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Comparable Equations 

The traditional proof that shallow water wave action is analogous to acoustic waves also illustrates the analogy to hold for only one and two-dimensional acoustic systems. The three-dimensional acoustic wave has no direct water wave analogy. If z is the water depth taken from its mean surface level while u and v are respectively particle velocities in the x and y directions parallel to the mean surface plane, the equations of continuity and motion can be written (Feather, 1961).

The two-dimensional acoustic wave equation has the same form.

By comparing the equations of continuity of the two systems, the water depth (z) corresponds to the air density (ρ) in analogy. The equations of motion show this analogy to be inversely related. This conclusion coincides with the wave speed dependency on water depth and air density.

If the water depth increases, the water wave speed also increases. Likewise, if the air density decreases, its wave speed increases (for constant pressure).

Fictitious Gas 

Courant and Friedrichs (1948) transformed the ‘water wave equations into a set of “fictitious gas” equations by means of a hypothetical density ρ equal to water density ρ times its depth z together with a gas pressure equal to the horizontal force exerted by a standing body of water. This “gas” had to have a specific heat of 2.0 to satisfy conditions of dynamic similitude.

Operational Analog Equation

Variation of Wave Speed

the acoustic wave speed in air is independent of frequency. The wave speed of water has be assumed to be also frequency independent for the analogy to hold true. The real wave speed of water does vary with fre-frequency within the very useful lower frequency range of 1.5 to 10.0 Hz.

Experiments (Figure 6) reveal the real wave speed (CR = fλ) as determined by standing wave measurements to approach the theoretical wave speed (CW= √ˉˉˉgh) as the frequency approaches zero. At 1.5 Hz, which is the lower limit of wave visual discernability, the real wave speed is within 95% of its theoretical value, Cw. As the frequency is increased, the real wave speed· slows down until a lower limit is reached. Above 10 Hz, the real wave speed remains constant at the value for the minimal water depth of 1/8 in. and seems to be independent of variations of real water depth, which has a maximum value of 3 in.

Empirical Wave Speed

The real wavelength (λR= CR/F), which is measured by a standing wave experiment, can be compared with the theoretical wavelength (λt = √ˉˉˉvgh/f) by noting their difference (Δλ = λt – λR) and its dependence on frequency and water depth.

The result (Figure 7) is that at depths below 1-1/2 in. this difference between wavelengths is equal to the water depth itself (Δλ = h) and is frequency independent throughout the useful range of frequencies. The greater water depths show a low-frequency deviation from 1this relationship but approach it as the frequency is increased. Because of the tendency of the real wave speed to approach the theoretical wave speed as the frequency is lowered, the wavelength difference must extrapolate to zero as the frequency approaches zero. This empirical relationship can be rewritten to provide an expression of real wave speed in terms of theoretical wave speed and water depth.

Analog to the Analog 

The real shallow water wave system with its frequency-dependent wave speed does not meet the acoustic analog requirements of a constant wave ratio between the two systems. A modeling relationship between the two types of wave systems has been developed which is in terms of only geometric, wave speed, and frequency ratios. If this is assumed to be the nature of modeling requirements of any two wave systems, the real water wave system can be set as a model to the theoretical water wave system by a similar relationship.

The geometric ratio of these two systems is taken to be unity since they both occupy the same physical layout. Real wave speed (CR) can be replaced by its empirical expression and a rearrangement of terms shows the theoretical water system frequency to be a function of water depth (h), acceleration of gravity (g) and the frequency applied to the real system.

The shorter-than-theoretical wavelength at any applied frequency of the real system appears in the theoretical system with that same length but as if being driven by a higher-than-real frequency. The acoustic analog equation can be rewritten in these ascertainable terms.

This expression can be reduced to simplest terms showing the analogous acoustic frequency as a function of geo-metric ratio, wave speed ratio and applied frequency of the real systems involved.