Acoustics

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9.11. Infinite exponential 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} e^{m x}

$S (x) = S_{T} e^{m x}$

(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

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

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

(9.40)

where

k = \frac{2 π}{λ} = \frac{ω}{c}

$k = \frac{2 π}{λ} = \frac{ω}{c}$

(9.41)

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 throat in m.
m is flare constant in m⁻¹. Obviously, m determines the magnitude of the second term of the equation above, which expresses the rate at which the sound pressure changes with distance down the horn. If m = 0, Eq. (9.40) becomes the equation for propagation in a cylindrical tube, i.e., a horn with zero flare.
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 exponential horn of any length is

\tilde{p} (x) = e^{- m x / 2} ({\tilde{p}}_{+} e^{- j k x \sqrt{1 - \frac{m^{2}}{4 k^{2}}}} + {\tilde{p}}_{-} e^{j k x \sqrt{1 - \frac{m^{2}}{4 k^{2}}}})

$\tilde{p} (x) = e^{- m x / 2} ({\tilde{p}}_{+} e^{- j k x \sqrt{1 - \frac{m^{2}}{4 k^{2}}}} + {\tilde{p}}_{-} e^{j k x \sqrt{1 - \frac{m^{2}}{4 k^{2}}}})$

(9.42)

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{e^{- m x / 2}}{ρ_{0} c} {{\tilde{p}}_{+} (\sqrt{1 - \frac{m^{2}}{4 k^{2}}} - j \frac{m}{2 k}) e^{- j k x \sqrt{1 - \frac{m^{2}}{4 k^{2}}}} - {\tilde{p}}_{-} (\sqrt{1 - \frac{m^{2}}{4 k^{2}}} + j \frac{m}{2 k}) e^{j k x \sqrt{1 - \frac{m^{2}}{4 k^{2}}}}} . \end{matrix}

$\begin{matrix} \tilde{u} (x) = - \frac{1}{j k ρ_{0} c} \frac{\partial}{\partial x} \tilde{p} (x) \\ = \frac{e^{- m x / 2}}{ρ_{0} c} {{\tilde{p}}_{+} (\sqrt{1 - \frac{m^{2}}{4 k^{2}}} - j \frac{m}{2 k}) e^{- j k x \sqrt{1 - \frac{m^{2}}{4 k^{2}}}} - {\tilde{p}}_{-} (\sqrt{1 - \frac{m^{2}}{4 k^{2}}} + j \frac{m}{2 k}) e^{j k x \sqrt{1 - \frac{m^{2}}{4 k^{2}}}}} . \end{matrix}$

(9.43)

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{m^{2}}{4 k^{2}}} - j \frac{m}{2 k}) m^{5} \cdot N^{- 1} \cdot S^{- 1} \\ = G_{A T} + j B_{A T} . \end{matrix}

$\begin{matrix} Y_{A T} = \frac{\tilde{U} (x_{T})}{\tilde{p} (x_{T})} = \frac{S_{T} \tilde{u} (x_{T})}{\tilde{p} (x_{T})} = \frac{S_{T}}{ρ_{0} c} (\sqrt{1 - \frac{m^{2}}{4 k^{2}}} - j \frac{m}{2 k}) m^{5} \cdot N^{- 1} \cdot S^{- 1} \\ = G_{A T} + j B_{A T} . \end{matrix}$

(9.44)

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

\begin{matrix} Z_{A T} = \frac{ρ_{0} c}{S_{T}} (\sqrt{1 - \frac{m^{2}}{4 k^{2}}} + j \frac{m}{2 k}) \\ = R_{A T} + j X_{A T} N \cdot s / m^{5} . \end{matrix}

$\begin{matrix} Z_{A T} = \frac{ρ_{0} c}{S_{T}} (\sqrt{1 - \frac{m^{2}}{4 k^{2}}} + j \frac{m}{2 k}) \\ = R_{A T} + j X_{A T} N \cdot s / m^{5} . \end{matrix}$

(9.45)

The real and imaginary parts of Z 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, unlike with the parabolic or conical horns, this impedance is independent of the distance x along the axis of the horn. Next, we shall examine how varying the flare constant m affects the acoustic impedance Z _AT.

Flare constant and throat impedance

When the flare constant m is greater than 4π divided by the wavelength (m > 2k, low frequencies), the acoustic resistance R _AT and the acoustic reactance X _AT, at the throat of the horn where the area is S _T, are

\begin{matrix} R_{A T} = 0 \\ X_{A T} = \frac{ρ_{0} c}{S_{T}} (\frac{m}{2 k} - \sqrt{\frac{m^{2}}{4 k^{2}} - 1}) . \end{matrix}

$\begin{matrix} R_{A T} = 0 \\ X_{A T} = \frac{ρ_{0} c}{S_{T}} (\frac{m}{2 k} - \sqrt{\frac{m^{2}}{4 k^{2}} - 1}) . \end{matrix}$

(9.46)

When the flare constant m equals 4π divided by the wavelength, the acoustic resistance, and reactance are

\begin{matrix} R_{A T} = 0 \\ X_{A T} = \frac{ρ_{0} c m}{2 k S_{T}} = \frac{ρ_{0} c}{S_{T}} . \end{matrix}

$\begin{matrix} R_{A T} = 0 \\ X_{A T} = \frac{ρ_{0} c m}{2 k S_{T}} = \frac{ρ_{0} c}{S_{T}} . \end{matrix}$

(9.47)

For all cases where m is less than 4π divided by the wavelength (m < 2k, high frequencies), the acoustic resistance and reactance at any point x along the horn where the cross-sectional area is S are

\begin{matrix} R_{A T} = \frac{ρ_{0} c}{S_{T}} \sqrt{1 - \frac{m^{2}}{4 k^{2}}} \\ X_{A T} = \frac{ρ_{0} c m}{2 k S_{T}} = \frac{ρ_{0} c^{2} m}{2 ω S_{T}} \equiv \frac{1}{ω C_{A T}} \end{matrix}

$\begin{matrix} R_{A T} = \frac{ρ_{0} c}{S_{T}} \sqrt{1 - \frac{m^{2}}{4 k^{2}}} \\ X_{A T} = \frac{ρ_{0} c m}{2 k S_{T}} = \frac{ρ_{0} c^{2} m}{2 ω S_{T}} \equiv \frac{1}{ω C_{A T}} \end{matrix}$

(9.48)

where C _AT = 2S _T/ρ ₀ c ² m.

For very high frequencies, the reactance approaches zero and the resistance approaches ρ ₀ c/S _T or ρ ₀ c/S in general. This is also the impedance for a plane progressive sound wave in a tube of uniform cross section S.

Cutoff frequency

The special case of m = 4π/λ occurs at a frequency that we shall designate as f _c, where

f_{c} = \frac{m c}{4 π} .

$f_{c} = \frac{m c}{4 π} .$

(9.49)

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 (see Eq. 9.46). The throat impedance of an infinite exponential horn is plotted in Fig. 9.9.

To obtain the acoustic impedance at the throat of the horn in terms of the cutoff frequency, we observe that f _c/f = m/2k. Substituting in Eq. (9.45) yields

Z_{A T} = \frac{ρ_{0} c}{S_{T}} (\sqrt{1 - {(\frac{f_{c}}{f})}^{2}} + j \frac{f_{c}}{f}) = R_{A T} + j X_{A T}

$Z_{A T} = \frac{ρ_{0} c}{S_{T}} (\sqrt{1 - {(\frac{f_{c}}{f})}^{2}} + j \frac{f_{c}}{f}) = R_{A T} + j X_{A T}$

(9.50)

where

S _T is throat area in m².
ρ ₀ c is characteristic impedance of air in rayls.
f _c is cutoff frequency.
f is driving frequency.

Graphs of two quantities A and B that are directly proportional to the resistive and reactive parts of the acoustic impedance at the throat of an infinitely long exponential horn are shown in Fig. 9.8. The quantities A and B also are directly proportional to the real and imaginary parts of the acoustic admittance at the throat. The relations among A, B, R _AT, X _AT, G _AT, and B _AT are given on the graph. When the frequency is greater than approximately double the cutoff frequency f _c, the throat impedance is substantially resistive and very near its maximum value in magnitude.

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Table of Contents for Acoustics

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9.11. Infinite exponential horn

Theoretical considerations

Throat impedance

Flare constant and throat impedance

Cutoff frequency

Table of Contents for
Acoustics