Tim Mellow

On the sound field of an oscillating disk in a finite open and closed circular baffle

Equations describing the radiation characteristics of a rigid disk in a finite open baffle are derived using a method similar to that used by Streng for a circular membrane based upon the dipole part of the Kirchhoff–Helmholtz boundary integral formula. In this case, however, a power series solution to the radiation integral is derived in order to eliminate the need for numerical integration. Hence, a set of simultaneous equations is obtained by simply equating the coefficients of the power series, which leads to two mathematical functions, one real and one imaginary, that can be applied to any radial velocity distribution. This provides an alternative method to obtain the sound scattered by a disk or the complementary hole in an infinite resilient screen according to Babinet’s principle. Using the principle of superposition (or Gutin concept), it is shown how the sound radiation characteristics of a disk radiating from just one side can be obtained by combining the radiation field of a disk in a finite b...

On the sound field of a resilient disk in an infinite baffle

The Rayleigh integral describing the near-field pressure of an axisymmetric planar monopole source with an arbitrary velocity distribution is solved with a method similar to that used by Mast and Yu [J. Acoust. Soc. Am. 118(6), 3457–3464 (2005)] for a rigid disk in an infinite baffle. The closed-form solution is in the form of a double expansion, which is valid for distances from the observation point to the center of the source that are greater than its radius. However, for the remaining immediate near field, the King integral is solved using a combination of Gegenbauer’s summation theorem and the Lommel expansion, resulting in a solution which is in the form of a triple expansion, reducing to a double expansion along the source’s axis of symmetry. These relatively compact solutions in analytic form do not require numerical integration and therefore present no numerical difficulties except for a singularity at the rim. As an example of a monopole source with an arbitrary velocity distribution, equations ...

On the sound field of a resilient disk in free space

Radiation characteristics are calculated for a circular planar sound source in free space with a uniform surface pressure distribution, which can be regarded as a freely suspended membrane with zero mass and stiffness. This idealized dipole source is shown to have closed form solutions for its far-field pressure response and radiation admittance. The latter is found to have a simple mathematical relationship with the radiation impedance of a rigid piston in an infinite baffle. Also, a single expansion is derived for the near-field pressure field, which degenerates to a closed form solution on the axis of symmetry. From the normal gradient of the surface pressure, the surface velocity is calculated. The near-field expression is then generalized to an arbitrary surface pressure distribution. It is shown how this can be used as a simplified solution for a rigid disk in free space or a more realistic sound source such as pre-tensioned membrane in free space with non-zero mass and a clamped rim.

On the sound field of a circular membrane in free space and an infinite baffle

An enhanced method for calculating the radiation characteristics of a tensioned circular membrane in free space is presented using an analytical solution to the infinite integral in the free-space Green’s function in cylindrical coordinates. This enables direct calculation of the surface pressure series coefficients by equating the coefficients of the resulting Bessel series in a set of simultaneous equations. Eliminating both numerical integration and least-squares minimization improves calculation speed and accuracy. An infinite baffle is introduced to provide an indication of what the theoretical limit of the bass performance would be using a very large enclosure. Furthermore, analytical solutions to the pressure field integrals are presented. A force transmission coefficient is introduced, which is the ratio of the total radiation impedance to the motional impedance. The motional, radiation, and diaphragm impedances of the damped membrane are calculated, together with the near- and far-field pressure ...

On the forces in single-ended and push-pull electret transducers

The equations for the electromechanical force conversion in single-ended and push-pull electret transducers are derived. Traditionally, the charge distribution has been modeled as a concentrated layer at an arbitrary distance from the surface of the dielectric. For the purpose of this analysis, a negative charge is assumed to be evenly distributed throughout the dielectric. The membrane has a conductive coating in which a positive charge is induced, giving an overall dipole charge. The resulting formulas are used to derive the voltage sensitivity of a microphone and the equivalent electrical circuit for the electromechanical transduction part of a microphone or loudspeaker. An equivalent external polarizing voltage is then derived that would produce the same driving force in a conventional electrostatic loudspeaker without a stored charge. The condition for the static stability of a circular electret membrane is also determined.

On the sound field of a shallow spherical shell in an infinite baffle.

A method is presented for calculating the far field sound radiation from a shallow spherical shell in an acoustic medium. The shell has a concentrated ring mass boundary condition at its perimeter representing a loudspeaker voice coil and is excited by a concentrated ring force exerted by the end of the voice coil. A Greens function is developed for a shallow spherical shell, which is based upon Reissners solution to the shell wave equation [Q. Appl. Math. 13, 279-290 (1955)]. The shell is then coupled to the surrounding acoustic medium using an eigenfunction expansion, with unknown coefficients, for its deflection. The resulting surface pressure distribution is solved using the King integral together with the free space Greens function in cylindrical coordinates. In order to eliminate the need for numerical integration, the radiation (coupling) integrals are solved analytically to yield fast converging expansions. Hence, a set of simultaneous equations is obtained which is solved for the coefficients of the eigenfunction expansion. These coefficients are finally used in formulas for the far field sound radiation.

A dipole loudspeaker with a balanced directivity pattern.

Analytical equations describing radiation characteristics of an oscillating ring in a circular finite baffle are derived, including the limiting case of a dipole point source at the center. An oscillating sphere would represent the ideal dipole source, having a constant directivity pattern at all frequencies, but would be inconvenient to realize especially in portable devices. It is found that a planar piston with uniform surface velocity but variable phase arranged to emulate the sphere does not have such a smooth on-axis response as the sphere. Instead a planar piston with the same phase distribution but uniform pressure represents an ideal planar source with a smooth on-axis response and near constant directivity. The surface velocity is plotted and it is then shown that a similar response can be achieved using a finite number of concentric rings based on this velocity distribution.

Comparison of spheroidal and eigenfunction-expansion trial functions for a membrane in an infinite baffle

The on-axis far-field pressure response of a circular membrane in an infinite baffle when driven by a uniformly distributed electrostatic force is calculated using two different trial functions for the surface velocity distribution. The first is an expansion based upon a solution to the free space wave equation in oblate spheroidal coordinates, which has already been derived in a previous paper [J. Acoust. Soc. Am. 120(5), 2460-2477 (2006)], and the second is a membrane eigenfunction expansion (or Bessel series), which is rigorously derived in this letter. Although the latter can be used as a basis for calculating a number of different radiation characteristics such as the radiation impedance or directivity, etc., only the on-axis far-field sound pressure is considered here. The results are compared and discussed.

Expansions for the radiation impedance of a rectangular piston in an infinite baffle

Relatively compact analytical expressions in the form of fast-convergin g expansions are derived for the radiation resistance and reactance of a rectangular rigid piston in an infinite rigid baffle, which are computationally efficient at high frequencies or large aspect ratios and yield simple approximations (asymptotic expressions) at low frequencies. Plots of the normalized radiation resistance and reactance are shown for various aspect ratios with constant width as well as constant area. Comparisons are also made with the impedance of an elliptic piston.

Ringing the changes: The potential for push‐pull electret transducers in cell phones.

Electret microphones are very common in mobile devices, but are invariably single‐ended in configuration. For moderate dynamic conditions this is not a problem, but under high‐sound pressure levels, better linearity can be obtained using a push‐pull arrangement. Recent progress in nano‐porous electret materials also opens up possibilities for electret loudspeakers. Such loudspeakers would offer potential benefits such as high‐conversion efficiency, no magnetic field, and an ultra‐thin package. However, there are product integration challenges. The authors describe a simulation model of a push‐pull electret transducer and discuss the conditions under which it is linear. It turns out that the relationship with a non‐electret electrostatic transducer is relatively simple.