Notes for ‘Acoustic directivity of rectangular pistons on prolate spheroids’

• A ‘prolate spheroid’ is obtained when you rotate an ellipse on its major axis - a symmetric egg basically.

• The coordinate system is defined by \(\xi, \eta, \phi\) , just like how in spherical coordinates there’s \(r,\theta,\phi\), except with the ranges (eqn 2):

\[ \begin{align}\begin{aligned} 1 \leq \xi < \infty\\-1 \leq \eta \leq 1\\ 0 \leq \phi \leq 2 \pi\end{aligned}\end{align} \]

Defined variables of interest

• The Directivity function is given by eqn. 15

\[f(\theta,\phi) = \sum_{m=0}^\infty \sum_{l=m}^\infty \frac{\epsilon_{m}S^{(1)}_{ml}(h,cos\theta)}{R^{(4)\prime}_{ml}(h, \xi) N_{ml}}i^{l+1} \times [\tilde I^{s}_{ml} cos m\phi + \tilde I^{c}_{ml} sin m\phi]\]

• \(\epsilon_{m}\) has a piecewise nature, where:

\[\begin{split}\epsilon_{m}=\begin{cases} 1 \quad &\text{if} \, m = 0 \\ 1 \quad &\text{if} \, m \neq 0 \\ \end{cases}\end{split}\]

• h, ‘size parameter’ \(h=kd/2\), where k is the wavenumber and d is the ‘interfocal distance of the generating ellipse’

• \(R^{(4)}_{ml}(h, \xi)\) : prolate spheroidal radial function of the 4th kind (eqn.5), where \(R^{(4)}_{ml}(h, \xi) = R^{(1)}_{ml}(h, \xi) - iR^{(2)}_{ml}(h, \xi)\). Defined in [2]

• \(S^{(1)}_{ml}(h, \eta)\): prolate spheroidal angle function of the 1st kind (eqn. 4). Defined in [2].

• \(N_{ml}\) : prolate spheroidal angle normalization factor (eqn. 11)

• \(A_{ml}, B_{ml}\) : unknown coefficients to be estimated – related to the boundary condition of ‘particle velocity at the spheroid-fluid interface’. Defined in [3].

• \(\tilde I^{s}_{ml} cos m\phi\) (eqn. 12) and \(\tilde I^{c}_{ml} sin m\phi\) ‘..define the size, shape, and location of the piston in the spheroisal baffle..’. In detail,

\[ \begin{align}\begin{aligned}\tilde I^{s}_{ml} = \int \int_{S_{i}} (\xi^{2}_{0} - \eta^{2})^{1/2}S^{(1)}_{ml}(h, \eta) cos m\phi \:d\eta d\phi\\\tilde I^{c}_{ml} = \int \int_{S_{i}} (\xi^{2}_{0} - \eta^{2})^{1/2}S^{(1)}_{ml}(h, \eta) sin m\phi \:d\eta d\phi\end{aligned}\end{align} \]

• The directivity itself is given by eqn. 16:

\[F(\theta, \phi) = f(\theta,\phi)/f(\theta_{0},\phi_{0})\]

where \(\theta_{0},\phi_{0}\) ‘define the direction of maximum response’

Notes

• \(R^{(1,2,4)}_{ml}(h, \xi)\) and \(S^{(1)}_{ml}(h, \eta)\) are

• There seem to be something relevant here (Scipy implementations).

References

1. Boisvert & Buren 2004, Acoustic directivity of rectangular pistons on prolate spheroids, JASA, 116, 1932 (2004); doi: 10.1121/1.1778840

2. 3. Flammer, Spheroidal Wave Functions, Stanford University Press, Stanford, CA, 1957

3. 10. 5. Boisvert and A. L. Van Buren, ‘‘Acoustic radiation impedance of rectangular pistons on prolate spheroids,’’ J. Acoust. Soc. Am. 111, 867–874 (2002)