon Not R Astron Soc Geophys Suppl, 3, 287-297 (1935) · Zbl 0013.18902
[14] Yang, X.; Dias, F.; Liao, S., On the steady-state resonant acoustic-gravity waves, J Fluid Mech, 849, 111-135 (2018) · Zbl 1415.76087
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.