Wave Impedance, Reflection from a Radius Mismatch

In the previous section, we identified the right- and left-traveling components of two key quantities describing wave propagation in an acoustic tube: the pressure $p(x,t)$ in the tube ( $p(x,t) = p^+(x - c/t) + p^-(x + c/t)$), and the volume velocity $u(x,t)$ in the tube ( $u(x,t) = u^+(x - c/t) + u^-(x + c/t)$). For the right- and left-traveling components, it turns out we can relate them using relatively simple formulas. Using a combination of calculus, Newton's laws of motion, and the law of conservation of matter, it can be shown that the right-traveling pressure and volume velocity components obey the following formula:

$$p^+\left(x - \frac{c}{t}\right) = R_A \, u^+\left(x - \frac{c}{t}\right),$$ (7)

where $R_A$ is the wave impedance in the tube, given by the following formula:

$$R_A = \frac{\rho c}{A},$$ (8)

where $\rho$ is the ambient fluid density in the tube, $c$ is the velocity of wave propagation (see Equation (5)), and $A$ is the cross-sectional area of the tube. Thus, the wave impedance $R_A$ relates pressure to volume-velocity everywhere along a plane wave traveling to the right along the axis of an acoustic tube having cross-sectional area $A$.

Similarly, for the left-traveling wave components, it may be shown that

$$p^-\left(x + \frac{c}{t}\right) = - R_A \, u^-\left(x + \frac{c}{t}\right).$$ (9)

It is next interesting to consider what happens to a traveling pressure waveform in an acoustic tube when it encounters a radius mismatch. In other words, what happens when the waveform is traveling through an initial tube with radius $r_1$, and all-of-a-sudden is transferred into a tube with a second disparate radius $r_2$? It turns out that part of the waveform will be reflected back into the first tube, and the strength of the reflection $k$ is given by the following formula:

$$k = \frac{R_2 - R_1}{R_1 + R_2},$$ (10)

where $R_1$ is the wave impedance in the first tube section, and $R_2$ is the wave impedance in the second tube section. Using the previous formulas, it may be further shown that the reflectance $k$ is also given by the following formula for cylindrical tubes:

$$k = \frac{r_1^2 - r_2^2}{r_1^2 + r_2^2},$$ (11)

where $r_1$ is the radius of the first tube, and $r_2$ is the radius of the second tube.