91影视

Spherical harmonics $Y_l^m\left(胃,蠒\right)=P_l^m\left(\cos 胃\right)e^{卤im蠒}$.

Any disturbance or energy transfer from one location to another is referred to as a wave. Each wave is guided by a mathematical formula. A wave can be either standing or stationary.

The wave equation is as given below.

${鈭�}^{2}u=\frac{1}{v^2}\frac{{鈭�}^{2}u}{鈭�t^2}$,

Write the wave equation in spherical coordinates in the form of spherical harmonics.

$u\left(r,胃,桅,t\right)=\left(\begin{array}{c}{j}_1\left(kr\right)\\ y_1\left(kr\right)\end{array}\right)P_l^m\cos 胃e^{卤im桅}\left(\begin{array}{c}\cos \left(kvt\right)\\ sin\left(kvt\right)\end{array}\right)$

Neglect $y_1\left(kr\right)$as inside the spherical cavity, it has no effect.

$u\left(r,胃,桅,t\right)=\left\{j_1\left(kr\right)\right\}{P}_l^m\cos 胃e^{卤im桅}\left\{\cos \left(kvt\right)\right\}$

Apply the boundary condition.

If $ka={位}_1$then the Bessel function equal to zero.

The frequency for normal mode is$蝇=2蟺谓$ .

Write the equation in terms of v.

$谓=\frac{蝇}{2蟺}$

Replace w by kv in the above equation.

$谓=\frac{kv}{2蟺}$ 鈥�.. (1)

Use the relation written below.

${位}_1=ka$

$k=\frac{{位}_1}{a}鈥�$ 鈥�.. (2)

Put equation (2) in (1).

$$\begin{gathered} 谓=\frac{\frac{{位}_1}{a}v}{2蟺} \\ =\frac{{位}_{1}v}{2蟺a} \end{gathered}$$

Hence the characteristic vibration frequencies of sound in a spherical cavity is $谓=\frac{{位}_{1}v}{2蟺a}$.