Ultrasonic scattering by a fluid cylinder of elliptic cross section, including viscous effects

Abstract

This paper analyzes acoustic scattering by a viscous compressible fluid cylinder of elliptic cross section submerged in an unbounded viscous nonheat-conducting compressible fluid medium. The classical method of eigenfunction expansion along with the appropriate wave field expansions and the pertinent boundary conditions are used to develop a solution in the form of infinite series involving Mathieu and modified Mathieu functions of complex arguments. The complications arising due to the nonorthogonality of angular Mathieu functions corresponding to distinct wave numbers in addition to the problems associated with the appearance of additional angular-dependent terms in the boundary conditions are all avoided in an elegant manner by expansion of the angular Mathieu functions in terms of transcendental functions and subsequent integration, leading to a linear set of independent equations in terms of the unknown scattering coefficients. A multiprecision code was developed for computing the Mathieu functions of complex argument in terms of complex Fourier coefficients that are themselves calculated by numerically solving appropriate sets of eigen-systems. The numerical results point to the imperative influence of fluid viscosity in notable reduction of pressure amplitudes at intermediate and high frequencies. They also reveal the central role of the cross sectional ellipticity in conjunction with the angle of incidence in altering the pressure directivities. Limiting cases are considered, and fair agreements with well-known solutions are obtained.

Optical radiation force (per–length) on an electrically conducting elliptical cylinder having a smooth or ribbed surface

“…Eqs. (18) and (20) correspond to the simplified expressions for the radiation force vector in each polarization scheme. The differential surface vector is expressed as…”

“…The aim of this work is therefore directed toward developing a semi-analytical formalism based on the partial wave series expansion method in cylindrical coordinates [10] without any approximations. Another method may consider a wave-basis using infinite series expansions in elliptical Mathieu functions [15][16][17][18][19][20][21], however, the generation and computation of the elliptical functions may not be a straightforward task in numerical implementation.…”

Free vibration analysis of infinitely long thick-walled hollow elliptical cylinder

“…5 into the Navier's equation of motion, Eq. 4, two sets of fully uncoupled wave equations can be obtained as [54,64] where Ω could be any of Ψ or Φ . Furthermore, k 2 c and k 2 s are given as [65] in which c p = ( + 2 )∕ and c s = ∕ are propagation velocities of distortional and dilatational waves in the solid medium.…”

“…Hasheminejad and Sanaei [54] used the theory of elasticity to study the ultrasonic scattering from a cylinder with the elliptical cross-section. Hayek and Boisvert [47] used higher-order shell theories that included the effect of rotary inertia and shear deformation along with benefiting from the Ritz approach to study the vibration of elliptical shells.…”

Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with arbitrary incidence

“…2 in Ref. [59], where the large resonances appear to be unphysical) and particular care should be given to check the adequate convergence of the series in the least-square sense and ensure that the wave-functions form a complete set on the surface of the elliptical cylinder. Furthermore, it has been mentioned in 58 "that the acoustic radiation torque acting on the elliptic cylinder in a standing wave field vanishes due to the special symmetry of the wave fields."…”