Resonances in a Spherical Cavity

Resonances exist inside a cavity of any shape. In this section the resonant frequencies of a spherical cavity are calculated. The resonant frequencies of a rectangular cavity are calculated in another section. The resonant frequencies of a cylindrical cavity are also fairly easy to calculate; for other shapes numerical techniques can be used.

Spherical Cavity Modes

The solution of the wave equation in spherical coordinates is given by equation (W9) in the Wave Spectra Section. For modes inside a cavity the spherical Hankel function is replaced by a spherical Bessel function of the 1st kind. For non-zero values of m two terms are combined to yield sinmφ or cosmφ variation, which for convenience are labeled by the indices -m and m. The sound pressure is then proportional to this modified form of equation (W9). The mean molecular velocity in the radial direction is proportional to the derivative of (W9) with respect to radius. The velocity must be zero at the cavity wall. For a sphere of radius a, the modal solutions consist of discrete values of k, the wave propagation constant, such that the velocity is zero at r=a. In other words, k is selected such that ka is a root of the first derivative of the spherical Bessel function.

There are an infinite number of roots for each value of the index n; a third index q denotes the root number, in order of ascending values. Each mode is then defined by the triad of indices n,m, and q. The resonant frequencies are independent of m. The roots were determined via a Matlab program, and the first few values are given in the table below:

Roots of the 1st derivative of the spherical Bessel function of the 1st kind

 n 1st root 2nd root 3rd root 4th root 5th root 0 4.4934 7.7253 10.9041 14.0662 17.2208 1 2.0816 5.9404 9.2058 12.4044 15.5792 2 3.3421 7.2899 10.6139 13.8461 17.0429 3 4.5141 8.5838 11.9727 15.2445 18.4681 4 5.6467 9.8404 13.2956 16.6093 19.8624

The lowest root, corresponding to the lowest resonant frequency, is the 1st root of the n=1 mode. The possible values of m in this case are -1, 0, and 1. The different values of m represent different rotations of the mode within the cavity. A contour plot of the sound pressure for the m=0 mode is shown in this figure [44kb]. The z-axis is vertical, and the pressure is rotationally symmetric around the z-axis. At one instant in time the pressure is positive in the upper half and negative in the lower half. The pressure is maximum at the top of the figure, and the red contour lines represent values descending to zero at the straight contour line at kz=0. The pressure is most negative at the bottom. The pressure then oscillates at the resonant frequency, such that the maximum pressure is at the bottom a half-cycle later.

The lowest mode contour plot is kind of boring, so just for fun I will throw in the contour plot [44 kb] of the n=2, m=0, q = 2 mode.

Comparison with Cubical Cavity Resonances

The resonant frequencies of a spherical cavity and a cubical cavity of equal volumes are shown in this figure [42kb]. The density of modes is a bit higher for the cube, but otherwise the behavior is similar. The intensity of a resonance depends on the cavity Q, so for similar walls the intensity will also be similar for a cube and a sphere.

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