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Operational Constraints  

The water wave system whose depth is less than 1/4 in. provides a frequency independent wave speed, thereby meeting the constant wave speed ratio requirement for the sound wave analogy. The range of applications of this very shallow water system is limited to the ordinary ripple tank experiments which demonstrate two dimensional wave interference and reflection characteristics due to single and multiple point sources.

One dimensional wave situations such as acoustic waves inside a pipe require that the pipe diameter be less than 1/4 wavelength. Similarly, entire Helmholtz apparatus must be within the 1/4 wavelength limit. The wavelength at 12.5 Hz in water 1/4 in. deep is 0.58 in. Therefore its 1/4 wavelength is too small to be physically practical due to excessive surface tension and viscosity effects, which cause great damping of wave action and distortions of the image on the viewing screen. The minimum channel width for satisfactory pipeline wave action is 1/4 in., which corresponds to a minimum wave length of 1 in. A comparative plot of wavelengths (Figure 8) and frequencies for different water depths shows an increasing available frequency range with increasing depth. Advantages of increased water depth are limited by increasing nonlinearity of wave speed changes. The compromise depth of 1 in. provides satisfactory experimental data while offering a relatively broad working frequency band of 1.5 to 15.0 Hz whose upper frequency 1/4 wavelength is 1/4 in while the water depth is within limits of the empirical wave speed equation.

Analog Equations  

Establishment of the 1 inch operating depth criterion allows development of a frequency conversion factor chart whose value depends on the applied water wave frequency, standard speed of sound in air, and the water wave speed. This factor is multiplied by the geometric ratio between the air and water systems to obtain the equivalent acoustic frequency. A plot of this factor (F) is found in Appendix A, page 69.

Equation (3) can be rewritten in terms of factor F:

An alternative to the frequency conversion factor approach in the determination of the equivalent acoustic frequency is to determine the equivalent acoustic wave length.

By using:the scaling equation can be rearranged to show the equivalent acoustic wavelength to be a function of the geometric ratio and the real water wavelength.

This wavelength relationship is independent of water and acoustic wave speeds and is good for any water depth. The only handicap is that the analogy is in terms of wavelengths rather than frequencies, meaning that the water wavelength must be determined for each analogous data point. If the wavelength analogy is employed, a section of the wave tank will have to be devoted to a standing wave fixture which, upon adjustment, will provide the wavelength of any applied frequency. A graph of real wavelengths and associated frequencies for the standard 1 in. water depth is also in Appendix A, page 69.  (shown above)