Change in Tube Area

A change in tube cross-sectional area creates a junction between two impedances $Z_1$ and $Z_2$.

Figure 5: Tube with a change in cross-sectional area.

The pressure and velocity at the junction is given by

$$p(J) = p_1^{+}(J) + p_1^{-}(J)$$

$$U(J) = U_1^{+}(J) + U_1^{-}(J)$$

$$= \frac{1}{Z_1}(p_1^{+}(J) - p_1^{-}(J))$$

where

$$p_1^{+}(J) = Z_1U_1^{+}, \quad p_1^{-}(J) = -Z_1U_1^{-}.$$

The new impedance at the junction $Z_2$, is given by

$$Z_2 = \frac{p(J)}{U(J)} = Z_1\frac{p_1^{+}(J) + p_1^{-}(J)}{p_1^{+}(J) - p_1^{-}(J)}.$$