Plane waves in a uniform section of tube

Consider an infinite cylindrical pipe whose central axis we label $z$. We will begin by treating loss-less plane wave propagation in such a pipe. The pressure on any plane perpendicular to $z$ is constant, so the pressure is simply a function of $z$ and time, $t$. The wave equation is then ([40] p.107)

$$\frac{\partial^2}{\partial z^2} p = \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2}$$ (2.1)

where $c \approx 343$ m/s is the speed of sound. The complex solution is

$$p(z,t) = \left(A e^{-i k z} + B e^{i k z}\right) e^{i \omega t}$$ (2.2)

where the real part of $p$ is the physically observable pressure variation from the equilibrium atmospheric pressure value. $A$ is the complex amplitude of the forward travelling pressure wave (in the positive direction along $z$) with the absolute value being the peak pressure in the forward wave and the real part defining the wave's phase by setting the value of the pressure at $z=0,t=0$. Similarly $B$ is the complex pressure amplitude for the backward travelling wave. $k$ is the wavenumber of the sinusoidal wave with the corresponding wavelength $\lambda = 2 \pi/k$. $\omega$ is the angular frequency given by $\omega = 2 \pi f$ where $f$ is the frequency of the wave.

To verify that equation (2.2) is indeed a solution we can substitute into equation (2.1) giving

$$-k^2 p = -\frac{\omega^2}{c^2} p.$$ (2.3)

The equality holds because $c =\omega/k =f \lambda$.

These pressure changes are compression waves; they correspond to changes in the typical movements of gas molecules in the direction of propagation. The motion of the molecules in the air may be modelled by considering the motion of an ``acoustic particle'' large enough to contain millions of molecules but small enough that the acoustic variables are constant throughout its volume [40] p.99. To find the volume velocity from the pressure we use the linear inviscid force equation ([40] p.104):

$$\rho \frac{\partial v}{\partial t} = - \nabla p.$$ (2.4)

Here $v$ is the acoustic particle velocity and $\nabla$ is the gradient operator given by

$$\nabla p = \hat{x} \frac{\partial p}{\partial x} + \hat{y} ... ...artial p}{\partial y} + \hat{z} \frac{\partial p}{\partial z}$$ (2.5)

where $\hat{x}$ is a unit vector along the $x$ direction etc. For plane waves the pressure varies only along the $z$ axis. The velocity is then a vector parallel to the $z$ axis. Like the pressure, the velocity will be sinusoidal with time dependence $\exp(i \omega t)$. Substituting equation (2.2) into equation (2.4) gives the $z$ component of the velocity as

$$v_z = \frac{1}{\rho \omega}\left(k A e^{-i k z} - k B e^{i k z}\right) e^{i \omega t}.$$ (2.6)

We will see when we come to consider ducts of varying cross-section, that it is useful to deal with a quantity called the volume velocity, defined as $U = v_z S$ where $S$ is the cross-sectional area of the pipe. The volume velocity is then

$$U(z,t) = \frac{S}{\rho c}\left(A e^{-i k z} - B e^{i k z}\right) e^{i \omega t}.$$ (2.7)

The acoustic impedance is then defined as being the ratio of the pressure and volume velocity. For forward travelling waves this is the characteristic impedance, $Z_c=\rho c /S$ and for backward travelling waves this is $-Z_c=-\rho c /S$ where $\rho=1.21$kgm$^{-3}$ is the equilibrium density of air. While the acoustic impedance of travelling plane waves is $\pm \rho c/S$, the acoustic impedance of the combination of forward and backward going waves must be calculated from the ratio of the pressure and velocity resulting from both:

$$Z = \frac{p}{U}=\frac{\rho c}{S}\frac{A e^{-i k z} + B e^{i k z}} {A e^{-i k z} - B e^{i k z}}.$$ (2.8)

So far we have provided a formula for the volume velocity and impedance at any point in a duct of constant cross-section provided $A$ and $B$ (the complex amplitude of the forward and backward travelling pressure waves at $z=0$) are known. Our eventual aim is to be able to work out the acoustic variables at one end of a series of concentric cylinders from the acoustic variables at the other end. The first step is to be able to project acoustic variables down a cylindrical section from an arbitrary $z$ axis position. Consider the cylindrical pipe in figure 2.1. Two planes are defined: plane 1 is at $z=z_1$ and plane 0 is a distance $d$ to the left of this. We want to know how to project the acoustic variables from plane 1 to plane 0.

Figure 2.1: Uniform pipe

Labelling $p^{(0)}$ as the pressure on plane 0, by substituting the $z$ coordinate of $z=z_1-d$ into (2.2) we get

$${p^{(0)} = A e^{-ik(z_1-d)} + B e^{ik(z_1-d)}}$$

$$= (Ae^{-ikz_1}+Be^{ikz_1})\cos(kd) + (Ae^{-ikz_1}-Be^{ikz_1})i\sin(kd).$$ (2.9)

Now the pressure on plane 0 can be found from $p^{(1)}$ and $U^{(1)}$, the pressure and volume velocity on plane 1.

$$p^{(0)} = \cos(kd) p^{(1)} + i \sin(kd) Z_c U^{(1)}$$ (2.10)

where $Z_c=\rho c /S$. Similarly, for the volume velocity on plane 0, putting $z=z_1-d$ into (2.7) gives:

$$U^{(0)} = i \sin(kd) Z_c^{-1} p^{(1)} + \cos(kd) U^{(1)}$$ (2.11)

The impedance, $Z^{(0)}$ on plane 0 can be obtained by dividing the pressure and volume velocity there:

$$Z^{(0)} = \frac{p^{(0)}}{U^{(0)}} = \frac{\cos(kd) p^{(1)} ... ...Z_c U^{(1)}} {i \sin(kd) Z_c^{-1} p^{(1)} + \cos(kd) U^{(1)}}$$ (2.12)

Dividing through by $U^{(1)}$ gives $Z^{(0)}$ in terms of $Z^{(1)}$, the impedance on plane 1:

$$Z^{(0)} = \frac{p^{(0)}}{U^{(0)}} =\frac{\cos(kd) Z^{(1)} + i \sin(kd) Z_c} {i \sin(kd) Z_c^{-1} Z^{(1)} + \cos(kd)}$$ (2.13)

Notice that if a travelling wave is present at plane 1 in say the positive $z$ direction only we get $B=0$ and $Z^{(1)}=Z_c$. Equation (2.13) then reduces to $Z^{(0)}=Z_c$ showing that the impedance is unchanged by projection along the cylinder. In general, however, the impedance will be changed by projection.