Multimodal radiation impedance of a duct terminated in an infinite baffle

In chapter 2 we defined the acoustic impedance in a duct as the ratio of the pressure amplitude to the volume velocity amplitude. We therefore wish to define the radiation impedance as the ratio of the pressure amplitude and the volume velocity amplitude on the radiating surface. An expression has been derived for the total pressure field in terms of the volume velocity amplitude. The next step is therefore to calculate the pressure amplitude of a particular mode at $z=0$ from the total pressure field. To do this we first express the pressure field there as the sum of the contributions of all the modes,

$$p(x,y,0) = \sum\limits_{m=0}^\infty P_m \psi_m(x,y).$$ (3.6)

Now we recall the orthogonality of the modes from equation (2.47):

$$\int\limits_S \psi_n \psi_m dS = S \delta_{nm}.$$ (3.7)

Multiplying $p$ by $\psi_n$ and integrating over the surface area therefore gives

$$\int_S p(x,y,0) \psi_n(x,y) dS = \sum\limits_{m=0}^\infty \int_S P_m \psi_m(x,y) \psi_n(x,y) dS = P_n S.$$ (3.8)

When this is rearranged, the pressure amplitude of the $n$th mode is given in terms of the total pressure field as

$$P_n = \frac{1}{S} \int_S p(x,y,0) \psi_n(x,y) dS.$$ (3.9)

Now we will work out how the pressure and velocity modes couple. Substituting equation (3.5), which gives the total pressure field due to the contribution of the velocity modes, into equation (3.9) gives

$$P_n = \frac{i\omega\rho}{2\pi S^2}\sum\limits_{m=0}U_m \int_S dS \int_S dS_0 \psi_m(x_0,y_0) \psi_n(x,y) \frac{e^{-ikh}}{h}.$$ (3.10)

This expression gives the pressure amplitude of the $n$th mode due to the contributions of all the velocity modes, not just the $n$th velocity mode. The modes are therefore coupled at the opening as was expected.

Looking back to equation (2.95), the relationship between the pressure and velocity modes was given as

$${\mathbf P} = Z {\mathbf U}$$ (3.11)

where $Z$ is known as the impedance matrix. ${\mathbf P}$ is the column vector whose $n$th element is the pressure amplitude of the $n$th mode, $P_n$. Similarly, ${\mathbf U}$ is the column vector whose $m$th element is the volume velocity amplitude of the $m$th mode, $U_m$. Expressing this in summation notation, we get

$$P_n = \sum\limits_{m=0}^\infty Z_{nm} U_m.$$ (3.12)

Comparison of equations (3.10) and (3.12) yields

$$Z_{nm} = \frac{i\omega\rho}{2\pi S^2} \int_S dS \int_S dS_0 \psi_m(x_0,y_0) \psi_n(x,y) \frac{e^{-ikh}}{h}$$ (3.13)

where $Z_{nm}$ is the $(n,m)$ element of the impedance matrix and gives the contribution to the $n$th pressure mode by the $m$th velocity mode. Because this is the impedance at the open end, we call the impedance matrix here the ``radiation impedance matrix''. Note that we are integrating twice: first we integrate to get the pressure field at $(x,y,0)$ due to the sum of all source elements and then we integrate this pressure at every point in the piston to isolate a single modal pressure amplitude component.