Loading & acoustics

Throat impedance

Acoustic load of a finite horn from areas and length, by conical-element transfer matrices and a piston mouth.

Sheet: 09 / 25
Units: Metric
Air: 20 °C

Fig. 1Throat acoustic impedance

ResistanceReactanceR, infinite horndrag on plot to read

Flare cut-off $f_c$: 89.0Hz
Mouth $ka=1$: 153Hz
Quarter wave $c/4L$: 71.5Hz
Throat, round: 101mm
Mouth, round: 71.4cm
Air volume: 120L

Fig. 2Equivalent round section

How it works

Enter a horn the way you would measure one: throat area, mouth area and axial length. The sheet computes the acoustic impedance the driver sees at the throat, normalised to $\rho c/S_t$ . A resistance near 1 means the horn presents the full characteristic load; reactance is energy sloshing back and forth without being radiated.

Transfer-matrix method

The flare is divided into short conical elements. Pressure $p$ and volume velocity $U$ at the two ends of an element are related by a two-by-two matrix, and impedances transform from the mouth end to the throat end as

\begin{bmatrix} p_1 \\ U_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix}\begin{bmatrix} p_2 \\ U_2 \end{bmatrix}, \qquad Z_1 = \frac{A\,Z_2 + B}{C\,Z_2 + D}

For a cone whose ends lie at distances $x_1$ and $x_2 = x_1 + L$ from its apex, with end areas $S_1$ and $S_2$ , the exact spherical-wave solution gives

\begin{aligned} A &= \frac{x_2}{x_1}\cos kL - \frac{\sin kL}{k x_1} & B &= j\,\frac{\rho c}{S_2}\,\frac{x_2}{x_1}\sin kL \\[4pt] C &= j\,\frac{S_1}{\rho c}\left[\left(\frac{x_2}{x_1}+\frac{1}{k^{2}x_1^{2}}\right)\sin kL - \frac{L}{k x_1^{2}}\cos kL\right] & D &= \frac{x_1}{x_2}\left(\cos kL + \frac{\sin kL}{k x_1}\right) \end{aligned}

Starting from the radiation impedance at the mouth, the calculator applies this transformation element by element until it reaches the throat. A conical horn needs one element; curved flares use 160.

Mouth termination and boundaries

The mouth is modelled as a rigid piston of equal area in an infinite baffle (2π steradians). For a mouth against one or two further boundaries, the mirror images are treated as extra pistons moving in phase, which is equivalent to a single piston of two or four times the area. This is accurate while the mouth is small compared with the wavelength, exactly where the boundaries matter.

Reading the plot

Below the flare cut-off the resistance collapses and the throat load is almost purely reactive.
Peaks and dips above cut-off are standing waves between mouth and throat; they are spaced about $c/2L$ apart.
The dashed curve is the same flare continued to infinity. A well-sized mouth makes the solid curve hug it.

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How it works

Transfer-matrix method

\begin{bmatrix} p_1 \\ U_1 \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix}\begin{bmatrix} p_2 \\ U_2 \end{bmatrix}, \qquad Z_1 = \frac{A\,Z_2 + B}{C\,Z_2 + D}

For a cone whose ends lie at distances $x_1$ and $x_2 = x_1 + L$ from its apex, with end areas $S_1$ and $S_2$ , the exact spherical-wave solution gives

\begin{aligned} A &= \frac{x_2}{x_1}\cos kL - \frac{\sin kL}{k x_1} & B &= j\,\frac{\rho c}{S_2}\,\frac{x_2}{x_1}\sin kL \\[4pt] C &= j\,\frac{S_1}{\rho c}\left[\left(\frac{x_2}{x_1}+\frac{1}{k^{2}x_1^{2}}\right)\sin kL - \frac{L}{k x_1^{2}}\cos kL\right] & D &= \frac{x_1}{x_2}\left(\cos kL + \frac{\sin kL}{k x_1}\right) \end{aligned}

Starting from the radiation impedance at the mouth, the calculator applies this transformation element by element until it reaches the throat. A conical horn needs one element; curved flares use 160.

Mouth termination and boundaries

Reading the plot

Below the flare cut-off the resistance collapses and the throat load is almost purely reactive.
Peaks and dips above cut-off are standing waves between mouth and throat; they are spaced about $c/2L$ apart.
The dashed curve is the same flare continued to infinity. A well-sized mouth makes the solid curve hug it.

Throat impedance

Inputs

How it works

Transfer-matrix method

Mouth termination and boundaries

Reading the plot

Inputs

How it works

Transfer-matrix method

Mouth termination and boundaries

Reading the plot

§ How it works

Transfer-matrix method

Mouth termination and boundaries

Reading the plot

§ How it works

Transfer-matrix method

Mouth termination and boundaries

Reading the plot

How it works

How it works