Propagation of Spherical Waves (Conical Tubes)

We have seen how a tract of cylindrical tube is governed by a partial differential equation such as (7) and, therefore, it admits exact simulation by means of a waveguide section. When the tube has a conical profile, the wave equation is no longer (1), but we can use the equation for propagation of spherical waves [27]:

$$\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\par... ...t) = \frac{1}{c^2} \frac{\partial^2 p(r,t)}{\partial t^2} \: ,$$ (75)

where $r$ is the distance from the cone apex.

In the equation (73) we can evidentiate a term in the first derivative, thus obtaining

$$\frac{\partial^2 p(r,t)}{\partial r^2} + \frac{2}{r}\frac{\... ...r} = \frac{1}{c^2} \frac{\partial^2 p(r,t)}{\partial t^2} \; .$$ (76)

If we recall equation (26) for lossy waveguides, we find some similarities. Indeed, we are going to show that, in the scalar case, the media described by (26) and (74) have structurally similar wave admittances.

Let us put a complex exponential eigensolution in (73), with an amplitude correction that accounts for energy conservation in spherical wavefronts. Since the area of such wavefront is proportional to $r^2$, such amplitude correction has to be inversely proportional to $r$, in such a way that the product intensity (that is the square of amplitude) by area is constant. The eigensolution is

$$p(r,t)=\frac{1}{r} e^{s t + v r} \; ,$$ (77)

where $s$ is the complex temporal frequency, and is the complex spatial frequency. By substitution of (75) in (73) we find the algebraic relation

$$v = \pm \frac{s}{c} \; .$$ (78)

So, even in this case the pressure can be expressed by the first of (4), where

$$\begin{array}{ll} p^+ = \frac{1}{r} e^{s(t - r/c)}; & p^- = \frac{1}{r} e^{s(t + r/c)} \; . \end{array}$$ (79)

Newton's second law

$$\frac{\partial u_r}{\partial t} = - \frac{1}{\rho} \frac{\partial p}{\partial r}$$ (80)

applied to (77) allows to express the particle velocity $u_r$ as

$$u_r(r,t) = \left( \frac{1}{ r s} \mp \frac{1}{c} \right) \frac{1}{\rho r} e^{s(t \pm r/c)} \; .$$ (81)

Therefore, the two wave components of the air flow are given by

$$\begin{array}{ll} u^+ = S \left( \frac{1}{r s} + \frac{1}{c}... ...}{c} \right) \frac{1}{\rho r} e^{s(t + r/c)} & \; , \end{array}$$ (82)

where $S$ is the area of the spherical shell outlined by the cone at point $r$.

We can define the two wave admittances

$$\begin{array}{l} \Gamma^+ = \Gamma(s) = \frac{u^+}{p^+} = G_... ...}{p^-} = G_0 \left( -1 + \frac{1}{s L} \right) \; , \end{array}$$ (83)

where $G_0 = \frac{S}{\rho c}$ is the admittance in the degenerate case of a null tapering angle, and $L=\frac{r}{c}$ is a shunt reactance accounting for conicity [46]. The wave admittance for the cone is $\Gamma(s)$, and $\Gamma^*(-s^*)$ is its paraconjugate in the analog domain. If we translate the equations into the discrete-time domain by bilinear transformation, we can check the validity of equations (21) for the case of the cone.

Wave propagation in conical ducts is not lossless, since . However, the medium is passive in the sense of section II, since the sum is positive semidefinite along the imaginary axis.

As compared to the lossy cylindrical tube, the expression for wave admittance is structurally unchanged, with the only exception of the sign inversion in the shunt inductance. This difference is justified by thinking of the shunt inductance as a representation of the signal that does not propagate along the waveguide. In the case of the lossy tube, such signal is dissipated into heat; in the case of the cone, it fills the shell that is formed by interfacing a planar wavefront with a spherical wavefront.

The discrete-time simulation of a length-$L_R$ cone tract having the (left) narrow end at distance $r_0$ from the apex is depicted in figure 8.

Figure: One-variable waveguide section for a length-$L_R$ conical tract