Acoustic Waves, Sound Intensity, and Acoustic Impedance

(Topic)

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

p0

(1)

and density

ρ0,

(2)

write

|---------------------| p(x,t) = p0 + p′(x, t), | -----------------------

(3)

and

|----------------------| |ρ(x,t) = ρ0 + ρ′(x,t). | -----------------------

(4)

The acoustic quantities

′ p

(5)

and

′ ρ

(6)

are assumed small compared with the equilibrium values.

u(x,t)

(7)

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

|--------------| |∂2p′ 2∂2p′ | |--2-= c ---2 , -∂t-------∂x----

(8)

where

|-----(----)---| | 2 ∂p | |c = --- . | --------∂ρ--s--

(9)

For an ideal gas undergoing small adiabatic disturbances,

|---∘-------| | γp0- | c = ρ . | ---------0--

(10)

A progressive plane wave has the characteristic specific acoustic impedance

|--------| z0-=-ρ0c.-

(11)

Pressure and particle velocity are then related by

|-′------| -p-=-z0u--

(12)

for a wave traveling in the positive

x

(13)

direction.

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

|----------------------| | 2 | |⟨I⟩ = prms = ρ0c u2rms.| --------ρ0c------------

(14)

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

Figure 1. A small fluid element is compressed and accelerated by pressure differences across its faces. Linear acoustics follows from mass conservation, momentum conservation, and the relation between pressure and density changes.

1 Equilibrium and acoustic perturbations

Let the undisturbed fluid be at rest:

u = 0. 0

(15)

Write the total density and pressure as

′ ρ = ρ0 + ρ ,

(16)

and

p = p0 + p′.

(17)

The acoustic approximation requires

|------------------------| ||ρ′| ≪ ρ0, |p′| ≪ p0,| -------------------------

(18)

and particle velocities small compared with the sound speed:

|--------| ||u | ≪ c.| ---------

(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

|----------------| |∂ρ ∂ (ρu) | |---+ ------ = 0.| -∂t-----∂x--------

(20)

Insert

ρ = ρ0 + ρ′.

(21)

Because

ρ 0

(22)

is constant,

= 0.

(23)

The product

ρ′u

(24)

is second order in the acoustic disturbance.

Neglecting it gives the linear continuity equation

|----------------| |∂ρ′ ∂u | |∂t- + ρ0∂x- = 0.| ------------------

(25)

A local velocity convergence therefore increases the density perturbation.

3 Momentum conservation

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

( ∂u ∂u ) ∂p ρ --- + u--- = − ---. ∂t ∂x ∂x

(26)

The convective term

uux

(27)

is second order.

Replacing

ρ

(28)

ρ0

(29)

in the first-order term gives

|--------------| | ∂u- ∂p′ | |ρ0∂t = − ∂x .| ----------------

(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,

|---------| p′ = c2ρ′, | -----------

(31)

where

|-----(----)---| | 2 ∂p | |c = --- . | --------∂ρ--s--

(32)

This Thermodynamic derivative is the square of the sound speed.

For an ideal gas,

−γ pρ = constant

(33)

for an adiabatic change.

Differentiating gives

|-----γp0--| |c2 = ----.| -------ρ0--|

(34)

5 Pressure wave equation

Use

p′ = c2ρ′.

(35)

The continuity equation becomes

|------------------| |∂p′ 2∂u | |--- + ρ0c ---= 0.| -∂t--------∂x-------

(36)

Differentiate with respect to time:

′ 2 ptt + ρ0c uxt = 0.

(37)

Differentiate the momentum equation with respect to

x :

(38)

′ ρ0uxt = − pxx.

(39)

Eliminating

uxt

(40)

gives

|------------| | ′ 2 ′ | -ptt =-cp-xx.-

(41)

6 Particle-velocity wave equation

Differentiate the momentum equation with respect to time:

′ ρ0utt = − pxt.

(42)

Differentiate the pressure form of continuity with respect to

x :

(43)

p′xt + ρ0c2uxx = 0.

(44)

Therefore

|------------| -utt =-c2uxx.-|

(45)

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

Figure 2. Linear acoustics couples three perturbation fields. Continuity connects density to particle-velocity gradients, momentum connects particle acceleration to pressure gradients, and the equation of state connects pressure and density.

7 Sinusoidal plane wave

Consider a positive-direction plane wave

|-′------------------| -p-=--^pcos(kx-−-ωt-).|

(46)

The wave equation requires

|--------| -ω-=-ck.-|

(47)

Assume

u = u^cos(kx − ωt ).

(48)

The momentum equation gives

ρ0ω ^u = k^p.

(49)

Since

ω ∕k = c,

(50)

|--------| |^p-= ρ0c.| -^u--------

(51)

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

8 Characteristic specific acoustic impedance

Define

|--------| z0-=-ρ0c.-

(52)

Then

|--------| |p′ = z0u| ----------

(53)

for a positive-direction progressive plane wave.

The SI units are

[z₀]	=	(54)
	= Pas∕m.	(55)

The unit

Pa s∕m

(56)

is also called the rayl.

For a negative-direction progressive wave,

|----------| p ′ = − z0u.| ------------

(57)

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

Figure 3. In a progressive plane wave traveling to the right, acoustic pressure and particle velocity are in phase. Their amplitude ratio is the characteristic specific acoustic impedance z₀ = ρ₀c.

9 Acoustic intensity

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

In one dimension the instantaneous intensity is

|----------------------| |I(x,t) = p′(x,t)u(x,t).| ------------------------

(58)

In three dimensions, the acoustic intensity vector is

|--------| |I = p′u. | ---------

(59)

Its SI unit is

|-------| W ∕m2. | ---------

(60)

A positive value of

I

(61)

means net energy flow in the positive

x

(62)

direction.

10 Average intensity of a harmonic plane wave

For a right-moving harmonic plane wave,

p′ = ^pcos(kx − ωt ),

(63)

and

u = u^cos(kx − ωt ).

(64)

Then

2 I = ^p ^ucos (kx − ωt ).

(65)

Since

1 ⟨cos2⟩ = -, 2

(66)

|------1---| ⟨I⟩ = -^p^u.| -------2----

(67)

Using

^p = z0^u,

(68)

|-------------------| | -^p2- 1- 2 | ⟨I⟩ = 2z = 2z0^u . | ---------0-----------

(69)

Since

^p prms = √--, 2

(70)

and

√^u-- urms = 2 ,

(71)

|----------------------| | p2 | |⟨I⟩ = -rms = ρ0c u2rms.| --------ρ0c------------

(72)

11 Complex-amplitude intensity

Let

[ ] p ′ = Re ^pe− iωt ,

(73)

and

[ ] u = Re ^ue−iωt .

(74)

Then the time-averaged intensity is

|----------------| | 1- ∗ | ⟨I⟩ = 2 Re (^p^u ).| ------------------

(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

|------1-----| wK = -ρ0u2.| -------2------

(76)

The compressional contribution is

|------------| | (p′)2 | |wP = ----2.| -------2ρ0c---

(77)

Thus

|--------------------| | 1 2 (p′)2 | |w = -ρ0u + -----2.| ------2-------2-ρ0c--

(78)

For a progressive plane wave,

′ p = ρ0cu.

(79)

Therefore

wK = wP

(80)

instantaneously.

The kinetic and compressional energy densities are equal.

13 Energy density and intensity

For a progressive plane wave,

I = p′u.

(81)

Using

p′ = ρ0cu,

(82)

I = ρ0cu2.

(83)

But

w = ρ0u2.

(84)

Therefore

|-------| I = cw | ---------

(85)

for an ideal progressive plane wave.

Likewise,

|------------| -⟨I⟩-=-c⟨w-⟩.-|

(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

ρ0 = 1.204kg ∕m3,

(87)

and

c = 343 m ∕s.

(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

p = 0.200Pa rms

(92)

in the same air.

Then

⟨I⟩	=	(93)
	≈ 9.69 × 10⁻⁵ W∕m².	(94)

The corresponding particle-velocity RMS amplitude is

u_rms	=	(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

P

(97)

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

4πr2.

(98)

Neglecting absorption,

|------------| | -P---| I (r) = 4πr2.| --------------

(99)

Therefore

|--------| |I ∝ -1 .| -----r2--|

(100)

Since

2 I ∝ p rms,

(101)

the far-field pressure amplitude scales approximately as

----------- | 1 | |prms ∝ --.| --------r--|

(102)

Figure 4. For an ideal point source in a lossless free field, the same acoustic power crosses increasingly large spherical surfaces. Intensity therefore falls as 1∕r², while pressure amplitude falls as 1∕r.

17 Example 3: spherical intensity

Suppose a source radiates

P = 1.00 W

(103)

uniformly.

r = 2.00 m,

(104)

I	=	(105)
	≈ 1.99 × 10⁻² W∕m².	(106)

r = 4.00 m,

(107)

the intensity is one fourth as large:

|----------------------| I ≈ 4.97 × 10 −3W ∕m2. | ------------------------

(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

|-------------(----)-----| | I | |LI = 10 log10 ---- dB. | ----------------Iref-------

(109)

For sound in air, a conventional reference intensity is

|------------------| I = 10 −12W ∕m2. | -ref----------------

(110)

An increase in intensity by a factor of

10

(111)

raises the level by

10dB.

(112)

A factor of

100

(113)

raises it by

20dB.

(114)

19 Sound pressure level

For sound pressure in air, define

|------------(-----)-----| | prms | Lp = 20log10 pref dB, | --------------------------

(115)

with the conventional reference pressure

|--------------| -pref-=-20-μPa.-|

(116)

The factor

20

(117)

rather than

10

(118)

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

2 I ∝ p rms.

(119)

20 Example 4: pressure level

For

p = 0.200 Pa, rms

(120)

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

Figure 5. The decibel scale is logarithmic. Multiplying intensity by ten adds 10 dB, while multiplying pressure amplitude by ten adds 20 dB when the acoustic impedance is unchanged.

21 Distance and level in a free field

For ideal spherical spreading,

I ∝ r−2.

(124)

If the distance doubles,

I2 1- I1 = 4 .

(125)

Thus

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

The familiar approximately

6 dB

(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:

|------| | ^p | z = -.| -----^u--

(129)

In general,

z

(130)

can be complex and position dependent.

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

|-------------| z = z0 = ρ0c. | ---------------

(131)

The characteristic value

z0

(132)

is real.

23 Acoustic impedance based on volume velocity

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

|--------| |U = Su, | ----------

(133)

where

S

(134)

is cross-sectional area.

The acoustic impedance is then

|--------| | ^p | |Z = ^-.| ------U--

(135)

For a uniform progressive plane wave in a duct,

|------ρ-c-| |Z0 = -0-.| -------S---|

(136)

Its SI units are

---------- | 3 | Pa-s∕m--.-

(137)

Thus:

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

Figure 8. Specific acoustic impedance relates pressure to particle velocity, while acoustic impedance relates pressure to volume velocity. For a uniform duct, the two differ by the cross-sectional area.

24 Normal-incidence interface

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

Let

z1 = ρ1c1,

(138)

and

z2 = ρ2c2.

(139)

Write the pressure fields as

pi = Piei(k1x− ωt),

(140)

pr = Prei(− k1x−ωt),

(141)

and

pt = Ptei(k2x− ωt).

(142)

For the corresponding particle velocities,

ui = pi, z1

(143)

ur = − pr, z1

(144)

and

ut = pt. z2

(145)

25 Boundary conditions at an ideal fluid interface

Pressure is continuous:

|-------------| Pi-+-Pr-=-Pt.--

(146)

Normal particle velocity is continuous:

|--------------| |Pi-−-Pr-= Pt.| | z1 z2 | ----------------

(147)

These two conditions determine reflection and transmission.

26 Pressure reflection coefficient

Define

|--------| | Pr | rp = --.| ------Pi--

(148)

Solving the boundary equations gives

|-------------| | z2 − z1 | rp = -------. | -----z2-+-z1--

(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

|-----P--| |tp = -t.| ------Pi--

(150)

Then

|------------| | --2z2-- | tp = z + z .| ------1----2--

(151)

The pressure amplitude can exceed the incident pressure amplitude when

z2 > z1.

(152)

This does not imply transmitted power greater than incident power.

Figure 6. At normal incidence, acoustic pressure and normal particle velocity are continuous across an ideal fluid interface. Impedance mismatch determines the reflected and transmitted pressure amplitudes.

28 Power reflection and transmission

For a plane wave,

|P-|2 ⟨I⟩ = 2z

(153)

when

P

(154)

is the complex peak pressure amplitude and

z

(155)

is real.

Therefore the reflected power fraction is

|----------| |ℛ = |rp|2.| -----------

(156)

The transmitted power fraction is

|------------| | z1 2 | |𝒯 = z2|tp| .| --------------

(157)

For real positive impedances,

|----------------| | --4z1z2--- | |𝒯 = (z + z )2 .| --------1----2---

(158)

Also,

|------------| -ℛ-+--𝒯-=--1.|

(159)

29 Impedance matching

|--------| -z1 =-z2,|

(160)

then

|-------| rp-=-0.-|

(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

z ≈ 413 Pa s∕m, air

(162)

and

z ≈ 1.48 × 106Pa s∕m. water

(163)

For a plane wave incident from air onto water,

r_p	=	(164)
	≈ 0.99944.	(165)

Thus

|--------------| |ℛ ≈ 0.99888. | ---------------

(166)

The transmitted fraction is only

|-----------------| 𝒯 ≈ 1.12 × 10−3. | ------------------

(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:

|------| |u = 0.| --------

(168)

This is the high-impedance limit:

z2 → ∞.

(169)

Then

|----------| |r → +1. | --p--------

(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

|------| p ′ = 0. --------

(171)

This corresponds to the low-impedance limit:

z → 0. 2

(172)

Then

|----------| -rp-→--− 1.|

(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.

Figure 7. A rigid termination is a particle-velocity node and pressure antinode. An ideal pressure-release termination is a pressure node and particle-velocity antinode.

33 Standing waves and local impedance

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

The ratio

z(x) = ^p(x)- ^u(x)

(174)

then depends on position.

It can also become complex.

Therefore

z0 = ρ0c

(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

′ p (x, t) = 2 ^pcos(kx) cos(ωt),

(176)

with the wall at

x = 0.

(177)

The associated particle velocity has the form

2 ^p u(x, t) = ---sin(kx) sin(ωt ). z0

(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:

|---------------------| Itotal =-I1 +-I2-+-⋅⋅⋅ .

(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

I,

(180)

then

Itotal = 2I.

(181)

The level increase is

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

Thus doubling acoustic intensity adds approximately

3 dB.

(184)

37 Connection to string impedance

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

∘ --- Zstring = T μ.

(185)

For a plane acoustic wave,

z0 = ρ0c.

(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

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

39 Practice exercises

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