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9.9. Infinite parabolic horn [11]

Theoretical considerations

A parabolic horn can either have a rectangular cross section with two parallel straight walls and two nonparallel straight walls, or a circular cross section with a curved wall following a parabola. The former gives more accurate results using the one-dimensional wave equation and is easier to construct, but in the figures we shall use the latter for convenience. 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}

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

(9.25)

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 parabolic horn is obtained by inserting S(x) from Eq. (9.25) into Eq. (2.27) to yield

(\frac{\partial^{2}}{\partial x^{2}} + \frac{1}{x} \frac{\partial}{\partial x} + k^{2}) \tilde{p} (x) = 0

$(\frac{\partial^{2}}{\partial x^{2}} + \frac{1}{x} \frac{\partial}{\partial x} + k^{2}) \tilde{p} (x) = 0$

(9.26)

where

k = \frac{2 π}{λ} = \frac{ω}{c}

$k = \frac{2 π}{λ} = \frac{ω}{c}$

(9.27)

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.

The general solution for the pressure in a parabolic horn of any length is

\tilde{p} (x) = {\tilde{p}}_{+} H_{0}^{(2)} (k x) + {\tilde{p}}_{-} H_{0}^{(1)} (k x)

$\tilde{p} (x) = {\tilde{p}}_{+} H_{0}^{(2)} (k x) + {\tilde{p}}_{-} H_{0}^{(1)} (k x)$

(9.28)

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}{j ρ_{0} c} ({\tilde{p}}_{+} H_{1}^{(2)} (k x) + {\tilde{p}}_{-} H_{1}^{(1)} (k x)) . \end{matrix}

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

(9.29)

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})} = j \frac{ρ_{0} c}{S_{T}} \frac{H_{0}^{(2)} (k x_{T})}{H_{1}^{(2)} (k x_{T})} \\ = \frac{ρ_{0} c}{S_{T}} (\frac{2}{π k x_{T} (J_{1}^{2} (k x_{T}) + Y_{1}^{2} (k x_{T}))} + j \frac{J_{0} (k x_{T}) J_{1} (k x_{T}) + Y_{0} (k x_{T}) Y_{1} (k x_{T})}{J_{1}^{2} (k x_{T}) + Y_{1}^{2} (k x_{T})}) N \cdot s / m^{5} \end{matrix}

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

(9.30)

where we have used the relationships of Eqs. (A2.75) and (A2.111) from Appendix II. This is the same as the radiation impedance of an infinitely long pulsating cylinder of radius x _T. If we equate the real and imaginary parts of the impedance, we find that the cutoff frequency occurs at kx _T = 0.268, which we shall designate as f _c, where

f_{c} = \frac{0.268 c}{2 π x_{T}} .

$f_{c} = \frac{0.268 c}{2 π x_{T}} .$

(9.31)

Figure 9.8 Plot of the quantities A and B, which are defined by the relations given on the graph.

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

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9.9. Infinite parabolic horn [11]

Theoretical considerations

Throat impedance

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Acoustics