11.9Project

For given dimensions of the length L and the cross-sectional area S of each pipe, write a computer program using the provided MATLAB function (in Sect. 11.8) to calculate and plot power transmission coefficient Tw of the pipeline as a function of frequency.

Hint: Doing this example by hand can be difficult because (1) formulas of acoustic impedance are in complex number format and (2) power transmission coefficients need to be calculated at multiple frequencies. You can use the MATLAB functions provided in Computer Code Section (Sect. 11.7) or any suitable programming language for this project.

11.10Homework Exercises

Exercise 11.1

In Pipe 2 as shown in the figure below, a pressure pf2(x, t) of a forward wave and a pressure pb2(x, t) of a backward ware are formulated in terms of the complex amplitudes of the pressures Pf2R and Pb2R at x ¼ x2 (RHS of Pipe 2) as:

p f 2ðx, tÞ ¼ 12 nP f 2Re jkðx x2Þejωt þ cco

pb2ðx, tÞ ¼ 12 nPb2Rejkðx x2Þejωt þ cco

Pipe 2

Show that:

Part a (Backward Plane Waves)

The complex amplitude of the pressure Pb2L of the forward wave pb2(x, t) at x ¼ x2 (RHS of Pipe 2) can be formulated in terms of the complex amplitude of the pressure Pb2R and length L2 as:

Part b (LHS of a Pipe)

The acoustic impedance Zc2L at x ¼ x1 (LHS of Pipe 2) can be formulated in terms of the complex amplitudes of the pressures Pf2R, Pb2R, the length L2, and the crosssectional area S2 as:

where Z f 2 ¼ ρoc.

S2

Exercise 11.2

The cross-sectional areas of Pipe 1 and Pipe 2 are S1 and S2, respectively, as shown in the figure below. The pressures of the forward wave and backward wave in Pipe 1 at x ¼ x2 (RHS of Pipe 1) are Pf1R and Pb1R, respectively. The pressures of the forward wave and backward wave in Pipe 2 at x ¼ x3 (RHS of Pipe 2) are Pf2R and Pb2R, respectively:

Show that the pressures Pf1R and Pb1R (RHS of Pipe 1) can be formulated in terms of the pressure Pf2R and Pb2R (RHS in Pipe 2) as:

Exercise 11.3 (Closed-Open Pipe)

A pipe has a length L and a cross-section area Si. The far end is open (Zo ¼ 0) and the near end is closed by a rigid cap (Z1 = 1):

(a)Calculate the eigenfrequencies of the pipe in terms of the pipe length L.

(b)Calculate the eigenmodes of the pressure and plot the first three modes of the pipe using the equation below:

12.1Pressure in a One-to-Two Pipe

The power transmission coefficients of a one-to-two pipe as shown below will be calculated and analyzed in this section.

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

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

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

1 = 1 + 1

=

= =

+

where:

Si: cross-section area of the inlet pipe

S1: cross-section area of outlet Pipe 1 (side branch) S2: cross-section area of outlet Pipe 2 (main pipe) pi: pressure of the incident wave in the inlet pipe

p1: pressure of the transmitted wave in outlet Pipe 1 (side branch) p2: pressure of transmitted wave to outlet Pipe 2 (main branch)

Pi ¼ PiR: complex pressure of the incident wave at the RHS of the inlet pipe

P1: complex pressure of the transmitted wave in the side branch pipe at the intersection

P2 ¼ P2L: complex pressure of the transmitted wave at the LHS of the outlet pipe Zi: acoustic impedance of the incident wave at the RHS of the inlet pipe

Z1: acoustic impedance of the transmitted wave inside the branch pipe at the intersection

Z2: acoustic impedance of the transmitted wave at the LHS of the outlet pipe

Pressures and Velocities in a One-to-Two Pipe

The incident and transmission pressures are forward plane waves (Pi, P1, and P2) and have the same wave function format. The pressure and velocity functions of an incident wave are shown below for reference: