Lecture 21-22: Sound Waves in Fluids

• Sound in ideal fluid• Sound in real fluid. Attenuation of the sound

waves

1

Sound in an ideal fluid• An oscillatory motion with small amplitude in a compressible fluid is called

a sound wave. At each point, a sound waves causes alternate compression or rarefaction.

• In ideal fluid, there are no dissipation processes (no viscosity and thermal conductivity). The governing equations are

.constant

,0div

,

S

vt

pvvtv

Any motion in ideal fluid is adiabatic.

2

We will assume that the oscillations are small. is small and all thermodynamic quantities might be split into the background (time-independent and uniform in space) and oscillatory (non-uniform) parts,

v

00 ,ppp

The linear (leading) terms of the governing equations are

.0div

,

0

0

vt

ptv

Unknowns: ,, pv

Two thermodynamic variables are sufficient to define the thermodynamic state of a single-phase fluid, say, density and entropy. But, in ideal fluid, entropy is constant, hence, there is only one independent thermodynamic variable, density. Variations of pressure,

S

pp

Equations of linear acoustics

adiabatic compressibility 3

It is easy to show that the oscillatory fluid motion defined by the linearized equations is irrotational (similar to the Lecture: Gravity Waves). Hence, it is convenient to introduce the velocity potential,

v

From the Euler’s equation,

tp

0

Then, the continuity equation can be re-written to

S

pcc

t

,022

2

wave equation:

Let us consider a plane wave,

Similar wave equations can be derived for the pressure and density.

tx,

.02

22

2

2

xc

t

To solve this equation, let us introduce two new variables, ctx

ctx

,

4

The wave equation turns into0

First integration gives

F

Second integration, 21 ff

Or, in terms of old variables ctxfctxf 21 are any functions

(defined by initial conditions)

21, ff

ctxf 1 is a travelling plane wave propagated in the positive direction of the x-axis

ctxf 2 is a travelling plane wave propagated in the negative direction of the x-axis

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Speed of sound in perfect gasEquation of state:

RTp

pV

gas constant

molecular weight

v

p

TST c

cppT

Rp

,

Hence,

TR

c

cc

V

p

Tc and in independent of pressure

6

Attenuation of sound waves+ viscosity and thermal conductivity

Governing equations:

k

iik x

vTSv

tS

T

vt

vvpvvtv

0

3

div

div

Split the variables into constant background and oscillating parts,

SSSTTT

ppp

00

00

,

,,

Velocity has only oscillating part, .v

7

Assuming that the oscillations are small, the equations can be linearised

.

,div

,div

TtS

T

vt

vvptv

00

0

0

0

3

For a plane wave, ,,,,,,,, txptxtxvv x 00

,

,

,~

2

2

00

0

2

2

0

0

xT

tS

T

xv

t

xv

xp

tv

x

xx

where 34~

8

TSSTpp ,,,

.

,

TTSS

S

TTpp

p

T

T

Let us choose density and temperature as two independent thermodynamic variables, hence,

Small variations in pressure and entropy can be expressed as

.

,

TTca

S

Taap

V

00

2

20

2

pTV

pT

Tp

VS

TS

Tc

Tp

a

,

,,

0

2 1Introducing the following notations:

Isothermal compressibility

thermal expansion coefficient

isochoric heat capacity

9

The governing equations are now reduced to

,

,

,~

2

2

002

0

2

2

022

0

0

xT

tT

ct

Ta

xv

t

xv

xT

ax

at

v

V

x

xx

Let us further consider only a harmonic (monochromatic) wave

kxtix eTv ,,

,

,

,~

TkTiciTa

ikvi

vkTikaikavi

V

x

xx

200

2

0

20

220

0

For such a wave,

This is the linear homogeneous system of algebraic equations. Non-zero solutions exist only if the determinant of the matrix of coefficients is equal to zero. 10

0

0

02

002

0

0222

0

kciTai

iik

ikaikaki

V

~ Expansion of this determinant gives the dispersion relation (relation between frequency ω and wavenumber k)

Wavenumber k is in general complex, ikk 0

xktixixkitikxti eeeee 00 χ defines the attenuation of sound waves

Let us assume that the dissipation is weak, i.e. , κ and χ are small.

Zeroth order of the dispersion relation,

~

00

0

0 20

200

220

200

2

02

00

20

200

akcakTa

cTa

k

akak

V

V

Or, vV cTakac 0222

022

Using thermodynamic identities, can be shown thatV

p

vp

c

cac

Tacc

22

022

,

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Using thermodynamic identities, can be shown that

V

pvp c

cacTacc 22

022 ,

The zeroth order of the dispersion relation reduces to

00 k

cck

This is because the dissipative effects are neglected, so the relation for the ideal fluid was obtained.

The first order gives

pV ccc11

34

2 30

2

The higher frequency the higher dissipation. All mechanisms of dissipation makes equal contribution.

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Absorption of sound at a solid wall• Strong absorption must occur when a sound wave is reflected from a solid

wall: • Near a solid wall, there is a periodically fluctuating temperature difference

between the fluid and the wall. At the wall itself, however, the temperatures of the wall and the adjoining fluid must be the same. As a result, a large temperature gradient is formed in a thin boundary layer of fluid, which results in a large dissipation by thermal conduction.

• For a similar reason, the fluid viscosity leads to strong absorption of sound when the wave is incident in an oblique direction. In this case the fluid velocity in the wave (in the direction of propagation) has a non-zero component tangential to the surface. At the surface itself, however, the fluid must completely ‘adhere.’ Hence a large velocity gradient must occur in the boundary layer of fluid, resulting in a large viscous dissipation.

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