Helmholtz Resonance - Quantitative Explanation

Quantitative Explanation

It can be shown that the resonant angular frequency is given by:

(rad/s),

where:

• (gamma) is the adiabatic index or ratio of specific heats. This value is usually 1.4 for air and diatomic gases.
• A is the cross-sectional area of the neck
• is the mass in the neck
• P₀ is the static pressure in the cavity
• V₀ is the static volume of the cavity

For cylindrical or rectangular necks, we have

,

where:

• L is the length of the neck
• is the volume of air in the neck

thus:

By the definition of density:, thus:

,

and

,

where:

• f_H is the resonant frequency (Hz)

The speed of sound in a gas is given by:

,

thus, the frequency of the resonance is:

The length of the neck appears in the denominator because the inertia of the air in the neck is proportional to the length. The volume of the cavity appears in the denominator because the spring constant of the air in the cavity is inversely proportional to its volume. The area of the neck matters for two reasons. Increasing the area of the neck increases the inertia of the air proportionately, but also decreases the velocity at which the air rushes in and out.

Depending on the exact shape of the hole, the relative thickness of the sheet with respect to the size of the hole and the size of the cavity, this formula can have limitations. More sophisticated formula can still be derived analytically, with similar physical explanations (although some differences matter). See for example the book by F. Mechels. Furthermore, if the mean flow over the resonator is high (typically with a Mach number above 0.3), some corrections must be applied.