Obtaining Bell Reflection Function

The relationship between traveling waves in adjacent sections may be written in matrix form as

$$\begin{bmatrix} p_n^+\\ p_n^-\\ \end{bmatrix} = \mathbf{A_n} \begin{bmatrix} p_{n+1}^+\\ p_{n+1}^-\\ \end{bmatrix},$$

where the scattering matrix is given by

$$\mathbf{A_n} = \begin{bmatrix} \frac{Z_n}{Z_{n+1}}\frac{Z_{n+1} +... ...*}{Z_{n+1}^*}\frac{Z_{n+1}^* + Z_n}{Z_n+Z_n^*}e^{-jkL_{n+1}}\\ \end{bmatrix},$$

for a section of length $L_{n}$ and with complex wave impedance $Z_n$.

For a model having $N$ sections, $N-1$ scattering matrices are multiplied,

$$\begin{bmatrix} p_1^+\\ p_1^-\\ \end{bmatrix} = \mathbf{A_1A_2\hdots A_{N-1}} \begin{bmatrix} p_{N}^+\\ p_{N}^-\\ \end{bmatrix},$$

to yield the model's single final scattering matrix

$$\mathbf{P} = \prod^{N-1}_{n=1}\mathbf{A_n},$$

relating the bell input and output traveling pressure waves.

The expression for the reflection function of the bell may be formed by taking the ratio of the wave reflected by the bell $p_1^-$ to the bell input wave $p_1^+$,

$$R_B = \frac{p_1^-}{p_1^+} = \lambda^2(\omega)\frac{p_N^+\mathbf{P_{2,1}} + p_N^-\mathbf{P_{2,2}}} {p_N^+\mathbf{P_{1,1}} + p_N^-\mathbf{P_{1,2}}}$$

$$= \lambda^2(\omega)\frac{\mathbf{P_{2,1}} + \mathbf{P_{2,2}}R_L(\omega)} {\mathbf{P_{1,1}} + \mathbf{P_{1,2}}R_L(\omega)},$$

where the final expression is obtained by incorporating an open end reflection at the termination of the $N^{\mbox{th}}$ section by substituting $p_N^- = p_N^+R_L(\omega)$, and by commuting round-trip propagation losses $\lambda^2(\omega)$.

Similarly, the bell transmission is given by the ratio of the wave radiated out the bell $p_N^+T_L(\omega)$, where $T_L\omega)$ is the open-end transmission function, to the bell's input $p_1^+$,

$$T_B(\omega) = \frac{p_N^+\lambda(\omega)T_L(\omega)}{p_1^+} = \frac{\lambda(\omega)T_L(\omega)}{\mathbf{P_{1,1}} + \mathbf{P_{1,2}}R_L(\omega)}.$$ (4)