This week we continue our Ripple Tank series with another snippet of an important thesis paper, one that launched the career path of ASC founder, president and TubeTrap inventor, Art Noxon, PE Acoustical

The inertance tube is not necessary for Helmholtz resonance since the port to the compliant volume has its own effective inertial length, L_o. A series of experiments was conducted in which the resonant frequencies for a variety of compliant volumes were determined while maintaining a constant port configuration. Analysis of the data established the effective inertial length of the port to be independent of compliant volume, as it is assumed to be. The test apparatus is illustrated in figure 11.

At one end of a 2-foot-long, 3/8 in. wide ripple channel is positioned the wave drive. The other end is left open. The port to the compliant volume is located about midlength and is simply a 1/4 in. gap in the otherwise continuous wall. Placing a second port in the other wall directly opposite the first and duplicating each compliant volume serves to enhance the action, particularly with the smaller, less power absorbing elements.

For frequencies above and below Helmholtz resonance, waves from the wave drive travel the full length of the tube and out its end. At resonance the impedance of the port is so low compared with the distributed impedance of the downstream remainder of the channel that practically all power is transmitted through the port into the resonating element, leaving the remaining half of the ripple channel water standing still.

The value of the compliant element is varied in this case by changing its available free water surface area. The resonant frequency should vary linearly with the inverse of the square root of the area, as defined by this rearrangement of the Helmholtz resonant frequency equation.

Figure 12 is the result of experimental data which have been re-plotted in terms of the theoretical linear relationship. The straight line which extrapolates to the zero point illustrates the close agreement between theory and measurement of this feature of the Helmholtz function. Its slope is the value of K.

The compliant element resonance is a Helmholtz function since it has an effective inertial length in series with a compliant area. The resonant frequency of any given element can be lowered by extending its effective inertial length. The inertance element is generally used for this purpose. Varying the port area of the compliance element also provides this function. Decreasing the port are increases the inertial length but also decreases the area available to contact the incident wave front. This reduced area decreases the amount of power transferred through the area.

The resonant frequency equation requires a given compliant element to have resonant frequencies directly proportional to the inverse of the square root of the total effective inertial length:

A series of ripple tank experiments was conducted to obtain an empirical relationship between the total inertial length and resonant frequencies. The Helmholtz resonator used is as Figure 10 depicts, and the test arrangement is similar to that for the compliance element of Figure 11 except that various inertial lengths are inserted between the wall openings and the compliance element port. As before, two resonators set opposite one another provide a more complete power transfer at resonance, particularly with the lower frequencies.

The effective inertial length (L_o) of the compliant element has to be initially determined by another arrangement of the resonant frequency equation.

The addition of inertance lengths in 1/2 in segments results in lowering of the resonant frequency. The 1/√¯¯L function of the inertial length is plotted against resonant frequency in Figure 13 and their linear relationship, which extrapolates to zero, is experimentally verified.

The Helmholtz resonance function of the ripple tank is thus in accord with its theoretical description by Equation (7) as demonstrated by the two sets of experiments which relate the resonant frequency to compliant areas and inertial lengths.

Example: An acoustic Helmholtz resonator has a volume V = 193 in.³, a throat area A = 1.97 in.², an effective inertial length L = 1.42 in. and resonant frequency of 192 Hz. Develop its ripple tank equivalent.

A compliant volume of 193 in.³ is too large for the ripple tank system, whose largest volumes are of the order of 4 in.³. A geometric ratio of 1/4 will reduce the volume to (1/4)³ X (193) = 3.0 in.³. The area is reduced to (1/4)² X (1. 97 in.² ) = 0.123 in.² which is a 1/8 in. wide tube area in 1 in. deep water. The effective entrance inertial length (L_o) must be obtained experimentally similar to that of Figure 13. An additional inertance length of 1/4 x 1.42 = .36 in.

The inertial element length of the real acoustic resonator was 0.25 in., which would be 1/16 in. in the ripple system. The degree of analogy between the two systems relative to entrance inertial length will be seen by comparing the effective entrance length (L_o) with its 1/16 in analogous length.