Multimodal equations in a uniform wave-guide

The inclusion of higher modes in horn acoustics has been studied recently [32,33,38,41,42]. In general, the pressure and velocity in a cylinder can be expressed in terms of the modes of the duct whose amplitude patterns have $n$ nodal circles and $l$ nodal diameters on a circular cross-section where $(n,l)=(0,0)$ is the plane wave mode. We are analysing axi-symmetric systems so we will treat axi-symmetric (nodal circle) modes only. The pressure and volume velocity are then vectors with a single subscript, $n$.

In the case of rectangular cross-section, the modes of the duct will have an integer number of nodal lines parallel to the $y$ axis and an integer number of nodal lines parallel to the $x$ axis. We will only treat systems that preserve symmetry about the central axis, so only modes with an even number of nodal lines in both dimensions need to be considered and the subscript $n = (n_x,n_y)$ will be used where there are $2n_x$ nodal lines parallel to the $y$ axis and $2n_y$ nodal lines parallel to the $x$ axis.

The pressure for each mode obeys the 3 dimensional wave equation:

$$\Delta p = \frac{1}{c^2} \frac{\partial^2 p}{\partial t^2}.$$ (2.22)

Here $\Delta$ is the Laplacian operator which may be expressed as the sum of the $z$ direction component and the component on the $x$-$y$ plane:

$$\Delta = \Delta_{\bot} + \frac{\partial^2}{\partial z^2}$$ (2.23)

where $\Delta_{\bot}$ is given in Cartesian coordinates as

$$\Delta_{\bot} = \left(\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}\right)$$ (2.24)

and in cylindrical polars as

$$\Delta_{\bot} = \left(\frac{\partial^2}{\partial r^2} +\frac... ... r} +\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}\right).$$ (2.25)

The wave equation can be solved by expressing the pressure as a sum of the contributions of the modes of the duct where each term is the multiplication of the profiles along $z$, $t$ and the $x$-$y$ plane.

We can then solve the problem by separation of variables ([43] pp.540-556). From Pagneux et al. [32] the pressure and axial velocity are:

$$p(x,y,z,t) = \sum\limits_{n=0}^\infty P_{n}(z)\psi_{n}(x,y)\exp(i \omega t),$$ (2.26)

$$v_z(x,y,z,t) = \frac{1}{S}\sum\limits_{n=0}^\infty U_{n}(z)\psi_{n}(x,y)\exp(i \omega t),$$ (2.27)

where $\psi_n$ is the pressure profile on the $x$-$y$ plane and $P_n(z)$ is the pressure profile of the $n$th mode along the length of the tube. Similarly, $U_n(z)$ is the axial volume velocity profile of the $n$th mode along the length of the tube. $P_n(z)$ and $U_n(z)$ are in general complex numbers to take phase into account. Note that although it is convenient to refer to $U_n(z)$ as the volume velocity, the net volume velocity is $U_0$; the other entries are the amplitudes of the axial velocity distributions multiplied by the surface area but have no net contribution to the volume velocity [32].

Equation (2.26) gives the pressure as a series of terms, $\psi_0$ being unity, and so the $n = 0$ contribution represents plane wave propagation while the other modes have a non-uniform pressure profile. Substituting $p$ from equation (2.26) into the wave equation (2.22) and dividing through by $p$ gives:

$$\frac{1}{\psi_n} \Delta_{\bot} \psi_n + \frac{1}{P_n} \frac{... ...frac{1}{c^2} \frac{\partial^2}{\partial t^2} \exp(i \omega t)$$ (2.28)

Since each term in this equation is a function of a different variable, each term must equal a constant. The differentiation with respect to $t$ is straightforward giving

$$\frac{1}{c^2}\frac{\partial^2}{\partial t^2} \exp(i \omega t) = -k^2 \exp(i \omega t)$$ (2.29)

with the eigenvalue $k=\omega/c$ being the free space wavenumber. Defining $k_n$ to be the wavenumber along $z$ (commonly referred to as the propagation factor):

$$\frac{\partial^2}{\partial z^2} P_n(z) = -k_n^2 P_n(z).$$ (2.30)

The transverse term gives

$$\Delta_{\bot} \psi_n(x,y) = -\alpha_n^2 \psi_n(x,y)$$ (2.31)

with $\alpha_n$ being the eigenvalue of the $n$th mode. Physically $\alpha_n$ is the wavenumber in the $x$-$y$ plane and is zero for plane wave propagation and is positive and real for the modes which feature nodal lines or circles.

It then follows from equation (2.28) that the wavenumbers follow the relation

$$k_n^2 = k^2 - \alpha_n^2.$$ (2.32)

The wavelength along $z$ will be $\lambda_n = 2\pi / k_n$.

Note that in a pipe of uniform cross-section the solution of equation (2.30) gives

$$P_n(z) = A_n e^{-i k_n z} + B_n e^{i k_n z}$$ (2.33)

To find the corresponding volume velocities we use the force equation (2.4). The $z$ component of the velocity is

$$\rho \frac{\partial v_z}{\partial t} = - \frac{\partial p}{\partial z}.$$ (2.34)

giving the corresponding axial volume velocity as

$$U_n(z) = \frac{k_n S}{k \rho c}(A_n e^{-i k_n z} - B_n e^{i k_n z}).$$ (2.35)

We can note from this that the characteristic impedance, defined to be the ratio of pressure and volume velocity of forward travelling waves, also depends on the mode number, $n$. For plane waves it was $Z_c=\rho c /S$ but for the $n$th mode this becomes:

$$Z_c = \frac{k \rho c}{k_n S}.$$ (2.36)

Assuming loss-less propagation $k_n$ is positive and real above cut-off point ( $k_c = \alpha_n$) so the pressure varies sinusoidally along the $z$ axis with a wavelength of $\lambda_n \geq \lambda$ where $\lambda = 2 \pi/k$ is the free-space wavelength. Below the cut-off point $k_n$ is negative and imaginary [42] so the pressure will be exponentially damped. Calculation of $k_n$ first requires calculation of $\alpha_n$ which in turn depends on the boundary conditions in equation (2.31) (and therefore on the geometry of the duct). It will be treated for both lossy and non-lossy propagation in section 2.4.1 for ducts of circular cross-section and in section 2.4.2 for rectangular ducts.

We have seen that plane waves travelling across a section of tube are only reflected when the cross-section changes. The plane wave pressure amplitude at any point in a uniform section of tube may be obtained from a known plane wave pressure amplitude at a single point simply by projection using equation (2.10). Each of the other modes of the pipe will also only be reflected where the cross-section changes and so may be treated independently in a section of uniform cross-section. The wavenumber along the $z$ axis becomes $k_n$ for the $n$th mode and, as discussed earlier in the section, the characteristic impedance of the $n$th mode is $Z_c = \pm k \rho c / k_n S$. Projection of pressure and volume velocity complex amplitudes of the $n$th mode then follow from equations (2.10) and (2.11).

$$P_n^{(0)} = \cos(k_n d) P_n^{(1)} + i \sin(k_n d) \left(\frac{k \rho c}{k_n S}\right) U_n^{(1)}.$$ (2.37)

$$U_n^{(0)} = i \sin(k_n d) \left(\frac{k_n S}{k \rho c}\right) P_n^{(1)} + \cos(k_n d) U_n^{(1)}.$$ (2.38)

Here $P_n^{(0)}$ and $U_n^{(0)}$ are the complex amplitudes of the pressure and volume velocity of the $n$th mode on plane 0 and $P_n^{(1)}$ and $U_n^{(1)}$ are the complex amplitudes of the pressure and volume velocity of the $n$th mode on plane 1 with the planes separated by a distance $d$ (see figure 2.1).

We define the pressure vector, $\mathbf{P}$, as a column vector consisting of the modal pressure amplitudes $P_n$ and $\mathbf{U}$ as a column vector of the corresponding $U_n$ values. A method of projecting the pressure and velocity vectors along a uniform section of tube will now be discussed.

In matrix notation the pressure vector on plane 0 is given in terms of the vectors on plane 1 by

$${\mathbf P}^{(0)} = D_1 {\mathbf P}^{(1)} + D_2 Z_c {\mathbf U}^{(1)}$$ (2.39)

where $D_1$, $D_2$ and $Z_c$ are diagonal matrices with the elements given by

$$D_1(n,m) = \left\{ \begin{array} {r@{\quad:\quad}l} \cos(k_n d) & n = m, \\ 0 & n \neq m, \end{array} \right.$$ (2.40)

$$D_2(n,m) = \left\{ \begin{array} {r@{\quad:\quad}l} i \sin(k_n d) & n = m, \\ 0 & n \neq m, \end{array} \right.$$ (2.41)

$$Z_c(n,m) = \left\{ \begin{array} {r@{\quad:\quad}l} k \rho c/ k_n S & n = m, \\ 0 & n \neq m. \end{array} \right.$$ (2.42)

Similarly the volume velocity on plane 0 is given in terms of the vectors on plane 1 as

$${\mathbf U}^{(0)} = D_2 Z_c^{-1} {\mathbf P}^{(1)} + D_1 {\mathbf U}^{(1)}.$$ (2.43)