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

$S\left(x\right)={S}_{T}{\left(\mathrm{cosh}\frac{x}{{x}_{T}}+\alpha \mathrm{sinh}\frac{x}{{x}_{T}}\right)}^{2}$

(9.51)

where S
_{
T
} is the area of the throat, which is located at x
=
0 and 0
≤
α
≤
1. We can vary the parameter α to create any profile between hyperbolic (α
=
0) and exponential (α
=
1). In the steady state, the Helmholtz equation for the hyperbolic horn is obtained by inserting S(x) from Eq. (9.51) into Eq. (2.27) to yield

$(\frac{{\partial}^{2}}{\partial {x}^{2}}$

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