4.3Acoustic Intensity

The instantaneous intensity Ii(t) of a sound wave is defined as the instantaneous rate of work done by one element of the fluid on an adjacent element per unit area.

Hence:

The acoustic intensity I is simply the time average of Ii(t). Hence:

Formulas of acoustic intensity I for traveling waves u (x, t)and standing waves us(x, t) will be developed based on the definition of acoustic intensity shown above. The following two formulas of pressure-velocity relationships will be derived in this section:

4.3.1Acoustic Intensity of BTW

The acoustic intensity I of traveling waves is:

I ¼ ρ1c p2 RMS

o

where the RMS pressure of traveling waves is:

where A is the amplitude of the traveling waves of the four basic traveling waves:

To validate the acoustic intensity formula for the forward traveling waves, the following forward traveling waves are shown as an example below:

pþðx, tÞ ¼ A ∙ cos ðωt kxÞ

Based on Euler’s force equation, the flow velocity is given by:

pþðx, tÞ ¼ A ∙ cos ðωt kxÞ ! uþðx, tÞ ¼ þ ρ1c pþðx, tÞ

o

Hence, the time average of instantaneous intensity can be written as:

To validate the acoustic intensity formula for backward traveling waves, the following backward traveling waves are shown as an example below:

p ðx, tÞ ¼ A ∙ cos ðωt þ kxÞ

Based on Euler’s force equation, the flow velocity is:

p ðx, tÞ ¼ A ∙ cos ðωt þ kxÞ ! u ðx, tÞ ¼ ρ1c p ðx, tÞ

o

Hence, the time average of instantaneous intensity can be written as:

4.3.2Acoustic Intensity of BSW

The acoustic intensity of a standing wave is:

Is ¼ 0

The following is an example to validate the above statement.

Using the following basic standing wave as an example yields:

psðx, tÞ ¼ AcosðωtÞ cos ðkxÞ

The flow velocity of a standing wave by Euler’s force equation is:

A

usðx, tÞ ¼ ρoc sin ðkxÞ sin ðωtÞ

The acoustic intensity can be calculated by using the pressure and the velocity:

4.4Specific Acoustic Impedance Expressed as Real Numbers

The real number specific acoustic impedance (real impedance) z is defined as the ratio of acoustic pressure to associated flow velocity as follows:

z pu

The usage of these formulas of real impedance for standing waves is limited because they are time functions. However, because the real impedance is easier to understand and formulate, we will focus on real impedance in this section. The specific acoustic impedance expressed as complex numbers (complex impedance) will be introduced in the next section (Sect. 4.5).

Formulas of real impedance for traveling waves p (x, t) and standing waves ps(x, t) will be developed based on the definition of real impedance shown above. The following is a summary of formulas of real impedance that will be derived in this section:

4.4.1Specific Acoustic Impedance of BTW

Specific acoustic impedance of traveling waves is denoted by:

z ¼ ρoc

For forward and backward traveling waves:

p ðx, tÞ ¼ A ∙ cos ðωt þ kxÞ pþðx, tÞ ¼ A ∙ cos ðωt kxÞ p ðx, tÞ ¼ A ∙ sin ðωt þ kxÞ pþðx, tÞ ¼ A ∙ sin ðωt kxÞ

Let’s use the following backward traveling wave as an example:

p ðx, tÞ ¼ A ∙ cos ðωt þ kxÞ

The corresponding flow velocity by Euler’s force equation is:

1

u ðx, tÞ ¼ ρoc p ðx, tÞ

Hence, specific acoustic impedance is by definition denoted as:

Similarly, for a forward traveling wave:

98 4 Acoustic Intensity and Specific Acoustic Impedance

With a given specific acoustic impedance, the flow velocity can be calculated

where rayl is a unit created in honor of John William Strutt, Baron Rayleigh.

The product has greater acoustic significance for the medium than ρo and c along and is also called the characteristic impedance of the medium. The value of characteristic impedance for air is 415 rayl.

4.4.2Specific Acoustic Impedance of BSW

The specific acoustic impedance of standing waves is designated as:

The following is an example to validate one of the above relationships between pressure and specific acoustic impedance.

Let’s use the following standing wave as an example. psðx, tÞ ¼ A cos ðωtÞ cos ðkxÞ

The flow velocity of a standing wave, by Euler’s force equation, is expressed as:

1

usðx, tÞ ¼ ρoc sin ðωtÞ sin ðkxÞ

Specific acoustic impedance is written as:

Note that the specific acoustic impedance defined in real p and u is a time function, cos(ωt) and sin(ωt), even at a fixed location x. Therefore, the uses of the

¼ 14 Ae jkx þ Aejkx ejωt þ Aejkx þ Ae jkx e jωt ¼ 12 ðp þ p Þ

The flow velocity of a standing wave, by Euler’s force equation, is expressed as:

Note that complex specific acoustic impedance is zero at cos(kx) ¼ 0 and is infinite at sin(kx) ¼ 0. (Let A+c ¼ A c ¼ A/2; A+s ¼ A s ¼ 0).

Example 4.6 (Traveling Wave with Known Pressure)

Use the following backward traveling wave to answer every question of this problem:

a)Obtain the expressions for the flow velocity at any point in space.

b)What is the RMS pressure of the wave at any point in space?

c)What is the RMS flow velocity of the wave at any point in space?

d)What is the acoustic intensity of the wave at any point in space?

e)What is the specific acoustic impedance of the wave at any point in space?

Example 4.6 Solution

A backward traveling wave can be represented as:

a)For a backward traveling wave, the velocity by Euler’s force equation is given by:

b) For a traveling wave, forward or backward, the RMS pressure P RMS is given by: P 2RMS ¼ 12 P2o, where Po is the amplitude of the pressure p (x, t)

Hence:

1

P RMS ¼ p jPoj

2

c) The RMS velocity U RMS is similar to the RMS pressure, given by:

Thus:

1

U RMS ¼ p jPoj

2ρoc

d) The intensity of backward traveling waves is:

I ¼ ρ 1c p2 RMS

o

where p2 RMS was calculated in part (b) as:

1

P RMS ¼ p jPoj

2

Hence:

e) The specific acoustic impedance is by definition given by:

z ¼ p ¼ p ¼ ρ0c u

Example 4.7 (Standing Wave Pattern)

Use the combination of the following waves to answer every question of this problem:

p ðx, tÞ ¼ Xo cos ðωt þ kxÞ; pþðx, tÞ ¼ Xo cos ðωt kxÞ

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

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.

Example 4.7 Solution

A standing wave pressure is constructed by the following two traveling waves with opposite traveling directions:

p ðx, tÞ ¼ Xo cos ðωt þ kxÞ pþðx, tÞ ¼ Xo cos ðωt kxÞ

a) A standing wave pressure can be obtained by:

ps ¼ p þ pþ ¼ Xo cos ðωt þ kxÞ þ Xo cos ðωt kxÞ ¼ 2Xo cos ðωtÞ cos ðkxÞ

b)The wave amplitude is 2Xo, and the wavelength is λ ¼ 2kπ:

c)The wave pattern when t ¼ 0 yields cos(ωt) ¼ 1:

=

−

d)The RMS pressure of a standing wave is:

e) The acoustic intensity of a standing wave is given by (see note for explanation):

1 Z T

Is ¼ T 0 pudt ¼ 0

f) Specific acoustic impedance is defined as:

z p ; z p ; z p u u u

Based on the formula for specific acoustic impedance of a standing wave: