Acoustics

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9.12. Infinite hyperbolic horn (hypex) [12]

Theoretical considerations

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

S (x) = S_{T} {(\cosh \frac{x}{x_{T}} + α \sinh \frac{x}{x_{T}})}^{2}

$S (x) = S_{T} {(\cosh \frac{x}{x_{T}} + α \sinh \frac{x}{x_{T}})}^{2}$

(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}} + \frac{2}{x_{T}} \cdot \frac{\sinh (x / x_{T}) + α \cosh (x / x_{T})}{\cosh (x / x_{T}) + α \sinh (x / x_{T})} \cdot \frac{\partial}{\partial x} + k^{2}) \tilde{p} (x) = 0

$(\frac{\partial^{2}}{\partial x^{2}} + \frac{2}{x_{T}} \cdot \frac{\sinh (x / x_{T}) + α \cosh (x / x_{T})}{\cosh (x / x_{T}) + α \sinh (x / x_{T})} \cdot \frac{\partial}{\partial x} + k^{2}) \tilde{p} (x) = 0$

(9.52)

where

k = \frac{2 π}{λ} = \frac{ω}{c}

$k = \frac{2 π}{λ} = \frac{ω}{c}$

(9.53)

and

$\tilde{p}$ $\tilde{p}$ is harmonically varying sound pressure at a point along the length of the horn in Pa. (It is assumed that the pressure is uniform across the cross section of the horn.)
c is speed of sound in m/s.
x _T is reference axial distance from the throat in m.
x is distance along the length of the horn from the throat in m.
α is parameter that never exceeds unity.
S _T is cross-sectional area of the throat in m².
S is cross-sectional area at x in m².

The general solution for the pressure in an hyperbolic horn of any length is

\tilde{p} (x) = \frac{1}{\cosh (x / x_{T}) + α \sinh (x / x_{T})} ({\tilde{p}}_{+} e^{- j k x \sqrt{1 - \frac{1}{k^{2} x_{T}^{2}}}} + {\tilde{p}}_{-} e^{j k x \sqrt{1 - \frac{1}{k^{2} x_{T}^{2}}}})

$\tilde{p} (x) = \frac{1}{\cosh (x / x_{T}) + α \sinh (x / x_{T})} ({\tilde{p}}_{+} e^{- j k x \sqrt{1 - \frac{1}{k^{2} x_{T}^{2}}}} + {\tilde{p}}_{-} e^{j k x \sqrt{1 - \frac{1}{k^{2} x_{T}^{2}}}})$

(9.54)

where

{\tilde{p}}_{+}

${\tilde{p}}_{+}$

denotes the pressure amplitude of the forward traveling wave and

{\tilde{p}}_{-}

${\tilde{p}}_{-}$

that of the backwards traveling wave. The tilde replaces the factor e ^jωt. Using Eq. (2.122), the velocity is given by

\tilde{u} (x) = - \frac{1}{j k ρ_{0} c} \frac{\partial}{\partial x} \tilde{p} (x) .

$\tilde{u} (x) = - \frac{1}{j k ρ_{0} c} \frac{\partial}{\partial x} \tilde{p} (x) .$

(9.55)

Throat impedance

Noting that in an infinite horn there are no reflections from the mouth, we set

{\tilde{p}}_{-} = 0

${\tilde{p}}_{-} = 0$

to obtain the acoustic throat admittance, which is the ratio of the volume velocity

\tilde{U}

$\tilde{U}$

to the pressure

\tilde{p}

$\tilde{p}$

at x = x _T, so that

\begin{matrix} Y_{A T} = \frac{\tilde{U} (x_{T})}{\tilde{p} (x_{T})} = \frac{S_{T} \tilde{u} (x_{T})}{\tilde{p} (x_{T})} = \frac{S_{T}}{ρ_{0} c} (\sqrt{1 - \frac{1}{k^{2} x_{T}^{2}}} - j \frac{α}{k x_{T}}) m^{5} / N \cdot s \\ = G_{A T} + j B_{A T} . \end{matrix}

$\begin{matrix} Y_{A T} = \frac{\tilde{U} (x_{T})}{\tilde{p} (x_{T})} = \frac{S_{T} \tilde{u} (x_{T})}{\tilde{p} (x_{T})} = \frac{S_{T}}{ρ_{0} c} (\sqrt{1 - \frac{1}{k^{2} x_{T}^{2}}} - j \frac{α}{k x_{T}}) m^{5} / N \cdot s \\ = G_{A T} + j B_{A T} . \end{matrix}$

(9.56)

The acoustic impedance Z _AT = 1/Y _AT at the throat is

\begin{matrix} Z_{A T} = \frac{ρ_{0} c}{S_{T}} {(\sqrt{1 - \frac{1}{k^{2} x_{T}^{2}}} - j \frac{α}{k x_{T}})}^{- 1} N \cdot s / m^{5} \\ = R_{A T} + j X_{A T} . \end{matrix}

$\begin{matrix} Z_{A T} = \frac{ρ_{0} c}{S_{T}} {(\sqrt{1 - \frac{1}{k^{2} x_{T}^{2}}} - j \frac{α}{k x_{T}})}^{- 1} N \cdot s / m^{5} \\ = R_{A T} + j X_{A T} . \end{matrix}$

(9.57)

The real and imaginary parts of Z _AT and Y _AT behave alike with frequency and differ only by the magnitude (S/ρ ₀ c)² and the sign of the imaginary part. Note also that like with an exponential horn, but unlike the parabolic or conical horns, this impedance is independent of the distance x along the axis of the horn.

Cutoff frequency

The special case of x _T = λ/2π occurs at a frequency that we shall designate f _c, where

f_{c} = \frac{c}{2 π x_{T}} .

$f_{c} = \frac{c}{2 π x_{T}} .$

(9.58)

This frequency f _c is called the cutoff frequency because, for frequencies lower than this, no power will be transmitted down the horn, i.e., the impedance at all positions along the horn is purely reactive. The throat impedance of an infinite hyperbolic horn is plotted in Fig. 9.9.

In Fig. 9.9, the throat impedances for the parabolic, conical, exponential, and hyperbolic horn types are shown. At very high frequencies, all these types behave about alike. At low frequencies, however, there are considerable differences. These differences can be shown by comparison of the throat impedances for the conical and hyperbolic horns with that for the exponential horn.

For all horns, the throat resistance is very low, or zero, below the cutoff frequency. Above the cutoff frequency, the specific throat resistance rises rapidly to its ultimate value of ρ ₀ c for those cases where the rate of taper is small near the throat of the horn. For example, the specific throat resistance for the hyperbolic horn reaches ρ ₀ c at about one-twentieth the frequency at which the specific throat resistance for the conical horn reaches this value. Similarly for the hyperbolic horn, the specific throat resistance approaches unity at about one-third the frequency for the exponential horn.

It would seem that for best loading conditions on the horn drive unit over the frequency range above the cutoff frequency, one should use the hyperbolic horn. However, it should also be remembered that the nonlinear distortion will be higher for the hyperbolic horn because the wave travels further in the horn before the pressure drops off owing to area increase than is the case for the other horns. For minimum distortion at given power per unit area, the conical horn is obviously the best of the three. The exponential horn is usually a satisfactory compromise in design because it falls between these two extremes.

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Table of Contents for Acoustics

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9.12. Infinite hyperbolic horn (hypex) [12]

Theoretical considerations

Throat impedance

Cutoff frequency

Table of Contents for
Acoustics