Acoustic Waves, Sound Intensity, and Acoustic Impedance

Acoustic Waves, Sound Intensity, and Acoustic Impedance

Sound is a mechanical disturbance that propagates through a material medium.

In a fluid, the simplest acoustic wave is a small longitudinal disturbance of:

• pressure,
• density,
• particle velocity.

The word particle here refers to a small material element of the fluid, not to an isolated microscopic molecule.

For a uniform fluid at equilibrium pressure

(1)

and density

(2)

write

(3)

and

(4)

The acoustic quantities

(5)

and

(6)

are assumed small compared with the equilibrium values.

If

(7)

is the longitudinal particle velocity, the linear acoustic equations lead to the wave equation

(8)

where

(9)

For an ideal gas undergoing small adiabatic disturbances,

(10)

A progressive plane wave has the characteristic specific acoustic impedance

(11)

Pressure and particle velocity are then related by

(12)

for a wave traveling in the positive

(13)

direction.

The time-averaged sound intensity of a harmonic plane wave can be written

(14)

These relations connect the mechanics of a fluid directly to measurable sound pressure and energy flow.

1 Equilibrium and acoustic perturbations

Let the undisturbed fluid be at rest:

(15)

Write the total density and pressure as

(16)

and

(17)

The acoustic approximation requires

(18)

and particle velocities small compared with the sound speed:

(19)

Products of two acoustic perturbations are then second order and are neglected in the linear theory.

2 Mass conservation

In one dimension, the exact continuity equation is

(20)

Insert

(21)

Because

(22)

is constant,

+ = 0.

(23)

The product

(24)

is second order in the acoustic disturbance.

Neglecting it gives the linear continuity equation

(25)

A local velocity convergence therefore increases the density perturbation.

3 Momentum conservation

For an inviscid fluid in one dimension, Euler’s equation is

(26)

The convective term

(27)

is second order.

Replacing

(28)

by

(29)

in the first-order term gives

(30)

Pressure gradients accelerate the fluid particles.

4 Equation of state and sound speed

For sufficiently rapid small-amplitude sound propagation, heat exchange over one acoustic cycle is often negligible.

The compression is then approximately adiabatic.

To first order,

(31)

where

(32)

This Thermodynamic derivative is the square of the sound speed.

For an ideal gas,

(33)

for an adiabatic change.

Differentiating gives

(34)

5 Pressure wave equation

Use

(35)

The continuity equation becomes

(36)

Differentiate with respect to time:

(37)

Differentiate the momentum equation with respect to

(38)

(39)

Eliminating

(40)

gives

(41)

6 Particle-velocity wave equation

Differentiate the momentum equation with respect to time:

(42)

Differentiate the pressure form of continuity with respect to

(43)

(44)

Therefore

(45)

Pressure and particle velocity both propagate at the same sound speed.

7 Sinusoidal plane wave

Consider a positive-direction plane wave

(46)

The wave equation requires

(47)

Assume

(48)

The momentum equation gives

(49)

Since

(50)

(51)

Thus pressure and particle velocity are in phase for a positive-direction progressive plane wave.

8 Characteristic specific acoustic impedance

Define

(52)

Then

(53)

for a positive-direction progressive plane wave.

The SI units are

[z0]	=	(54)
	= Pas∕m.	(55)

The unit

(56)

is also called the rayl.

For a negative-direction progressive wave,

(57)

The minus sign reflects the reversed direction of the particle-velocity field for a given pressure phase.

9 Acoustic intensity

Acoustic intensity is the rate of acoustic energy flow per unit area.

In one dimension the instantaneous intensity is

(58)

In three dimensions, the acoustic intensity vector is

(59)

Its SI unit is

(60)

A positive value of

(61)

means net energy flow in the positive

(62)

direction.

10 Average intensity of a harmonic plane wave

For a right-moving harmonic plane wave,

(63)

and

(64)

Then

(65)

Since

(66)

(67)

Using

(68)

(69)

Since

(70)

and

(71)

(72)

11 Complex-amplitude intensity

Let

(73)

and

(74)

Then the time-averaged intensity is

(75)

This form remains useful when pressure and particle velocity are not exactly in phase.

12 Acoustic energy density

The instantaneous acoustic energy density has two contributions.

The kinetic contribution is

(76)

The compressional contribution is

(77)

Thus

(78)

For a progressive plane wave,

(79)

Therefore

(80)

instantaneously.

The kinetic and compressional energy densities are equal.

13 Energy density and intensity

For a progressive plane wave,

(81)

Using

(82)

(83)

But

(84)

Therefore

(85)

for an ideal progressive plane wave.

Likewise,

(86)

This is the acoustic analogue of energy density multiplied by transport speed.

14 Example 1: characteristic impedance of air

Take representative room-temperature values

(87)

and

(88)

Then

z0	= ρ0c	(89)
	= (1.204)(343)	(90)
	≈ 413 Pas∕m.	(91)

This is the characteristic specific acoustic impedance of the air for the assumed conditions.

15 Example 2: intensity from sound pressure

Let

(92)

in the same air.

Then

⟨I⟩	=	(93)
	≈ 9.69 × 10−5 W∕m2.	(94)

The corresponding particle-velocity RMS amplitude is

urms	=	(95)
	≈ 4.84 × 10−4 m∕s.	(96)

The material particles move much more slowly than the wave propagates.

16 Spherical spreading

A point-like source radiating acoustic Power

(97)

uniformly in all directions distributes that power over a sphere of area

(98)

Neglecting absorption,

(99)

Therefore

(100)

Since

(101)

the far-field pressure amplitude scales approximately as

(102)

17 Example 3: spherical intensity

Suppose a source radiates

(103)

uniformly.

At

(104)

I	=	(105)
	≈ 1.99 × 10−2 W∕m2.	(106)

At

(107)

the intensity is one fourth as large:

(108)

Doubling distance reduces free-field intensity by a factor of four.

18 Sound intensity level

Because audible intensities span many orders of magnitude, a logarithmic scale is useful.

Define the sound intensity level

(109)

For sound in air, a conventional reference intensity is

(110)

An increase in intensity by a factor of

(111)

raises the level by

(112)

A factor of

(113)

raises it by

(114)

19 Sound pressure level

For sound pressure in air, define

(115)

with the conventional reference pressure

(116)

The factor

(117)

rather than

(118)

appears because intensity is proportional to pressure squared when the impedance is fixed:

(119)

20 Example 4: pressure level

For

(120)

Lp	= 20 log 10	(121)
	= 20 log 10(104)	(122)
	= 80.0 dB.	(123)

21 Distance and level in a free field

For ideal spherical spreading,

(124)

If the distance doubles,

(125)

Thus

LI,2 − LI,1	= 10 log 10	(126)
	≈−6.02 dB.	(127)

The familiar approximately

(128)

drop per distance doubling is a consequence of ideal spherical spreading, not a universal law for every acoustic environment.

22 Specific acoustic impedance

The ratio of complex acoustic pressure to complex particle velocity at a point is the specific acoustic impedance:

(129)

In general,

(130)

can be complex and position dependent.

For a positive-direction progressive plane wave in a lossless fluid,

(131)

The characteristic value

(132)

is real.

23 Acoustic impedance based on volume velocity

In ducts and acoustic networks, it is often convenient to use the volume velocity

(133)

where

(134)

is cross-sectional area.

The acoustic impedance is then

(135)

For a uniform progressive plane wave in a duct,

(136)

Its SI units are

(137)

Thus:

• specific acoustic impedance uses particle velocity,
• acoustic impedance uses volume velocity.

24 Normal-incidence interface

Consider a plane wave normally incident on a boundary between two lossless fluids.

Let

(138)

and

(139)

Write the pressure fields as

(140)

(141)

and

(142)

For the corresponding particle velocities,

(143)

(144)

and

(145)

25 Boundary conditions at an ideal fluid interface

Pressure is continuous:

(146)

Normal particle velocity is continuous:

(147)

These two conditions determine reflection and transmission.

26 Pressure reflection coefficient

Define

(148)

Solving the boundary equations gives

(149)

Notice the sign convention compared with the string displacement coefficient in M06-09.

The difference arises because pressure is a force-like variable, while the reflected particle velocity changes sign with propagation direction.

27 Pressure transmission coefficient

Define

(150)

Then

(151)

The pressure amplitude can exceed the incident pressure amplitude when

(152)

This does not imply transmitted power greater than incident power.

28 Power reflection and transmission

For a plane wave,

(153)

when

(154)

is the complex peak pressure amplitude and

(155)

is real.

Therefore the reflected power fraction is

(156)

The transmitted power fraction is

(157)

For real positive impedances,

(158)

Also,

(159)

29 Impedance matching

If

(160)

then

(161)

There is no reflected plane wave at normal incidence in the ideal model.

Thus, as in the string problem, impedance matching is the reflectionless condition.

30 Example 5: air-to-water impedance mismatch

Use representative values

(162)

and

(163)

For a plane wave incident from air onto water,

rp	=	(164)
	≈ 0.99944.	(165)

Thus

(166)

The transmitted fraction is only

(167)

The large impedance mismatch explains why nearly all normally incident acoustic power is reflected in this idealized calculation.

31 Rigid boundary

A rigid boundary has zero normal particle velocity:

(168)

This is the high-impedance limit:

(169)

Then

(170)

Pressure reflects without inversion.

The reflected particle velocity is inverted.

At the rigid boundary:

• particle velocity has a node,
• acoustic pressure has an antinode.

32 Pressure-release boundary

An ideal pressure-release boundary has

(171)

This corresponds to the low-impedance limit:

(172)

Then

(173)

Pressure reflects with inversion.

At the boundary:

• acoustic pressure has a node,
• particle velocity has an antinode.

An open pipe end is often approximated as pressure release in elementary models, although real open ends require an end correction.

33 Standing waves and local impedance

When incident and reflected waves coexist, pressure and particle velocity can acquire different spatial phases.

The ratio

(174)

then depends on position.

It can also become complex.

Therefore

(175)

should not be applied blindly to a standing-wave field.

It is the characteristic impedance of a progressive plane wave in the medium.

34 Pressure and velocity nodes

At a rigid wall, a standing wave may be written locally in the form

(176)

with the wall at

(177)

The associated particle velocity has the form

(178)

Thus pressure and particle-velocity nodes are separated spatially by one quarter wavelength.

This geometry is central to acoustic resonators.

35 Coherent and incoherent sound addition

Acoustic intensities cannot always be added by simply adding decibel values.

For mutually incoherent sources, time-averaged intensities add:

(179)

The total level is then computed after the intensities are summed.

For coherent sources, acoustic pressures must be added with their relative phases before calculating intensity.

Interference can produce enhancement or cancellation.

36 Two equal incoherent sources

If two independent sources each produce the same intensity

(180)

then

(181)

The level increase is

ΔL	= 10 log 10(2)	(182)
	≈ 3.01 dB.	(183)

Thus doubling acoustic intensity adds approximately

(184)

37 Connection to string impedance

M06-09 defined the characteristic mechanical impedance of a transverse string as

(185)

For a plane acoustic wave,

(186)

Both express the same general idea:

A wave impedance relates a force-like field to a velocity-like field for a traveling wave.

For the string, the force-like quantity is transverse force.

For acoustics, it is pressure.

The exact reflection sign depends on which field amplitude is chosen as the reflection variable.

38 Common mistakes

1. Confusing sound speed with fluid-particle speed.
2. Treating acoustic pressure p′ as the total thermodynamic pressure p.
3. Retaining second-order terms while claiming to use linear acoustics.
4. Using an isothermal derivative where the adiabatic sound-speed derivative is required.
5. Forgetting that a right-moving plane wave has p′ = ρ₀cu.
6. Forgetting the minus sign between pressure and particle velocity for a left-moving plane wave.
7. Writing intensity as pressure alone rather than pressure times particle velocity.
8. Using peak pressure in an RMS intensity formula without the factor of two.
9. Confusing specific acoustic impedance p∕u with acoustic impedance p∕U.
10. Forgetting the cross-sectional area when converting from specific acoustic impedance to duct acoustic impedance.
11. Adding decibel values directly instead of adding physical intensities when appropriate.
12. Forgetting that pressure-amplitude transmission can exceed one without violating energy conservation.
13. Using the string displacement reflection sign for acoustic pressure without re-deriving the boundary conditions.
14. Calling an open pipe end exactly pressure release without recognizing the end-correction approximation.
15. Applying z₀ = ρ₀c locally to an arbitrary standing-wave field.

39 Practice exercises

1. Linearize the one-dimensional continuity equation.
2. Linearize Euler’s equation for an inviscid fluid initially at rest.
3. Combine continuity, momentum, and p′ = c²ρ′ to derive the pressure wave equation.
4. Derive the particle-velocity wave equation.
5. Starting from the momentum equation, derive p∕u = ρ₀c for a right-moving plane wave.
6. Derive the average intensity formulas in terms of peak and RMS quantities.
7. Derive the acoustic energy density and show that kinetic and compressional contributions are equal for a progressive plane wave.
8. Derive the inverse-square intensity law for an isotropic point source.
9. Show why doubling distance reduces free-field level by approximately 6 dB.
10. Derive the relation between sound intensity level and sound pressure level for fixed acoustic impedance.
11. Derive the normal-incidence pressure reflection and transmission coefficients at a fluid interface.
12. Derive the power coefficients and verify ℛ + 𝒯 = 1.
13. Reproduce the air-to-water numerical example.
14. Derive the pressure and particle-velocity behavior at rigid and pressure-release boundaries.
15. Explain why the local impedance of a standing wave need not equal the characteristic impedance ρ₀c.

40 Summary

Linear acoustics begins with

(187)

(188)

and

(189)

These give

(190)

and

(191)

For a progressive plane wave,

(192)

and

(193)

for positive-direction propagation.

The average intensity is

(194)

Sound levels use logarithmic scales:

(195)

and

(196)

At a normal-incidence interface,

(197)

and

(198)

The power coefficients are

(199)

and

(200)

Acoustic impedance therefore unifies progressive-wave field ratios, interface reflection, power transmission, and resonant boundary behavior.

References

References