Non-Radial Harmonic Vibrations within a Conical Horn

Simple harmonic vibrations satisfying the equation $\phi ={\mathrm{}\left(\mathrm{kr}\right)\mathrm{}}^{-\frac{1}{2}}{J}_{m+\frac{1}{2}}\mathrm{}\left(\mathrm{kr}\right)\mathrm{}{P}_m\left(cos\theta \right)cos\sigma t$ are studied. In this equation, $\phi$ is the velocity potential, $J$ and $P$ denote Bessel and Legendre functions respectively, and $r$ and $\theta$ are polar coordinates. The parameter $m$ specifying the orders of the Bessel and Legendre functions is determined so that the vibrations satisfy the boundary conditions for a conical horn. This is possible by means of a new expansion for $P_m\mathrm{}\left(x\right)\mathrm{}$ which is herein developed. With the assumption of a loop at the opening of the horn and by the aid of an asymptotic expansion for $J_{m+\frac{1}{2}}\mathrm{}\left(z\right)\mathrm{}$, numerical values are computed, for horns of various angles (2° to 30°) and for two types of vibration, of the ratios $\frac{n}{n_0}$ of several frequencies to the fundamental frequency $n_0$, and of the ratios $\frac{\lambda }{d}$ of the corresponding wave-lengths to the diameter of the horn at the opening. The nature of these two types of vibration is indicated by figures.