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CHAPTER 4

Wave Channel System

Distributed impedance occurs whenever one dimension of a wave system meets or exceeds 1/4 wavelength of the lowest applied frequency. Both the acoustical pipeline and wave channel systems have the distributed impedance characteristic in common and the degree of analogy between these two systems is the object of this chapter.

Distributed Impedance

Terms of Impedance 

Wave systems of the distributed impedance type have values for resistance, compliance and inertance in terms of per unit length. An expression for resistance is not developed here since it is relatively insignificant in acoustic and wave channel action.

Distributed compliance of the wave channel is a function of water surface width (w) and specific weight, ρg, but can be rewritten in terms of wave speed C_W =  √¯¯gh, density ρ and an area S = w x h formed by the surface water width and depth.

Distributed inertance of the wave channel is a function of its density and the fluid cross-sectional area (A_W)

An expression for the value of distributed impedance (Z) can be found from transmission line theory and is in terms of compliance and inductance (Kinsler and Frey, 1950:209).

Variable Impedance

The effective rea (S) of distributed compliance is defined in terms independent of the fluid area (A) associated with distributed inertance. This allows the impedance and compliance of the wave channel to be independently adjusted by varying these respective areas. Compliance area (S = wh) is a function of two independent variables. The depth term (h) results from the wave speed (C_W =  √¯¯gh) substitution and is held constant to isolate the dependence of compliance on free surface width (w). The compliance and inertance equations can each be rearranged into terms of some constant times the independent variable.

The top row of Figure 14 shows channel section views that accommodate constant inertance with variable compliance. The inertance area is held constant and shown darkened while compliance area (S) is varied and shown by dotted lines. The lower three views suggest how to vary the channel inertance while holding the compliance constant.

Rectangular Channel Requirement 

The area associated with inertance and compliance within an acoustic pipeline are necessarily one and the same while that of the water wave channel are not. Geometric similitude requirements between these areas dictate that the water compliance area must equal its inertance area.

The shape of the inertance area is not defined while that of compliance is defined as the area below the water surface width for the distance of its depth, which is rectangular in shape. It is the unspecified inertance area which  must assume that shape.

The definition of the compliance area establishes the “proper” water wave channel shape for acoustic analog situations to be rectangular. Any other arrangement will be, by analog, inconsistent with acoustic pipeline capabilities. An application for the water wave channel’s capacity to provide independent adjustments of the inertance and compliance values of a distributed impedance system has yet to be determined.

Acoustic Analogy