3.8Homework Exercises

Exercise 3.1 A complex exponential function p(x, t) is given as:

pðx, tÞ ¼ e jðωtþkx π2Þ ½CEP&

Show that when ωk ¼ c, the above complex exponential function p(x, t) satisfies the one-dimensional acoustic wave equation:

∂2 pðx, tÞ ¼ 1 ∂2 pðx, tÞ ∂x2 c2 ∂t2

Show that when ωk ¼ c, the above complex exponential function p(x, t) satisfies the one-dimensional acoustic wave equation:

∂2 pðx, tÞ ¼ 1 ∂2 pðx, tÞ ∂x2 c2 ∂t2

Exercise 3.3 A complex exponential function p(x, t) is give as:

Show that when ωk ¼ c, the above complex exponential function p(x, t) satisfies the one-dimensional acoustic wave equation:

∂2 pðx, tÞ ¼ 1 ∂2 pðx, tÞ ∂x2 c2 ∂t2

Exercise 3.4 A backward traveling wave p (x, t) with a phase shift (Form 2:REP) is given as:

p ðx, tÞ ¼ A cos ðωt þ kx þ ϕ Þ

where A is an amplitude. Show that if ωk ¼ c, the above trigonometric cosine function p(x, t) satisfies the one-dimensional acoustic wave equation:

∂2 pðx, tÞ ¼ 1 ∂2 pðx, tÞ ∂x2 c2 ∂t2

Exercise 3.5 Prove that the following relationship between a standing wave and the combination of a forward wave and a backward wave is true:

sin ðωtÞ sin ðkxÞ ¼ 21 ð cos ðωt þ kxÞ cos ðωt kxÞÞ

Exercise 3.6 Prove that the following relationship between a standing wave and the combination of a forward wave and a backward wave is true:

b)Show that the harmonic function p(x, t) can be constructed by adding two trigonometric cosine functions as:

Form 2.

Exercise 3.10 The following is a harmonic function p(x, t) of one-dimensional sound pressure in Cartesian coordinates:

hi

pðx, tÞ ¼ Re jP cos ðkxÞe jðωtÞ

where P is a real number.

(a)Show that the harmonic function p(x, t) can be constructed by multiplying two trigonometric cosine functions:

pðx, tÞ ¼ P sin ðωtÞ cos ðkxÞ

(b)Show that the harmonic function p(x, t) can be constructed by adding two trigonometric sine functions:

pðx, tÞ ¼ 2P ½ sin ðωt þ kxÞ þ sin ðωt kxÞ&

Note that each term in the above function has no phase and can be considered as Form 1.

Exercise 3.11 The following is a harmonic function p(x, t) of one-dimensional sound pressure in Cartesian coordinates:

where A is a real number.

(a)Show that the harmonic function p(x, t) can be constructed by multiplying two trigonometric cosine functions:

pðx, tÞ ¼ A cos ðkxÞ cos ðωtÞ

(b)Show that the harmonic function p(x, t) can be constructed by adding two trigonometric cosine functions:

Exercise 3.12 Use the following combination of a forward wave p+(x, t) and a backward wave p (x, t) to answer every question of this problem:

pþðx, tÞ ¼ sin ðωt kxÞ

a)Calculate the standing wave (real number) produced by the forward wave and the backward.

b)Find the locations of the peaks and valleys in terms of wavelength.

Chapter 4

Acoustic Intensity and Specific Acoustic

Impedance

In the previous chapter, we constructed the sound pressures p(x, t) of both traveling waves and standing waves based on the acoustic wave equation. In this chapter, you will learn how to calculate flow velocity u(x, t), acoustic intensity, and specific acoustic impedance for given sound pressure. These quantities are commonly used in the analysis of acoustics and noise control. This chapter is organized as follows:

First, you will learn how to calculate the flow velocity u(x, t) for a given sound pressure p(x, t) using Euler’s force equation that was derived in Chap. 2. This process can be reversed to calculate the sound pressure p(x, t) for a given flow velocity u(x, t) using Euler’s force equation as shown in the figure below:

Euler′s Force Equation:

Pressure-Velocity Relationship

1

H. Lin et al., Lecture Notes on Acoustics and Noise Control, https://doi.org/10.1007/978-3-030-88213-6_4

Second, you will learn how to calculate acoustic intensity I from sound pressure p(x, t) and its corresponding flow velocity u(x, t). The acoustic intensity I is simply the average of the sound pressure p(x, t) multiplied by the flow velocity u(x, t) over the period T as shown in the figure above. The formula of acoustic intensity I for both traveling waves and standing waves will be derived.

Third, you will learn how to calculate specific acoustic impedance z as real numbers. The real number specific acoustic impedance z is simply the sound pressure p(x, t) divided by the flow velocity u(x, t) as shown in the figure above. The formulas of specific acoustic impedance z for both traveling waves and standing waves will be derived. Note that the specific acoustic impedance can be formulated in either the real number format or the complex number format. Formulations of the real number specific acoustic impedance z for traveling waves and standing waves are shown in the table below and will be developed in Sect. 4.4.

Fourth, the acoustic impedance z expressed as complex numbers will be introduced in Sect. 4.5. This is an advanced topic but is important for analyzing acoustic waves in pipes. The formulas of complex number acoustic impedance will be discussed and derived. An application of complex number acoustic impedance for filter designs is applied in Chaps. 12 and 13.

4.1Pressure-Velocity Relationship

The relationship between p(x, t) and u(x, t), based on Euler’s force equation, is:

∂∂x pðx, tÞ ¼ ρo ∂∂t uðx, tÞ

Formulas of pressure-velocity relationships for traveling waves u (x, t)and standing waves us(x, t) will be developed based on Euler’s force equation shown above. The following is a summary of formulas of pressure-velocity relationships that will be derived in this section:

Note that the flow velocity u (x, t) above was referred to as vf in Chap. 2. The u (x, t) is the macroscopic flow velocity vf and should not be confused with the microscopic colliding velocity vc. Also, (BTW) indicates basic traveling waves, and BSW indicates basic standing waves as discussed in Chap. 3.

4.1.1Pressure-Velocity Relationships for BTW

Even though there are no linear relationships between pressure and velocity for general complex waves, there are linear relationships between pressure and velocity for forward and backward traveling waves. The formulations are shown below.

Forward Traveling Waves

pþðx, tÞ ¼ cos ðωt kx þ ϕÞ

Or:

1

! uþðx, tÞ ¼ þ ρoc pþðx, tÞ

! pþðx, tÞ ¼ ρoc uþðx, tÞ

Backward Traveling Waves

p ðx, tÞ ¼ cos ðωt þ kx þ ϕÞ

Or:

1

! u ðx, tÞ ¼ ρoc p ðx, tÞ ! p ðx, tÞ ¼ ρoc u ðx, tÞ

The following is an example of how to prove the formulations above. The pressure of a backward traveling wave is:

According to Euler’s force equation, the velocity can be calculated by first taking the derivative of the pressure with respect to x and then taking its integration with respect to time t as:

∂

∂x p ðx, tÞ ¼ k sin ðωt þ kx þ ϕÞ

So, for backward traveling waves, the pressure and velocity relationship is:

1

u ðx, tÞ ¼ ρoc p ðx, tÞ

Similarly, the relationship between pressure and velocity for forward traveling waves can be derived as:

1

uþðx, tÞ ¼ ρoc pþðx, tÞ

Therefore, we get the relationship between pressure and velocity as:

4.1.2Pressure-Velocity Relationships for BSW

The pressure-velocity relationships for standing waves are different from the pressure-velocity relationships for the previous traveling waves.

The relationships between pressure and velocity for basic standing waves are:

1

psðx, tÞ ¼ cos ðωtÞ cos ðkxÞ ! uðx, tÞ ¼ ρoc sin ðωtÞ sin ðkxÞ

1

psðx, tÞ ¼ sin ðωtÞ sin ðkxÞ ! uðx, tÞ ¼ ρoc cos ðωtÞ cos ðkxÞ

1

psðx, tÞ ¼ sin ðωtÞ cos ðkxÞ ! uðx, tÞ ¼ ρoc cos ðωtÞ sin ðkxÞ

1

psðx, tÞ ¼ cos ðωtÞ sin ðkxÞ ! uðx, tÞ ¼ ρoc sin ðωtÞ cos ðkxÞ

The following is an example to validate the statement above.

Example 4.1

Given a basic standing wave with two cosine functions:

psðx, tÞ ¼ cos ðωtÞ cos ðkxÞ

Find the corresponding particle velocity using Euler’s force equation.

Example 4.1 Solution

The flow velocity can be calculated using Euler’s force equation:

1k

¼ρo ω sin ðkxÞ sin ðωtÞ

1

¼ ρoc sin ðkxÞ sin ðωtÞ

From the results above, we can conclude that:

1

usðx, tÞ ¼6 ρoc pðx, tÞ

The following is another example of validating these pressure-velocity relationships.

Example 4.2

Given a basic standing wave with cosine and sine functions: psðx, tÞ ¼ cos ðωtÞ sin ðkxÞ

Find the corresponding particle velocity using Euler’s force equation.

Example 4.2 Solution

The flow velocity can be calculated using Euler’s force equation:

4.1.3Pressure-Velocity Relationships in Complex Function Form

The general complex solution of sound pressure for acoustic waves is:

The general complex solution of the corresponding velocity can be expressed as:

where each term of the velocity can be calculated from the above pressure equation using Euler’s force equation:

1

u ðx, tÞ ¼ ρoc p ðx, tÞ

The following two examples prove the formula above by substituting one term of the complex pressure p into the Euler’s force equation to calculate one term of the complex velocity u .

Example 4.3

Assume that a pressure function is constructed with only the first term of the general solution of sound pressure as shown at the beginning of this section and that the other coefficients (complex numbers) are all zero. In this case:

p 2 ðx, tÞ ¼ e jðωtþkxÞ

Use Euler’s force equation to get the flow velocity u2(x, t) from the given pressure p2(x, t).

Example 4.3 Solution