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114	4 Acoustic Intensity and Speciﬁc Acoustic Impedance
	p ðx, tÞ ¼ A sin ðωt þ kxÞ ¼	1	jAe jðωtþkxÞ þ jAe jðωtþkxÞ

		2
	pþðx, tÞ ¼ A sin ðωt kxÞ ¼		1		jAe jðωt kxÞ jAe jðωt kxÞ

				2
	p ðx, tÞ ¼ A sin ðωt þ kxÞ ¼	1	jAe jðωtþkxÞ þ jAe jðωtþkxÞ

		2

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

b)What are the wave amplitude and the wavelength of the standing wave?

c)Sketch the resulting wave pattern and indicate the location of peaks and valleys in terms of wavelength.

d)Calculate the root-mean-square (RMS) pressure at x ¼ πk.

e)Calculate the acoustic intensity of the standing wave at any point in space.

f)Calculate the speciﬁc acoustic impedance of the standing wave at any point in space.

(Answers):

π p

i. (a) 2A cos (ωt) cos (kx); (b) 2A, 2 ; (c) N/A; (d) 2jAj; (e) 0;

(f) z

c cos ðωtÞ

cos ðkxÞ

; z

jρ

c cot kx ; z

jρ

c cot

sin ðωtÞ

sin ðkxÞ

ii. (a) 2A sin (ωt) sin (kx); (b) 2A,

2π

; (c) N/A; (d) 0; (e) 0;

sin ðωtÞ

sin ðkxÞ

(f) z ¼ ρoc cos ðωtÞ

cos ðkxÞ

; z ¼ jρoc tan ðkxÞ; z

¼ jρoc tan ðkxÞ

iii. (a) 2A sin (ωt) cos (kx); (b) 2A,

2π

; (c) N/A; (d) p2jAj; (e) 0;

(f) z

sin ðωtÞ

cos ðkxÞ

; z

jρ

c cot

; z

jρ

c cot

cos ðωtÞ

sin ðkxÞ

iv. (a) 2A cos (ωt) sin (kx); (b) 2A,

2π

; (c) N/A; (d) 0; (e) 0;

(f) z

cos ðωtÞ

sin ðkxÞ

; z

jρ

c tan

; z

jρ

c tan

sin ðωtÞ

cos ðkxÞ

4.8References

4.8.1Derivatives of Trigonometric and Complex Exponential Functions

Derivatives of trigonometric functions and complex exponential functions:

4.8 References

115

cos

Þ&& ¼

sin

Þ&

df ðxÞ

sin

Þ&

f 0

sin

Þ&& ¼ þ

cos

Þ&

df ðxÞ

¼ þ

cos

f 0

ð Þ&

e f ðxÞ

df ðxÞ

e f ðxÞ f 0

dx h

Reversed:

cos ½ f ðxÞ& f 0ðxÞdx ¼ þ sin ½ f ðxÞ&

sin ½ f ðxÞ& f 0ðxÞdx ¼ cos ½ f ðxÞ&

ef ðxÞ f 0ðxÞdx ¼ e f ðxÞ

4.8.2Trigonometric Integrals

Some trigonometric integrals are:

2π cos ðθÞdθ ¼ Z0

2π sin ðθÞdθ ¼ 0

2π cos ðθÞ sin ðθÞdθ ¼ 0

2π

cos

2θ

cos 2ðθÞdθ ¼

Þ dθ

2π

¼ 2π

2 Z0

2π

2 Z0

2π

cos ð2θÞdθ ¼

1dθ þ 2π

116 4 Acoustic Intensity and Speciﬁc Acoustic Impedance

2π

cos

2θ

sin 2ðθÞdθ

2 ð

Þ dθ

2π

¼ 2π 2 Z0

1dθ 2π 2 Z0

cos ð2θÞdθ ¼ 2

Change 2π to period T:

T Z

cosðωtÞdt ¼ T Z

cos

T t

dt ¼ T Z

sin

t dt ¼ 0

sinðωtÞdt ¼ T Z

2π

T Z0

cos ðωtÞ sin ðωtÞdt ¼ 0

cos

2ωt

1 1

4π

cos2ðωtÞdt ¼

dt ¼

T Z

1dt þ

cos

t dt ¼

cos 2ωt

dt ¼

1 1

4π

sin2ðωtÞdt ¼

T Z0

1dt

cos

t dt ¼

Chapter 5

Solutions of Spherical Wave Equation

In the previous chapter, formulas for sound pressure, ﬂow velocity, acoustic intensity, and speciﬁc acoustic impedance of plane waves were formulated in Cartesian coordinates. In this chapter, formulas for these properties will be developed in spherical coordinates.

The formulations of these properties in a spherical coordinate system are more useful than in a Cartesian coordinate system in terms of applications and numerical calculations because any vibrating surface can be treated as a point source. A point source radiates sound in radial directions and can be easily formulated in a spherical coordinate system.

The acoustic wave solutions in Cartesian coordinates were derived in the previous chapter. The acoustic wave solutions in spherical coordinates will be derived in this chapter. The following is a summary table of the acoustic wave solutions in both Cartesian and spherical coordinate systems:

General forms

Cartesian coordinates

Spherical coordinates

Position vector: r

r ¼ xex þ yey þ zez

r ¼ rer þ θeθ þ ϕeϕ

Gradient operator:

∂

eθ

eφ

∂x

∂y

∂z

∂r

∂θ

r2 sin θ

∂φ

The Laplacian

∂2

Dim

2 ∂2

∂

Dim

operator: 2

¼ ∂x

¼ ∂r

þ r

∂r

∂ !u

uðx, tÞ ¼ ρo

R ∂x pðx, tÞ dt

uðr, tÞ ¼ ρo

R ∂r pðr, tÞ dt

Euler’s force equation:

∂

¼ 0

∂t

Wave equations:

∂2

pðx, tÞ ¼

pðx, tÞ

½rpðr, tÞ& ¼

½rpðr, tÞ&

1 ∂2

∂x2

∂t2

∂r2

∂t2

p ¼

∂t2

p (x,

rp(r, t) ¼ A cos (ωt kr + θ)

t) ¼ A cos (ωt kx + θ )

Form 2 : REP

(continued)

117

H. Lin et al., Lecture Notes on Acoustics and Noise Control, https://doi.org/10.1007/978-3-030-88213-6_5

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