The reflection pressure is a backward plane wave (Pr) and is shown below for reference:

•Note that pressure and velocity are functions of space and time. Also, the real

pressure, pi(x, t), is the addition of a complex conjugate pair of the pressures

12 piðx, tÞ þ pi ðx, tÞ&.

Remarks

•The incident acoustic impedance and reflected backward acoustic impedance have a simple relationship:

Zr = 2Zi

•Because transmission pressures P1 and P2 are outward only (no return), acoustic impedance Z1(=ρoc/S1) and Z2(=ρoc/S2) are real numbers.

•Because the incident and reflected pressures Pi and Pr are individual plane waves

12.1.1 Equivalent Acoustic Impedance of a One-to-Two Pipe

The formulas for a one-to-two pipe can be compared to the formulas for a one-to-one pipe as listed below:

=

Formulas 8A–8B (Equivalent Acoustic Impedance of a One-to-Two Pipe)

The dimensions and acoustic impedance of a one-to-two pipe are given as follows:

• Cross-section areas of the pipes are Si, S1, and S2.

• Acoustic impedances of the pipe are Zi, Zr,Z1, and Z2:

1 = 1 + 1

=

=

where Zo is the equivalent acoustic impedance. This Zo is the combined impedance of Z1 and Z2 and is formulated as:

Proof 12: Solution

The pressure and velocity functions in the previous section are general solutions for plane wave equations. The exact reflection and transmission pressures can be solved by the boundary conditions at the intersection.

The boundary of the intersection of three pipes can be considered as a point, and the following conditions must be satisfied:

(a) All pressures are equal in amplitude at the intersection (related to Newton’s third law):

The acoustic impedance (Z) gives the relationship between pressure and velocity

Use the above acoustic impedance to replace U with P/Z in Eq. (2) to obtain:

Pi 2 Pr = P1 þ P2

Zi Zi Z1 Z2

Since P1 ¼ P2, P1 can be replaced with P2 in Eq. (3) to reduce one unknown variable ( P2):

Define the equivalent acoustic impedance Zo as the combined impedance of Z1 and Z2 and computed as:

Based on the equivalent acoustic impedance Zo defined above, Eq. (4) becomes:

Combining Eq. (1) and Eq. (5) yields Formula 8:

Note that by defining the equivalent acoustic impedance Zo as Formula 12, Formula 8 can be used to calculate pressures Pr and P2 for the one-to-two pipe. The definition of this new acoustic impedance Zo may be used in many outward pipes in a later section.