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

$S\left(x\right)={S}_{T}{\left(x/{x}_{T}\right)}^{2}$

(9.32)

where S
_{
T
} is the area of the throat, which is located at a distance x
=
x
_{
T
} ahead of the apex at x
=
0. In the steady state, the Helmholtz equation for the conical horn is obtained by inserting S(x) from Eq. (9.32) into Eq. (2.27) to yield

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

(9.33)

where

$k=\frac{2\pi}{\lambda}=\frac{\omega}{c}$

(9.34)

and

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