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Harmonics from a magic carpet

Published online by Cambridge University Press:  28 January 2021

Thomas E. Dobra Open the ORCID record for Thomas E. Dobra [Opens in a new window] ,
Andrew G. W. Lawrie Open the ORCID record for Andrew G. W. Lawrie [Opens in a new window]  and
Stuart B. Dalziel Open the ORCID record for Stuart B. Dalziel [Opens in a new window]
Thomas E. Dobra
Affiliation:
Hele-Shaw Laboratory, University of Bristol, University Walk, BristolBS8 1TR, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, UK
Andrew G. W. Lawrie*
Affiliation:
Hele-Shaw Laboratory, University of Bristol, University Walk, BristolBS8 1TR, UK
Stuart B. Dalziel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, UK
*
†Email address for correspondence: andrew.lawrie@bristol.ac.uk
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Abstract

We present a novel theoretical framework for the emission and absorption of two-dimensional internal waves in a density stratified medium. Our approach uses a weakly nonlinear perturbation expansion of a streamfunction field that exposes the harmonic structure emitted from a flexible boundary of infinite extent. We report the discovery of a special symmetry in polychromatic waves that share a common horizontal component of phase velocity. Under these conditions, there can be no wave–wave interactions in the domain interior, and therefore all harmonic generation is from the boundary. By activating polychromatic waves on this same flexible surface, we then consider the equivalent inverse problems of emission of a prescribed harmonic signature and absorption of wave energy from a given flow field. Specialising to monochromatic waves, to calculate the amplitudes and phases of the harmonics generated by a monochromatic boundary displacement and to find the explicit form of the absorbing boundary condition for a monochromatic internal wave, we present algorithms that refine lengthy algebraic processes down to a set of executable instructions valid for arbitrary order in the small parameter of the expansion. Finally, we compare our theoretical predictions up to third order with a sophisticated, digitally controlled experimental realisation that we call a ‘magic carpet’, and we find that harmonic analysis of the flow field convincingly supports our theory.

JFM classification

Geophysical and Geological Flows: Internal waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 911 , 25 March 2021 , A29
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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