Chapter 3

HELMHOLTZ SYSTEM

When all dimensions of a wave system are shorter than 1/4 wavelength of the applied frequencies, the impedance the system offers to oncoming waves is considered in terms of discrete values called “lumped  impedances.” Both the ripple tank and acoustic wave systems have Helmholtz type resonance action and utilize the lumped impedance viewpoint.

LUMPED IMPEDANCE 

Elements of Impedance 

Wave systems have impedance definable in terms of resistance, compliance and inertance: Resistance reduces the wave intensity over its length of travel. Its contribution to power dissipation is generally negligible in both the acoustic (Kinsler and Frey, 1950:205) and ripple tank Helmholtz resonance considerations.

Compliance (C) is the measure of apparent softness a discrete element presents to a wave pressure front. It is defined (Kinsler and Frey, 1950:203) as the volume displacement ΔV per change in pressure, ΔP. The property of the gravity wave that can be identified as a compliant element is a free surfaced column of liquid of surface area S and height h. Ripple tank compliance is in terms of free surface area and weight density.

Inertance (M) is the measure of mass loading a discrete element imposes on the wave pressure front. It is defined (Kinsler and Frey, 1950:203) as the effective inertial mass per unit area (m) divided by its area of transmission (A). The part of the gravity wave identifiable as the inertance element is a tube of cross-sectional area (A) and total effective length (L). Ripple tank inertance is in terms of liquid density (ρ), effective tube length, and cross-sectional area.

Ripple Wave Impedance 

The pressure required to displace an elemental volume (X) of fluid through a particular area (A) can be expressed as a sum of the pressures exerted by the discrete impedance elements which are affected by that volumetric displacement.

The compliance pressure is a direct function of the volumetric fluid velocity and flow cross-sectional area. The inertance pressure is due to the acceleration of the fluid through the flow area. The sum of these pressures can be written in terms of volumetric displacement, velocity and acceleration.

If the applied pressure is a sinusoidal function ( P = p·e^-jωt ), which is generally the case for both acoustic and ripple waves, the solution of the pressure equation is of the standard form.

The impedance of a fluid system to a pressure front is defined as the ratio of applied pressure to the volumetric velocity. An expression can be written in terms of the pressure equation solution:

Minimum impedance occurs at any given point where the reactance term (ωM – 1/ωC) is zero. The frequency of pressure oscillation associated with minimum impedance can be expressed in terms of inertance and compliance values for that point.

Helmholtz Resonance

The Helmholtz resonance is due to the combined effects of an inertial element in series with a compliance element. Minimum impedance at the entrance of this combination of elements will occur at a frequency given by Equation (6) which can be rewritten in terms of ripple tank quantities:

Variable Wave Speed 

The Helmholtz resonant frequency equation can be rewritten in terms of the wave speed   which presents the of a dependence on wave speed.

The real wave speed is less than the theoretical wave speed by a frequency dependent term (C_R = C_w – fh). If the resonance is truly a function of wave speed, a slower wave speed will result in a lower resonant frequency than predicted. The resonance equation can be rearranged into wave speed dependent and independent terms. The effective inertial length (L) is the sum of the real tube length(ℓ) and the end correction length (L_o) due to the entrance and exit effects.

By plotting the real inertance length (ℓ) against the applied (C/f = λ) wavelength squared term for both the real and theoretical wave speeds of an actual Helmholtz resonance experiment the curves of Figure 9 are obtained. A linear relationship is found to exist between the inertance length and the square of the theoretical wavelength and not the real wave speed. The Helmholtz resonance frequency is not dependent on real wave speed.

Acoustic Analogy 

An analogy between ripple tank and acoustic wave systems has been established in terms of frequency, geometric, and wave speed ratios.

This relationship is also derivable by using Helmholtz resonant frequency equations for air and the water ripple systems. Both systems have the same equations of the same form if considered in terms of wave speed (C), inertance area (A), inertance length (L), and compliance volume (V).

The ratio of frequencies results in a series of ratios of similar terms which, if expressed in terms of geometric ratios, yields the original analogy equation.

The ripple tank model of an acoustic Helmholtz resonant system is geometrically proportional in terms of compliant volume, inertance length, and entrance area. The equivalent frequency of the acoustic system is proportional to that of the ripple system through their wave speed and geometric ratios.