The equation describing the cross-sectional area S(x) as a function of the distance x along the axis is

$S\left(x\right)={S}_{T}{e}^{mx}$

(9.39)

where S
_{
T
} is the area of the throat, which is located at x
=
0. In the steady state, the Helmholtz equation for the exponential horn is obtained by inserting S(x) from Eq. (9.39) into Eq. (2.27) to yield

$\left(\frac{{\partial}^{2}}{\partial {x}^{2}}+m\frac{\partial}{\partial x}+{k}^{2}\right)\tilde{p}\left(x\right)=\text{0}$

(9.40)

where

$k=\frac{2\pi}{}$

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