Multimodal reflectance matrix

A formula relating the forward and backward components of the volume velocity amplitude vector in terms of the impedance matrix was quoted in Pagneux et al. [32] p.2046. Here we show the derivation for the pressure amplitude vector. The first step is to express the total pressure amplitude vector ${\mathbf P}$ as the sum of the forward going (${\mathbf P_+}$) and backward going (${\mathbf P_-}$) components:

$${\mathbf P} = {\mathbf P_+} + {\mathbf P_-}.$$ (6.1)

Now the total volume velocity is expressed using the same notation for forward and backward components:

$${\mathbf U} = {\mathbf U_+} + {\mathbf U_-}.$$ (6.2)

Recalling the characteristic impedance of higher modes from equation (2.36), the ratio of the $n$th element in the forward going pressure vector to the $n$th element in the forward going volume velocity vector is $k \rho c / k_n S$. Using $Z_c$, the diagonal characteristic impedance matrix defined in equation (2.42):

$${\mathbf U_+} = Z_c^{-1} {\mathbf P_+}.$$ (6.3)

Similarly for the backward going waves,

$${\mathbf U_-} = - Z_c^{-1} {\mathbf P_-}.$$ (6.4)

Defining the impedance matrix at a particular point as $Z$ with ${\mathbf P} = Z {\mathbf U}$ we get

$${\mathbf P_+} + {\mathbf P_-} = Z Z_c^{-1} ({\mathbf P_+} - {\mathbf P_-})$$ (6.5)

which may be rearranged to give

$$(Z Z_c^{-1} + I) {\mathbf P_-} = (Z Z_c^{-1} - I) {\mathbf P_+}$$ (6.6)

so the result is

$${\mathbf P_-} = {\mathcal R} {\mathbf P_+}$$ (6.7)

where ${\mathcal R}$ is the reflectance matrix:

$${\mathcal R}(\omega)=\left(Z Z_c^{-1} + I \right)^{-1} \left(Z Z_c^{-1} - I \right).$$ (6.8)

Notice that this is a correction to the reflectance matrix quoted in [41]. The correction arises because $A B A^{-1} \neq B$ in general, even when $A$ is a diagonal matrix. The correction only has an effect on the non-diagonal entries in ${\mathcal R}$. The graphs presented in [41] are of the ${\mathcal R}_{00}$ element and are unaffected by the correction.