Image Conditions for Elliptical-Coordinate Separation-of-Variables Acoustic Multiple Scattering Models with Perfectly Reflecting Flat Boundaries: Application to in Situ Tunable Noise Barriers

Summary Two-dimensional time-harmonic multiple scattering problems are addressed for a finite number of elliptical objects placed in wedge-shaped acoustic domains including half-plane and right-angled corners. The method of separation of variables in conjunction with the addition theorems is employed in the elliptical coordinates. The wavefunctions are represented in terms of radial and angular Mathieu functions. The method of images is applied to consider the effect of the infinitely long flat boundaries which are perfectly reflecting: either rigid or pressure release. The wedge angle is $\pi/n$ rad with integer $n$; image ellipses must be appropriately rotated to realise the mirror reflection. Then, the ‘image conditions’ are developed to reduce the number of unknowns by expressing the unknown expansion coefficients of image scattered fields in terms of real counterparts. Use of image conditions, therefore, leads to the $4n^2$-fold reduction in the size of a matrix for direct solvers and $2n$-times faster computation in building the system of linear equations than the approach without using them. Multiple scattering models using image conditions are formulated for rigid, pressure release and fluid ellipses under either plane- or cylindrical-wave incidence, and are numerically validated by the boundary element method. Furthermore, potential applications are presented: arrays of elliptically shaped scatterers make in situ tunable noise barriers by rotating scatterers. Finally, polar-coordinate image conditions (for circular objects) are also discussed when coordinates local to circles are also rotated. In Appendix, analytic formulae are provided, which permits the elliptical-coordinate addition theorems used in this article to be calculated by summation instead of numerical integration.

Certain Results on (p,q)-Hermite Based Apostol Type Frobenius-Euler Polynomials

In the present paper, the (p,q)-Hermite based Apostol type Frobenius-Euler polynomials and numbers are firstly considered and then diverse basic identities and properties for the mentioned polynomials and numbers, including addition theorems, difference equations, integral representations, derivative properties, recurrence relations. Moreover, we provide summation formulas and relations associated with the Stirling numbers of the second kind.

Image conditions and addition theorems for prolate and oblate spheroidal-coordinate separation-of-variables acoustic multiple scattering models with perfectly-reflecting flat surfaces

Three-dimensional time-harmonic acoustic multiple scattering problems are considered for a finite number of prolate and oblate spheroidal objects adjacent to flat surfaces. Wave propagation by spheroids is modelled by the method of separation of variables equipped with the addition theorems in the spheroidal coordinates. The effect of flat surfaces is accounted for by using the method of images; hence, the flat surfaces are of (semi-)infinite extent and perfectly reflecting: either rigid or pressure release. Wedge-shaped acoustic domains are constructed including half-space and right-angled corners with the wedge angle of $\pi /n$ rad with positive integer $n$. First, Euler angles are implemented to rotate image spheroids to realize the mirror reflection. Then, the ‘image conditions’ are developed to reduce the number of unknowns by expressing the unknown expansion coefficients of image-scattered fields in terms of real counterparts. Use of image conditions to 2D wedges, therefore, leads to the $4n^2$-fold reduction in the size of a matrix for direct solvers and $2n$-times faster computation than the approach without using them; for 3D wedges, the savings are $16n^2$-fold and $4n$-times, respectively. Multiple scattering models (MSMs) are also formulated for fluid, rigid and pressure-release spheroids under either plane- or spherical-wave incidence; novel addition theorems are also derived for spheroidal wavefunctions by using two rotations of spherical wavefunctions and a $z$-axis translation in-between, which is shown numerically more efficient than other addition theorems based on an arbitrary-direction translation and a single rotation. Finally, MSMs using image conditions are numerically validated by the boundary element method for a configuration populated with both prolate and oblate spheroids.