Absorbent Strip in a Hard Baffle Wall, with Mathieu Functions

→ See also Mechel  (1997), for notations and relations of Mathieu functions.

The sound field around a locally reacting strip with normalised admittance G in a hard baffle wall can be formulated as a boundary value problem with exact solutions in elliptic-hyperbolic cylinder co-ordinates ( $\rho,\vartheta)$. The co-ordinate curves are confocal ellipses and orthogonal confocal hyperbolic branches. The radial and azimuthal eigenfunctions in these co-ordinates are Mathieu functions.

Transformation between Cartesian $x=x_{1}=c\cdot\cosh\,\rho\cdot\cos\,\vartheta$?,

and elliptic-hyperbolic co-ordinates: $y=x_{2}=c\cdot\sinh\,\rho\cdot\sin\,\vartheta$?, (1)

The common foci are at $x=\pm c$. $z=x_{3}=z$?.

The boundary surface of the absorbent strip is at $\rho=0$, the focus distance is $c=a/2$, and the boundaries of the baffle wall are at $\vartheta=0$ and $\vartheta=\pi$. The Helmholtz differential equation (wave equation) $(\Delta+k_{0}^{2})\, u=0$in elliptic-hyperbolic...