A Three-Parameter Broadband Model

Подпись: Z = R0 - i Подпись: X-1 + X1 ш ш Подпись: (10.3)

A simple model for the broadband impedance boundary condition is to take the resistance R to be a constant and the reactance X to be given by Eq. (10.2). That is,

where R0 > 0, X-1 < 0, X1 > 0.

Подпись: (10.4)

For this model, the time-domain impedance boundary condition may be written as follows:

It is straightforward to show, by assuming time dependence of the form e~imt for all variables, that Eq. (10.4) is equivalent to the frequency-domain impedance boundary condition with impedance given by Eq. (10.3).

Подпись: (10.5a) (10.5b)

For a single-frequency sound field of frequency Q (Q > 0), this three-parameter model may still be used. Let Z0 = R0 – iX0 be the impedance of the acoustic liner at frequency Q, then the following choice of X-1 and X1 will transform the broadband model to a single-frequency model.