On long-wave sound scattering by a Rankine vortex: Non-resonant and resonant cases

Sound scattering by a vortex: case of exponentially decaying velocity

“…Owing to this, Aharonov & Bohm (1959), Berry et al (1980), Sakov (1993, Coste et al (1999) and Kopiev & Belyaev (2010) included it in the incident field so that…”

“…Pitaevskii (1959), Fetter (1964), Ferziger (1974), O'Shea (1975), Candel (1979), Fabrikant (1983), Kopiev & Leontiev (1987) and Colonius et al (1994) obtained a cylindrical wave (1.4) with scattering amplitude having a singularity in the direction . Sakov (1993), Ford & Smith (1999), Howe (1999) and Kopiev & Belyaev (2010) obtained a field that was regular but non-decaying at infinity, so that We will refer to the expressions corresponding to two previous equations as follows: the former as expression (i), and the latter as expression (ii); exact equations will be written out in § 5. Much earlier Aharonov & Bohm (1959) and Berry et al (1980) obtained an expression very similar to (ii), yet for a special case beyond the Born approximation.…”

“…having a singularity in the direction θ = 0. Sakov (1993), Ford & Smith (1999), Howe (1999) and Kopiev & Belyaev (2010) obtained a field that was regular but non-decaying at infinity, so that max θ |U scat | → const as kr → ∞.…”

“…For both cases they obtained regular solutions having the form (1.4); however, in the limit L → ∞ corresponding to case PWPV both were similar to the incorrect expression (i) rather than to (ii). Reinschke et al (1997) assumed the fluid to rest in the region r > L. Kopiev & Belyaev (2010) proposed replacing the plane wave (1.3) with a cylindrical wave emitted by a point source at a finite distance from the vortex centre. They obtained expression (ii) in a limited region beyond the Born approximation; however, it is not obvious whether their solution decomposed into a Fourier series by angle θ was a purely outgoing wave (1.4) in the far field.…”

“…The ill-posed nature of the problem for case PWPV means that a uniform plane wave (1.3) scattering by an axisymmetric vortex (1.6) with non-zero total circulation (1.7) is ambiguous. The observed scattered field in a physical or numerical experiment will depend on the actual incident field and flow and may be different for different implementations (see discussion by Kopiev & Belyaev 2010). Some researchers who recognised that this problem is ill-posed proposed changing the physical statement of the problem, precisely, to confine the region of interaction, in order to make use of the radiation condition.…”