Method to determine cutoff frequencies for acoustic waves propagating in nonisothermal media

A method to determine cutoff frequencies for linear acoustic waves propagating in nonisothermal media is introduced. The developed method is based on wave variable transformations that lead to Klein-Gordon equations, and the oscillation theorem is applied to obtain the turning point frequencies. Physical arguments are used to justify the choice of the largest turning point frequency as the cutoff frequency. The method is used to derive the cutoff frequencies in nonisothermal media modeled by exponential and power law temperature gradients, for which the cutoffs cannot be obtained based on known analytical solutions. An interesting result is that the acoustic cutoff frequencies calculated by the method are local quantities that vary in the media, and that their specific values at a given height determine the frequency that acoustic waves must have in order to be propagating at this height. To extend this physical interpretation of the acoustic cutoff frequency to nonisothermal media of arbitrary temperature gradients, a generalized version of the method applicable to these media is also presented.

Figure 1

The speed of sound $c_s$ normalized by $c_{s0}$ is plotted as a function of height $z∕z_0$ for the exponential model $\left(\exp \right)$ and the power law models with $m=1$, 2, 3, 4, and 5; note that all the models start at $z=z_0$.

Figure 2

Normalized velocity (a) and pressure (b) perturbations are plotted versus height in the nonisothermal model with the power law temperature distribution with $m=1$. The three considered cases correspond to $\omega =2{\Omega }_0$ (solid line), $\omega ={\Omega }_0$ (dashed line), and $\omega ={\Omega }_0∕2$ (dotted line). In all these cases, the wave propagation begins at the height $z=z_0$.

Figure 3

Normalized velocity (a) and pressure (b) perturbations are plotted versus height in the nonisothermal model with the power law temperature distribution with $m=3$. The three considered cases correspond to $\omega =50{\Omega }_0$ (solid line), $\omega =25{\Omega }_0$ (dashed line), and $\omega =10{\Omega }_0$ (dotted line). In all these cases, the wave propagation begins at the height $z=z_0$.

Figure 4

Normalized velocity and pressure perturbations are plotted versus height in the nonisothermal model with the exponential temperature distribution. The frequency of acoustic waves is $\omega =50{\Omega }_0$ and the wave propagation begins at the height $z=z_0$.

Figure 5

The acoustic cutoff frequency ${\Omega }_{\mathrm{ac}}$ normalized by ${\Omega }_0$ is plotted as a function of height $z∕z_0$ for the exponential model $\left(\exp \right)$ and the power law models with $m=1,2,3,4$, and 5; note that all the models start at $z=z_0$.