Figure 1 shows the geometry of an idealised Helmholtz resonator. A vessel with rigid walls encloses a volume $V$, and is connected to the outside world via a neck of length $L$ and cross-sectional area $S$. As usual, we assume that the pressure and density take the forms $p(t)=p_0+p'(t)$ and $\rho(t)=\rho_0+\rho'(t)$, where subscripts 0 denote the steady atmospheric pressure and density, and primes denote the acoustic perturbations, assumed to be small in comparison. We consider the case where the acoustic pressure is $p’_1(t)$ just outside the neck of the vessel, and we aim to calculate the corresponding pressure $p’_2(t)$ inside the volume. We define the mass flow rate inwards through the neck to be $q(t)$.
Conservation of mass requires that
$$q=V\dfrac{\partial \rho’}{\partial t} \tag{1}$$
so for a harmonic disturbance at angular frequency $\omega$,
$$q=V i \omega \rho’ . \tag{2}$$
This density change produces a proportional change in pressure, and we already know from section 4.1.1 that the constant of proportionality is the square of the speed of sound $c$:
$$p’_2=c^2 \rho’_2 = \dfrac{c^2 q}{i \omega V} . \tag{3}$$
Now if the “plug” of air in the neck moves as a rigid body, Newton’s law requires
$$\left( p’_1 – p’_2 \right) S = \rho_0 L S \dfrac{\partial u}{\partial t} \tag{4}$$
where $u$ is the velocity of the plug into the neck. But $q=\rho_o S u$, so
$$p’_1 = \dfrac{c^2 q}{i \omega V} + \dfrac{L}{S} i \omega q . \tag{5}$$
Thus
$$q= -\dfrac{i \omega S}{L} \dfrac{p’_1}{\left[\omega^2 -\frac{c^2 S}{VL}\right]} \tag{6}$$
and so finally
$$p’_2= -\dfrac{c^2 S}{VL} \dfrac{p’_1}{\left[\omega^2 -\frac{c^2 S}{VL}\right]} . \tag{7}$$
Resonance obviously occurs when
$$\omega = c \sqrt{\frac{S}{VL}} .\tag{8}$$
In this very simple theoretical model the response is infinite at resonance. As usual, the amplitude is limited in practice by damping, which in this case arises from viscous losses and from sound radiation.
To calculate a Helmholtz resonance frequency more accurately we would need to include an end correction: the effective mass of air moving in and around the neck is a little larger than the geometric dimensions of the neck itself, because of the kinetic energy involved in the air flow near the ends of the neck. This is usually allowed for by adding a small extra length to the real length. The magnitude of the required correction depends on the geometry around the opening, and requires numerical calculation. Two simple idealised cases give an idea of the magnitude: they involve a circular opening of radius $a$, and a neck which is either “flanged” or “unflanged”, as sketched in Fig. 2.
One consequence is that even a thin-walled vessel with an opening can show a Helmholtz resonance: an example is sketched in Fig. 3. For this case there is a “flange” on both sides of the opening, so the effective neck length for a circular opening of radius $a$ will be $L \approx 2 \times 0.85 a=1.7 a$.
