imaginary number; when kr 1, ϕ 0, z ¼ ρoc is a real number and is the same as a plane wave.

5.7Homework Exercises

Exercise 5.1

Acoustic pressure magnitude P0 of a spherical wave is given as 4.149 [Pa] at 1 [m] from the center of a source. If the wavelength is 0.1 [m], answer the following questions, and show units in the Meter-Kilogram-Second (MKS) system:

a)What is the particle velocity magnitude at any point in space?

b)What is the acoustic pressure magnitude at any point in space?

c)What is the acoustic intensity at 1 and 25 [m] from the center of the source?

d)What is the phase difference between the pressure and flow velocity of the sphere wave at 1 and 25 [m]?

Use 415 [rayls] for the characteristic impedance (ρ0c) of air. (Answers):

p

b)The pressure magnitude at r: Pr ¼ 4:149r [Pa]

c)The acoustic intensity at r ¼ 1 [m]: 0.02074 [ w/m2]

The acoustic intensity at r ¼ 25 [m]: 0.00003318 [w/m2]

d) The phase difference at r ¼ 1 [m]: ϕ1 ¼ 0.01591 rad ¼ 0.9118 [Deg] The phase difference at r ¼ 25 [m]: ϕ25 ¼ 0.00064 rad ¼ 0.0365 [Deg]

Exercise 5.2

The acoustic pressure magnitude from a spherical source, measured at 10 m from the center of the source, is 0.8 Pa. If the frequency of radiation is 250 Hz:

a)What is the surface velocity magnitude of the source if the source radius is 0.05 [m]?

b)What is the acoustic pressure magnitude on the surface of the source?

Use 415 [rayls] for the characteristic impedance (ρ0c) of air and 340 ms for the speed of sound (c) in air.

(Answers): (a) 1:713 ms ; (b) 160[Pa]

Exercise 5.3

Flow velocity magnitude U0 of a spherical wave is given as 0.02 [m/s] at 2 [m] from the center of a source. If the frequency is 1700 [Hz].

a)What is the flow velocity magnitude at any point in space?

b)What is the acoustic pressure magnitude at any point in space?

c)What is the acoustic intensity at 5 [m] from the center of the source?

Exercise 5.4

Derive the complex flow velocity u(r, t) from the following complex sound pressure p(r, t) using Euler’s force equation:

Hint: The flow velocity u(r, t) can be obtained by finding the integral of the derivative of the pressure p(r, t) in the Euler’s force equation shown below:

Exercise 5.5

Derive the complex flow velocity u(r, t) from the following complex sound pressure p(r, t) using Euler’s force equation:

To obtain the formulas for point sources, we will follow the three formulas as described below:

Formula 1: Formula for a pulsating spherical source with an arbitrary radius a Formula 2: Formula for a pulsating spherical source with a small radius (a 1) Formula 3: Formula for a point source with a zero radius (a ¼ 0)

Each formula is developed based on the previous formula. For example, Formula 3 is based on Formula 2; Formula 2 is based on Formula 1; Formula 1 is based on the general spherical wave derived in the previous chapter.

The following is a summary of formulations of acoustic waves from a spherical source that will be derived in this chapter: