Computer Program Section in Sect. 11.7. This method will plot the power transmission coefficient vs. the frequency of the pipeline.

Section 11.7 shows three MATLAB functions that can be used for the numerical analysis of pipelines described in the previous section (Sect. 11.6). These functions are based on Formulas 1, 4, 5, and 6 derived in this chapter.

Section 11.8 includes a numerical project to calculate and plot the power transmission coefficient vs. the frequency of a pipeline with an arbitrary number of pipes in series. The procedures for this project are similar to the procedures described in the Numerical Method Section (Sect. 11.6). The solutions are not provided.

11.1Complex Amplitude of Pressure and Acoustic Impedance

We will define the complex amplitude of pressure in this section.

11.1.1 Definition of Complex Amplitude of Pressure

Complex amplitudes of sound pressure are used for frequency domain analysis of pipes. For frequency domain analysis of pipes, all time-related parts must be eliminated. Therefore, complex amplitudes must have the following two properties:

•Complex amplitudes of pressure are time-independent.

•Complex amplitudes of pressure are location-dependent.

Based on these two properties, the complex amplitude Pf xo of pressure pf(x, t) of a forward wave can be demonstrated as:

Pf xo =Aþe jkxo

Note that:

• The time-dependent term e jωt is not included in the complex amplitude of pressure Pf xo .

•The complex amplitude of pressure Pf xo is the amplitude of pressure at location x ¼ xo where e jkðx xoÞ ¼ 1.

We can reverse the procedures above to obtain the pressure pf(x, t) for a given complex amplitude Pf xo at x ¼ xo. It is important to indicate the location of the complex amplitude of pressure when handling any complex amplitudes.

11.1.2 Definition of Acoustic Impedance

Three types of impedances are defined below and will be explained in the following sections:

Three Types of Impedances

Three types of impedances are defined below and will be explained in the following sections:

Specific acoustic impedance:

In the impedances above, P is the complex amplitude of pressure, V is the complex amplitude of velocity, S is the cross-section area of a pipe, and U is defined as the complex amplitude of velocity V multiplied by the cross-section area S.

Complex Acoustic Impedance

Based on the definition of acoustic impedance shown above, acoustic impedance is a complex number because both pressure and velocity are formulated in complex.

Plane waves in a pipe usually result from the addition of incident waves (forward) and reflected waves (backward). The combined acoustic impedance Zcof the combined (forward and backward waves) is a complex number and can be formulated as shown below:

Proof of Formula 1C

Based on Formulas 1D and 1E, the acoustic impedance Zc2R at the RHS of Pipe 2 (as an example) is defined as:

Based on Formulas 1A and 1B, the equation above becomes:

11.1.3 Transfer Pressure

Formulas 2A–2G

The complex amplitude of the pressure Pf2L of the forward wave pf2(x, t) at x ¼ x1 (LHS of Pipe 2) is related to the complex amplitude of the pressure Pf2R of the forward wave pf2(x, t) at x ¼ x2 (RHS of Pipe 2) in Formula 2A below. The complex amplitude of the pressure Pb2L of the backward wave pb2(x, t) at x ¼ x1 (LHS of Pipe 2) is related to the complex amplitude of the pressure Pb2R of the backward wave pb2(x, t) at x ¼ x2 (RHS of Pipe 2) in Formula 2B:

Pipe 2

,

Proof of Formulas 2A and 2C

In Pipe 2 as shown in the figure above, the pressure pf2(x, t) of a forward wave and the pressure pb2(x, t) of a backward wave are formulated in terms of complex amplitudes of the pressures Pf2L and Pb2L at x ¼ x1 (LHS of Pipe 2) as:

pf 2ðx, tÞ ¼ 12 nP f 2Le jkðx x1Þejωt þ cco

pb2ðx, tÞ ¼ 12 nPb2Lejkðx x1Þejωt þ cco

The pressure pf2(x, t) of the forward wave as a function of space and time in terms of the complex amplitude of pressure Pf2L at x ¼ x1 (LHS of Pipe 2) is:

p f 2ðx, tÞ ¼ 12 nP f 2Le jkðx x1Þejωt þ cco

The pressure pf2(x, t) of the forward wave as a function of space and time in terms of the complex amplitude of pressure Pf2R at x ¼ x2 (RHS of Pipe 2) is:

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

Comparing the two identical equations above yields:

Pf 2Le jkðx x1Þ ¼ P f 2Re jkðx x2Þ

!P f 2L ¼ P f 2Rejkðx2 x1Þ ¼ P f 2RejkL2

!P f 2R ¼ P f 2Le jkðx2 x1Þ ¼ P f 2Le jkL2

Proof of Formulas 2E and 2F

Based on Formula 1D, Pc2L ¼ Pf2L + Pb2L and Pc2R ¼ Pf2R + Pb2R, and using Formula 2C–2D or 2A–2B gets: