Inductance method

The inductance method due to Kergomard and Garcia [57] is reviewed here. The multimodal method was treated and the results used to get a polynomial for the frequency dependent inductance, $L$, so that the formula

$$p_1 = p_2 + i \omega L U, \hspace{1cm} U_1 = U_2 = U$$ (C.1)

matches the multimodal treatment when $0.05 < R_1/R_2 < 0.95$ and $0.001 < kR_2 < 3.5$. Here $p_1$ is the pressure immediately to the left of the discontinuity and $p_2$ is the pressure immediately to the right of the discontinuity. $U$ is the volume velocity, matched on both sides as with the plane wave approximation. The pressures and volume velocity are scalars here, not vectors since the multimodal effects are represented by the inductance.

Now we will work out the reflectance for a plane wave incident on a discontinuity between two infinite cylinders. We do this by expressing both the equations in (C.1) in terms of forward and backward going waves and solving. The pressure on the left is the sum of incident and reflected waves: $p_1 = p_1^{+} + p_1^{-}$. The pressure on the right is simply the transmitted term $p_2 = p_2^{+}$.

The volume velocity on the left is then

$$U_1 = (p_1^{+} - p_1^{-})(S_1/\rho c)$$ (C.2)

while on the right the volume velocity is

$$U_2 = p_2^{+}(S_2/\rho c).$$ (C.3)

From equation (C.1) the volume velocity is the same on each side of the discontinuity for this method giving:

$$p_2^{+} = (p_1^{+} - p_1^{-})(S_1/S_2).$$ (C.4)

Now we turn to the first part of equation (C.1). Substituting in the pressure as the sum of the forward and backward going waves and putting $U = U_1 = (p_1^{+} - p_1^{-})(S_1/\rho c)$ gives

$$p_1^{+} + p_1^{-} = p_2^{+} + i \omega L (p_1^{+} - p_1^{-})(S_1/\rho c).$$ (C.5)

In order to work out the reflectance we need to remove $p_2^{+}$ to obtain an expression featuring only $p_1^{+}$ and $p_1^{-}$. We therefore substitute $p_2^{+}$ from equation () into equation (C.5).

$$p_1^{+} + p_1^{-} = (p_1^{+} - p_1^{-})(S_1/S_2) + i \omega L (p_1^{+} - p_1^{-})(S_1/\rho c).$$ (C.6)

Rearranging we get the reflection coefficient, the ratio of the reflected and incident waves:

$$\frac{p_1^{-}}{p_1^{+}} = \frac{(S_1/S_2) - 1 + i \omega L (S_1/\rho c)} {(S_1/S_2) + 1 + i \omega L (S_1/\rho c)}.$$ (C.7)

As with the multimodal method this may be expressed in terms of dimensionless variables $kR_1$ and $\beta = R_1/R_2$;

$$\frac{p_1^{-}}{p_1^{+}} = \frac{\beta^2 - 1 + i \pi (kR_1) (L R_1/\rho)} {\beta^2 + 1 + i \pi (kR_1) (L R_1/\rho)}$$ (C.8)

where $L R_1/\rho$ is a function of the dimensionless variables and is tabulated in table 2 (A) of [57]. This formula is used to calculate the inductance method reflectance shown in figure 6.1 of section 6.4.