The uniform channel of infinite length is frequency indiscriminate. However, if its impedance at some point is abruptly changed, part of the input power is reflected back upstream. If two or more discontinuities occur within a few wavelengths of each other, the reflected waves interact with the discontinuities and the oncoming wave train to create the effect of wavelength-sensitive chokes and passes to the flow of power down the tube.
The filter network behavior is readily demonstrated by arranging a long channel wave guide with an approximate 2 in. space about midway as illustrated in Figure 18. A variety of discontinuities can be positioned in the space. The wave drive is located at one end of
the channel and the other end is left open.
A plot of the author’s subjective estimate of transmitted power passed through various filter networks at corresponding wavelengths is in Figures 19 through 23. They show the ripple tank system to be wavelength sensitive through the third harmonic due to inertance as well as compliance discontinuities and also to the lower frequency Helmholtz type of resonances. The higher harmonic frequency selectivity and Helmholtz resonance capabilities of the ripple tank carry the analogy to acoustic wave beyond the limitations that plague the systems of electrical analog (Kinsler and Frey, 1950).
A channel which is normally 1 in. wide with a 50% enlarged section 2 in. in length has the power attenuation characteristics of Figure 19. The initial low frequency partial choke (H) is a Helmholtz resonance due to the discrete compliant value of the enlarged volume. There is a full choke at the first harmonic, a full pass with the second harmonic and a partial choke of the third while higher frequencies freely pass.
This is equivalent to an acoustic pipeline whose diameter is abruptly enlarged over a section of known length. The electrical analog to this is a capacitor shunted across a transmission line, which has no Helmholtz-type reaction, no second harmonic pass and chokes instead of passes upper harmonic frequencies (Kinsler and Frey, 1950)
A 2 in. long filter section in the middle of an otherwise uniform channel of 1 in. width is an abrupt 50% reduction in width which produces the power attenuation curve of Figure 20. There is a full choke at the first harmonic and full pass at the second with a partial choke for the third and higher harmonics.
This is equivalent to an acoustic pipeline whose diameter is abruptly reduced for a section of its length. The electric analog for this is an inductance placed in series with the transmission line (Kinsler and Frey, 1950), which does not produce the second harmonic pass.
The abrupt discontinuities that produce the power attenuation curve of Figure 21 are two sets of walls perpendicular to the wave direction, each of which presents a 50% area reduction. A combination of the filtering effects of the previous two examples is observed. A low-frequency Helmholtz choke is due to the entrapped volume between the pair of walls. A full choke occurs with the first harmonic resonance of the filter section and a full pass with its second. There is a partial choke with third harmonic and higher wavelengths.
This is equivalent to installing a pair of ported baffles in an acoustic pipeline.
Elimination of compliance for a 2 in. section of the water wave channel is accomplished by eliminating its free surface. A short section of 1 in. square pipe set into the filter section of the water wave channel provides the power attenuation curve of Figure 22. There is a first harmonic full choke, a full pass for the second harmonic, and a full choke of the third and above.
Because the compliance area does not equal the inertial area, there is no acoustic analog to this experiment.
Changing the value of inertance while retaining the value of compliance of the water wave channel is accomplished by increasing its underwater wall width. The underwater width is increased 50% and the free surface width is unchanged by the filter element which produces the power attenuation curve of Figure 23, There is a full choke with the first harmonic and a full pass with the second. There is a partial choke at the third harmonic wavelength and full pass for anything above.
Because the compliance area does not equal the inertial area there is no acoustic analog to this experiment.
The behavior of the wave channel filter networks is directly analogous to that noted in comparable acoustic systems (Kinsler and Frey, 1950:225). The size of a filter system consisting of side branch resonator tubes as compared with the size of an equivalent frequency sensitive Helmholtz system is a particular illustration of the required size difference between the two systems.
The analogy of the ripple system might well be applied to the design of acoustic mufflers as well as speaking tubes in which engine noises are to be damped and the voice frequencies are passed. Frequency selective ear plugs constitute such systems.
Two Dimensional Analogy
Consideration of two dimensional wave system analogy is not included in this paper as it is readily available (French, 1966).