THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

7.11 The Helmholtz Resonator 147Because f is periodic with angular frequency ω, δ = Be iωt must be the solutionwhere B exists as a complex constant. Then inserting the solution into Equation(7.62) results in(−Bω 2 m + iBωR r + BS)e iωt = APe iωtSolving for B provides the following complex displacement:δ =[iω R r + iAPe iωt(mω − S ω)] (7.63)Differentiating Equation (7.63) yields the complex speed of the fluid in the resonator’sneck,⃗u =R r + iAPe iωt(mω − S ω) (7.64)The impedance found from the relation Z m = f/⃗u, and making use of Equations(7.61) and (7.64) yields(Z m = R r + i mω − S )(7.65)ωwhich means that the mechanical reactance isX m = mω − S ωResonance occurs when the reactance becomes zero, i.e.,√ √√Sω 0 =m = ρ 0 c 2 A 2 /V A= c(7.66)ρ 0 L ′ A L ′ VWe note that the resonance frequency depends on the volume of the cavity, noton the shape. Experimentation with differently shaped resonators having the sameS/L ′ V ratios indicate identical resonant frequencies. This holds true as long asall dimensions of the cavity are appreciably less than a single wavelength and theopening is quite small. There are additional frequencies in Helmholtz resonatorshigher than that given by Equation (7.66), which arise from standing waves in thecavity rather than from the oscillatory motion of the mass of air in the neck. Theseovertone frequencies are not harmonics of the driving fundamental, and the firstovertone may be several times the frequency of the fundamental.The sharpness (or narrowness of spread) of the resonance of a Helmholtz resonatoris define by the quality factor Q,√Q = mω ( )0L′ 3= 2π V(7.67)R rAAssumptions have been made, there are no losses other than those arising fromacoustic radiation and the termination of the neck is flanged.