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

7.11 The Helmholtz Resonator 1457.11 The Helmholtz ResonatorMany analyses of acoustic devices become simplified with the assumption that thewavelength in the propagation fluid is much greater than the dimensions of thedevices. If the wavelength is indeed much larger in all dimensions, the acousticvariable may be time varying, but it is virtually independent of the distance withinthe confines of the device. In such a case, the device can be viewed as a harmonicoscillator with one degree of freedom; and in the long-wavelength limit, such adevice can be considered a lumped acoustic element. An example of this typeof device is the Helmholtz resonator, illustrated in Figure 7.6 (three types areshown). The resonator is a rigid-walled cavity of volume V , with a neck of area Aand length L.According to theory, if λ ≫ L, the fluid in the neck behaves somewhat as asolid plug which vibrates back and forth. As the fluid (usually air) moves back andforth, the acoustic energy converts to heat as a result of friction along the neck.These losses can be increased by placing a light porous material across the neckor by placing the material inside the volume. Maximum sound absorption occursat the resonant frequency of the mass of air in the neck with the “spring” suppliedby the air resistance inside the enclosed volume. Very little sound is absorbed atother frequencies. However, for necks greater than 1 cm in diameter, the viscouslosses are considerably less than those associated with radiation. For the purposeof this analysis we can ignore the effects of viscosity in analyzing the Helmholtzresonator viewed as an analogous spring-mass system. The air in the neck has atotal effective massm = ρ 0 AL ′ (7.56)where the effective neck length L ′ is longer than the physical length L becauseof its radiation mass loading. We have seen earlier in this chapter that at lowfrequencies a circular opening of radius a is loaded with a radiation mass equalto that of the fluid occupying area πa 2 ; and effective length 8a/(3π) = 0.85a ifterminated in a wide flange and 0.6a for an unflanged terminal. We assume themass loading at the inner end of the neck is equivalent to a flanged termination.Figure 7.6. Three simple Helmholtz resonators.