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Analog Proofs

The ability of the ripple tank system to model I acoustic and other wave type systems is generally acknowledged and is commonly used in most elementary physics classes to demonstrate the basic nature of waves.

The proof of the analogy is usually offered in terms of comparison. Since the equations of continuity and motion between the two systems have similar terms in similar places they have analogous behavior.
A more rigorous proof is presented here which is based on the theory of similitude for fluid systems.

Similitude Proof

Conditions of Similitude 

Two independent fluid systems are analogous if they are geometrically, kinematically and dynamically similar. Conditions of geometric similitude (G) require that corresponding physical lengths of the two systems (A and B) will differ only by some constant proportion. Kinematic similitude exists if the velocity ratios of the two fluids at corresponding geometric points have the same ratio (K). Conditions of dynamic similitude are met if the ratio of forces of the two systems is a constant (D) at corresponding geometric points between the two systems I (Eshbach, 1952:6-63).

From Newton’s second law of motion the dynamic similitude ratio can be written as

By developing the relationships among the pertinent forces involved the dynamic similitude condition and hence the analogy between the acoustic and shallow water systems is established.

Dynamic Forces 

For any fluid, the force which contributes to its motion will be the sum of its gravity (Fg), viscous (Fv), surface tension (Ft) and elastic (Fe) forces.

The primary force for shallow water which contributes to its motion is due to gravity. As with the force on the face of a dam having area A, width w, water depth h and water density p, it is given by

For shallow water gravity wave action, the effects of surface tension and elastic forces are assumed negligible. The effect of viscosity is found to also be negligible in comparison with that of gravity by calculating values for a typical situation. The viscous force involved with a 1 foot wide sheet of water 1/2 in. deep whose free surface velocity is 1 foot/sec is about 5.8 x 10^-4 pounds:

The gravity force due to the same sheet of water whose depth is 1/2 in is calculated to be 1.3 pounds, which is about 2,000 times greater than the viscous force.

For sound propagation the viscous, surface tension, and gravity forces of air are assumed negligible in comparison with the effect of its elastic force, which is in terms of its bulk modulus of elasticity β, cross-section area A and volumetric strain dV/V.

The energy per unit volume is constant for air waves even though it varies in form between kinetic and potential (elastic). Equating these two forms results in an expression of volumetric strain in terms already established.

The air elastic force equation can now be rewritten in these more convenient terms including the air flow velocity (v).

The inertial force of either type of fluid is in terms of its specific gravity (p), flow cross-section area (A), and velocity (v).

Similitude and Wave Speed 

By equating the ratios of the pertinent forces, viz., dynamic to gravitational in the case of the water waves and dynamic to elastic for acoustic waves, we get

Cancellation of common terms shows that conditions of dynamic and kinematic similitude are simultaneously satisfied.

Since the wave speed of a shallow water system is

and the wave speed in air is

the condition of dynamic similitude establishes that the wave speed ratio for the two systems must be of constant proportion

In any wave action system, the wave speed (C) can also be expressed in terms of wave length (λ) and frequency (f) by C = fλ, so the ratio of wave speeds of the two systems can be in terms of their frequency and wave length ratios.

The ratio of wavelength to characteristic geometric length is the same for both and since f = C/λ,

Mach and Froude Numbers 

The dynamic similitude condition also leads to the following relationship between the Froude number (F) of the water system and the Mach number (M) of its acoustic counterpart.

Rearranging the static and inertial force terms shows the Froude number equal to the Mach number.