Under the assumption that the dimensions of the sound hole are much sm than the
ID: 3308718 • Letter: U
Question
Under the assumption that the dimensions of the sound hole are much sm than the acoustic wavelength, the following equation holds hsmal mair (t) -CAP(t), where mair (t) is the rate of mass flow through the sound hole, C is called the conductance of the sound hole and P(t) is the pressure difference across the hole. Further mair(t) = AAV(t), where (t) is the rate of volum e flow through the sound hole and po is the density of air (i) If the volume of the air cavity of the resonator is V, and the sound hole has a conductance C, find its Helmholtz resonance frequency using the above equations. (Hint : Use B=pocy (ii) Let us now apply the formula you derived to the case of a violin. The total conductance C for the f-holes of a violin is found to be given by: C =- 0.185L 0.192 + 1.053 where Ly is the length of the sound hole and hsh is the thickness of the top plate of the violin at the location of the sound hole. For a typical violin, these values are obtained as L1 = 68 min and hah-2.5 mm. The volunne of the violin body V = 1.82 litres. Use these paramieters to estimate the Helmholtz resonance frequency of the violin. The experimental value for the resonance frequency is 280 Hz. Do the calculated and measured values agree? If not, why?Explanation / Answer
The relationship between the rate of mass flow and the rate of volume flow is already given. So write the rate of volume flow V'' (P).
V'' = (C/rho0)P
=(C/rho0) (P - dP)
The small change of pressure, dP resulting in the small change of volume, dV of air inside the volume,V is given by bulk modulus as dP = (beta)*dV.
Therefore,
V'' = (C/rho0)*(P - (beta)*dV)
Compare this relation with the Helmholtz resonator, V'' = (1/I)P - (1/K I)*dV where resonator frequency,
f = sqrt(1/K I)
We get Helmholtz frequency, f = sqrt[(beta*C)/rho0]
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