4.2.3 The Webster horn equation

We are interested in acoustic waves in a horn with a varying cross-sectional area $S(x)$. We can give a simple derivation of the governing equation, based on assuming that the acoustic disturbance still takes the form of an approximately plane wave. This approximation is good when the section varies slowly, but for accurate answers it needs more care as the bell is approached and the horn flares more abruptly. More detail on this, and many other aspects of the acoustics of horns, can be found in the books by Fletcher and Rossing [1], and Chaigne and Kergomard [2].

We follow the same steps as in section 4.1.1. Consider a small element of the air inside a pipe with cross-sectional area $S(x)$, lying between positions $x$ and $x+\delta x$, as sketched in Fig. 1. As a result of the sound wave, the particles that were initially at the ends of this element are displaced by a small distance, to $x+\xi(x)$ and $x + \delta x + \xi(x + \delta x)$ respectively.

Figure 1. A small element of the acoustic wavefield in a tube with varying cross-section, showing particle displacements.

We can apply conservation of mass to this element of air: the product of density and volume must remain constant, so

$$\rho_0 S \delta x = (\rho_0 + \rho’) \left[ S(x) \delta x – S(x) \xi(x) +S(x+ \delta x) \xi(x+ \delta x) \right] $$

$$\approx (\rho_0 + \rho’) \left[ S \delta x + \frac{\partial}{\partial x} \left(S \xi \right) \delta x \right] . \tag{1}$$

Cancelling $\delta x$ and ignoring products of small quantities, this reduces to

$$0 \approx S \rho’ + \rho_0 \frac{\partial}{\partial x} \left(S \xi \right) . \tag{2}$$

As before, we can assume that $p’ = c^2 \rho’$ in terms of the pressure change $p’$, and so

$$p’ = -\dfrac{c^2 \rho_0}{S}\frac{\partial}{\partial x} \left(S \xi \right) . \tag{3}$$

We can obtain a second relation between $p’$ and $\xi$ by applying Newton’s law to our small volume. The pressure acts on the two end faces, giving a net force which must be balanced by mass times acceleration. However, because of the varying cross-section there is also a contribution to the force from the pressure acting on the sloping walls of the element. This can be conveniently expressed as the mean pressure multiplied by the projected area of the element side-walls in the axial direction. The resulting equation is

$$S(x) [p_0+p'(x)] -S(x+ \delta x) [p_0+p'(x+ \delta x)] + $$

$$(p_0 +p’) [S(x+ \delta x) -S(x)]= \rho_0 S \delta x \frac{\partial^2 \xi}{\partial t^2} \tag{4}$$

so that

$$- S\frac{\partial p’}{\partial x} \approx \rho_0 S \frac{\partial^2 \xi}{\partial t^2} .\tag{5}$$

Differentiating this with respect to $x$ and then using eq. (3), we obtain

$$\dfrac{\partial}{\partial x} \left( S\frac{\partial p’}{\partial x} \right) = -\rho_0 \frac{\partial^2}{\partial t^2} \left[ \frac{\partial }{\partial x} (S \xi) \right]= \dfrac{S}{c^2} \frac{\partial^2 p’}{\partial t^2} \tag{6}$$

which is the Webster horn equation.

We can learn something useful by making a change of variables in this equation: set $p’=\psi(x) S^{-1/2}$, and also also write the area in terms of a radius via $S(x)=\pi a^2(x)$. Finally, we are interested in modes and therefore we can assume harmonic time dependence $e^{i \omega t}$. Equation (6) can then be rearranged into a form first suggested by Benade and Jansson [3]:

$$\dfrac{d^2 \psi}{dx^2} + \left( \frac{\omega^2}{c^2} – \frac{1}{a} \dfrac{d^2a}{dx^2} \right) \psi = 0. \tag{7}$$

This is the familiar simple harmonic equation. If the term in brackets is positive, it has sinusoidal solutions which suggest local behaviour corresponding to travelling waves. However, if that bracketed term is negative, the solutions give exponential growth or decay, suggesting that the local behaviour might be evanescent in character. For a given frequency $\omega$, if this bracketed term reaches zero at some position along the tube, that gives a limit beyond which waves should not propagate.

In order to produce the numerical example shown in the main text of section 4.2, a broadly plausible bore profile $a(x)$ was chosen (based on suggestions from the literature [1]), and then a very simple finite-difference approach was used. The pressure was discretised with a chosen step length, and the derivatives on the left-hand side of eq. (6) were expressed via central differences of these discrete values. Blocked and free boundary conditions were respectively imposed at the two ends of the tube. The resulting discrete equations were used to populate a stiffness matrix, whose eigenvalues and eigenvectors were then computed. Convergence tests were used to find an appropriate step length, giving a good compromise between accuracy and conditioning of the matrix.

The specifics of the chosen bore profile are as follows. A straight tube of length 0.5 m and radius 1 cm was joined to a second 0.5 m length of tube in which the radius varied according to a power law with distance from the bell (a so-called Bessel horn [1]). The chosen power was -0.85, so that the behaviour was a little less extreme than a hyperbola. The function was truncated 5 mm before the singularity to give a finite bell of plausible-looking shape. With these parameters, the fundamental frequency was around 122 Hz.

The function $\frac{1}{a} \frac{d^2a}{dx^2}$ appearing in eq. (7) was, of course, zero throughout the straight section of pipe. Then, after a blip at the junction with the flaring pipe, it was monotonically increasing all the way to the bell. This function was used to compute the critical position along the bore based on the natural frequency of each mode, corresponding to the transition of behaviour of eq. (7). These are the positions plotted as vertical lines in Fig. 15 of section 4.2. The systematic variation of these lines with mode number is a direct consequence of the monotonic function.

[1] Neville H Fletcher and Thomas D Rossing; “The physics of musical instruments”, Springer-Verlag (Second edition 1998)

[2] Antoine Chaigne and Jean Kergomard; “Acoustics of musical instruments”, Springer/ASA press (2013)

[3] Arthur H. Benade and Erik V. Jansson; On plane and spherical waves in horns with nonuniform flare, Acustica 31, 80–98 (1974).