Author:
FORD RUPERT,LLEWELLYN SMITH STEFAN G.
Abstract
We investigate the scattering of a plane acoustic wave by an axisymmetric vortex in
two dimensions. We consider vortices with localized vorticity, arbitrary circulation and
small Mach number. The wavelength of the acoustic waves is assumed to be much
longer than the scale of the vortex. This enables us to define two asymptotic regions:
an inner, vortical region, and an outer, wave region. The solution is then developed
in the two regions using matched asymptotic expansions, with the Mach number
as the expansion parameter. The leading-order scattered wave field consists of two
components. One component arises from the interaction in the vortical region, and
takes the form of a dipolar wave. The other component arises from the interaction in
the wave region. For an incident wave with wavenumber k propagating in the positive
X-direction, a steepest descents analysis shows that, in the far-field limit, the leading-order scattered field takes the form
i(π−θ)eikX+½cosθcot(½θ)
(2π/kR)1/2ei(kR−π/4),
where θ is the usual polar angle. This expression is not valid in a parabolic region
centred on the positive X-axis, where kRθ2=O(1).
A different asymptotic solution is appropriate in this region. The two solutions match onto
each other to give a leading-order scattering amplitude that is finite and single-valued
everywhere, and that vanishes along the X-axis. The next term in the expansion in
Mach number has a non-zero far-field response along the X-axis.
Publisher
Cambridge University Press (CUP)
Subject
Mechanical Engineering,Mechanics of Materials,Condensed Matter Physics
Cited by
55 articles.
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