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9.10. Infinite conical horn

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} {(x / x_{T})}^{2}

$S (x) = S_{T} {(x / x_{T})}^{2}$

(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

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

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

(9.33)

where

k = \frac{2 π}{λ} = \frac{ω}{c}

$k = \frac{2 π}{λ} = \frac{ω}{c}$

(9.34)

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 is distance along the length of the horn from the apex in m.
x _T is distance from the apex to the throat in m.
S _T is cross-sectional area of the throat in m².
S is cross-sectional area at x in m².
Figure 9.9 Plot of normalized throat impedances for infinite parabolic (1), conical (2), exponential (3), and hyperbolic (4) horns using Eqs. (9.30, 9.37, 9.45, and 9.57) respectively. Real impedances S _T R _AT/(ρ ₀ c) are represented by solid curves and the imaginary impedances S _T X _AT/(ρ ₀ c) are represented by dashed curves. The value of α for the hyperbolic horn is ½. The cutoff frequencies of the parabolic, conical, exponential, and hyperbolic horns are 1182 Hz, 792 Hz, 337 Hz, and 399 Hz respectively.
The general solution for the pressure in a conical horn of any length is

\tilde{p} (x) = {\tilde{p}}_{+} \frac{e^{- j k x}}{x} + {\tilde{p}}_{-} \frac{e^{j k x}}{x}

$\tilde{p} (x) = {\tilde{p}}_{+} \frac{e^{- j k x}}{x} + {\tilde{p}}_{-} \frac{e^{j k x}}{x}$

(9.35)

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

\begin{matrix} \tilde{u} (x) = - \frac{1}{j k ρ_{0} c} \frac{\partial}{\partial x} \tilde{p} (x) \\ = \frac{1}{ρ_{0} c} {{\tilde{p}}_{+} (1 - \frac{j}{k x}) \frac{e^{- j k x}}{x} - {\tilde{p}}_{-} (1 + \frac{j}{k x}) \frac{e^{j k x}}{x}} . \end{matrix}

$\begin{matrix} \tilde{u} (x) = - \frac{1}{j k ρ_{0} c} \frac{\partial}{\partial x} \tilde{p} (x) \\ = \frac{1}{ρ_{0} c} {{\tilde{p}}_{+} (1 - \frac{j}{k x}) \frac{e^{- j k x}}{x} - {\tilde{p}}_{-} (1 + \frac{j}{k x}) \frac{e^{j k x}}{x}} . \end{matrix}$

(9.36)

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 impedance, which is the ratio of the pressure

\tilde{p}

$\tilde{p}$

to the volume velocity

\tilde{U}

$\tilde{U}$

at x = x _T, so that

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

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

(9.37)

This is the same as the radiation impedance of a pulsating sphere of radius x _T. The special case of kx _T = 1 occurs at the cutoff frequency, which we shall designate as f _c, where

f_{c} = \frac{c}{2 π x_{1}} .

$f_{c} = \frac{c}{2 π x_{1}} .$

(9.38)

The throat impedance of an infinite conical horn is plotted in Fig. 9.9.

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Table of Contents for Acoustics

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9.10. Infinite conical horn

Theoretical considerations

Throat impedance

Table of Contents for
Acoustics