Conical Acoustic Tubes

The conical acoustic tube is a one-dimensional waveguide which propagates circular sections of spherical pressure waves in place of the plane wave which traverses a cylindrical acoustic tube [22,352]. The wave equation in the spherically symmetric case is given by

$$c^2 p''_x = \ddot{p}_x \protect$$ (C.165)

where

$$\begin{array}{rclrcl} c & \isdef & \mbox{sound speed} & \qquad p_x & \isdef & x\, p(t,x)\\ [10pt] {\dot{p}_x} & \isdef & \dfrac{\partial}{\partial t}p_x(t,x) & \qquad p'_x & \isdef & \dfrac{\partial}{\partial x}p_x(t,x) \end{array}$$

and $p(t,x)$ is the pressure at time $t$ and radial position $x$ along the cone axis (or wall). In terms of $p$ (rather than $p_x=xp$ ), Eq.(C.166) expands to Webster's horn equation [360]:

$$\frac{1}{c^2} \ddot{p} \eqsp \frac{1}{A}\left(A p'\right)' \eqsp p'' + \frac{A'}{A}p'$$

where $A=\alpha x^2$ is the area of the spherical wavefront, so that $A'/A=2/x$ (as discussed further in §C.18.4 below).

Spherical coordinates are appropriate because simple closed-form solutions to the wave equation are only possible when the forced boundary conditions lie along coordinate planes. In the case of a cone, the boundary conditions lie along a conical section of a sphere. It can be seen that the wave equation in a cone is identical to the wave equation in a cylinder, except that $p$ is replaced by $xp$ . Thus, the solution is a superposition of left- and right-going traveling wave components, scaled by $1/x$ :

$$p(t,x) = \frac{ f\left(t-\frac{x}{c}\right)}{x} + \frac{ g\left(t+\frac{x}{c} \right)}{x} \protect$$ (C.166)

where $f(\cdot)$ and $g(\cdot)$ are arbitrary twice-differentiable continuous functions that are made specific by the boundary conditions. A function of $(t-x/c)$ may be interpreted as a fixed waveshape traveling to the right, (i.e., in the positive $x$ direction), with speed $c$ . Similarly, a function of $(t+x/c)$ may be seen as a wave traveling to the left (negative $x$ direction) at $c$ meters per second. The point $x=0$ corresponds to the tip of the cone (center of the sphere), and $p(t,x)$ may be singular there.

In cylindrical tubes, the velocity wave is in phase with the pressure wave. This is not the case with conical or more general tubes. The traveling velocity may be computed from the corresponding traveling pressure by dividing by the wave impedance. However, this impedance is frequency-dependent in cones, as we'll derive below in §C.18.4.