Consider now a *junction* between two of the uniform acoustic tubes in the concatenated tube model shown in Figure 1.1(b). The wave speeds in all the tubes are assumed to be constant, and equal to , so that the discrete-time representation of any single tube will have the form of the pair of digital delay lines shown in Figure 1.2(b). At the junction between the th and th tubes (of cross-sectional areas
and
respectively), for
, we will then have a pressure and a velocity on either side; we will write these pressure/velocity pairs as
, and
respectively--see Figure 1.3(a). Continuity arguments (or conservation laws) dictate that these quantities should remain unchanged as we pass through the boundary between the two tubes, and thus

Note that we have dropped the arguments and , since the relationships of (1.5) hold instantaneously, and only at the tube boundaries.

As per (1.2a) and (1.2b), the pressures and velocities can be split into leftward- and rightward-traveling waves as

where , the admittance of the th tube, is defined by

It is then possible, using (1.6) to rewrite (1.5) purely in terms of the wave variables, as

where is defined by

Here we have written a formula for calculating the pressure waves and

Suppose that we define a set of *power-normalized* wave variables by

Then the scattering operation (1.8) can be written, in matrix form, as

Because the are bounded in magnitude by 1, it is easy to see that scattering, in this case, corresponds to an orthogonal matrix transformation applied to the input wave variables.