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Physics is not my field and I'm a bit over my head. I need the formula for a computer program related to musical instruments--this is the only thing holding me up. It seems like it would be easy to derive from the standard formula, but it's not--at least not for me.

The only solutions I've been able to find in the literature are for special cases (2 identical necks, or 2 necks the same length), There is lots of "hand waving" about the analogy with electrical impedance, which I know how to calculate: R = 1 / ( 1/R1 + 1/R2 + ... 1/Rn) where R1...R are resistors connected in parallel. But I have not been able to find a formula for the general case in a publically-accessible paper.

Unfortunately, the term for acoustical impedance, (A * A) / m, has been factored out of the very simple equation for approximating the resonant frequency of a Helmholtz resonator:

F = (c / 2 * pi) * SQRT( A / (V * L'))

where c is the speed of sound, A is thecross-section area of neck, V is volume in the cavity, and L' is effective length of the neck.

Effective length is the fly in the ointment. It is give by L' = L + (k * a), where L is the physical length, a is the neck radius, and k is a empirically-determined constant. So even if all necks are the same physical length, their effective length will differ if their diameters differ. Somehow, effective length needs to be applied to area for each individual neck.

Backing up to the formula for angular frequency -- from which the standard formula is derived--makes for a messy equation with variables that are inconvenient to measure or not that signficant (i.e. changes in barometric pressure do not signficantly affect resonant frequency):

O = SQRT( g * (A * A) / m * P / V)

O (omega) is angular frequency in rad/s, g (gamma) is adiabatic index, m is mass of air in the neck, V is static pressure and A is as defined above. It seems to me if the single neck solution doesn't require

the adiabatic index or staic pressure, the multi-neck solution shouldn't either. From a practical standpoint, the only external variable significantly affecting the resonant frequency of a given Helmholtz resonator in air is the termperature (represented in the standard formula by the speed of sound).

Any help towards a practical formula would be greatly appreciated.