Three-dimensional wave equation The one-dimensional wave...

Noise Engineering / Aeroacoustics - 1 -

Three-dimensional wave equation

❖The one-dimensional wave equation

Flow variables

• Pressure :

• Density :

• Velocity :

(It is constrained to the x-direction)

Consider a control volume of unit length in the y and z

directions and of width δx.

),( txpp

),( tx

),( txvv

txuvtxv ,0,0,,

Control volumey

xu xu

x xx

x




Since mass must be conserved, the change of mass within

the control volume in unit time is related to the difference

between the rate of inflow and outflow of the fluid crossing

the end faces of element (Continuity)

The small quantities can be negligible. Then we have

Continuity equation.

txxutxuxt

,, 00

00

x

u

t




Similarly a linearized form of the equation of conservation

of momentum, when all products of the small quantities

have been neglected.

Differentiate the linearized continuity equation with respect

to t, and subtract from the momentum equation

differentiated with respect to the x.

),(),(0 txxptxpxt

u

00

x

p

t

u

02

2

2

2

x

p

t




In general an increase of pressure tends to increase the

density and so the gradient is positive constant, c.

Now, wave equation is derived.

2cp

01

2

2

2

2

2

x

p

t

p

c




A general solution of the one-dimensional wave equation

The pressure perturbation maintains the same form, but

travels at speed c.

ctxgctxftxp ,

p

x

0ct

0, tp 0, ttp




Harmonic waves

• One particular form of waves is described by the solution of wave

equation when f and g are harmonic functions

where, ω is the frequency of the wave (rad./sec.)

2π/ω is the period of the wave

• In the plane wave,

Then it becomes

cxticxti BeAetxp //,

cxtiAetxp /,

),(, 0 txuctxp



❖The basic equation

Conservation of mass

• Consider mass conservation for a small volume δV enclosed by

surface S.

• The rate of increase of mass within the volume δV must be equal to

the rate at which mass flows into the volume across the bounding

surface

n

S

VdSt

VS

nv

0




Conservation of mass

• Gauss’s theorem can be used to rewrite the surface integral, the

linearized equation is derived.

• In tensor notation,

0 div0

v

t

00

i

i

x

v

t

n

S

V




Momentum equation

• The rate at which the momentum increases in unit volume of

weakly disturbed matter must be equal to the force applied to that

unit volume by neighboring material.

Force applied to unit volume by neighboring material

Momentum increased

in unit volume of weakly

disturbed matter

• In tensor notation, n

S

V

0 grad0

p

t

v

00

i

i

x

p

t

v




Wave equation

• From the manipulation of mass conservation equation and

momentum equation, wave equation is derived.

(Differentiate with respect to the t )

(Divergence of equation)

• As a result, we can derive the wave equation.

ii

i

xx

p

t

v

0 0

tx

v

t i

i

00

01 2

2

2

2

p

t

p

c




Wave equation

• (Proof) the velocity field satisfies the wave equation

01

01

01

222

2

2

2

0

2

02

2

2

'2'

2

2

2

ii

ii

ii

vvtc

t

v

t

v

tc

px

pxtc

t

v

x

p i

i

0

'




The velocity potential

• From momentum equation, taking curl both sides. Then we have

• Define vorticity :

• If the fluid is initially at rest, the flow is irrotational.

• The forces arising from the pressure gradient are conservative and

act through the mass center of a particle. They can not induce spin

or vorticity.

0'0

pv

t

v

0

t





• Define velocity potential :

• Momentum equation becomes

• It means sum of pressure perturbation and is a constant and

can be set to zero.

v

0'0

p

t

t

0

tp

0'





• Using velocity potential, the wave equation becomes

• Laplace equation

(1) Incompressible flow

(2) Slow acoustic motion

(3) In the vicinity of a singularity

• The wave equation limits to Laplace’s equation for 3 cases.

01 2

2

2

2

tc

c

0

2

2

t

22

2

2

0 rt

02




Differences between Aerodynamics and Acoustics

• Steady Aerodynamics (Bernoulli equation)

• Acoustics (Unsteady Bernoulli)

.2

1 2 constup

tp

0'




The energetics of acoustic motion

• Multiply the quantity to the acoustic wave equation

for the potential.

• The first term :

• The second term :

tc // 2

0

022

2

2

2

0

cttc

2

0

22

2

0

2

2

2

0

2

'

2

1

c

p

ttcttct

2

021

2

00

2

0

22

2

0

' vt

vpx

xxtxtxxxtc

tc

i

i

iiiiii




The energetics of acoustic motion

• Re-write the acoustic wave equation for the potential

• Definition

(1) Potential energy :

(2) Kinetic energy :

(3) Energy flux :

• Energy conservation

0'2

1'

2

1 2

02

0

2

i

i

vpx

vc

p

t

2

0

2'

2

1

c

pep

2

02

1vek

vpI

'

Ix

Iee

t i

ikp



❖Some simple three-dimensional wave fields

General plane wave

• General form of waves

• The pressure perturbation corresponds to waves propagating along

the one direction

where k,

is called the wave number vector.

ctxgctxftxp

,

xktiAetxp ,

321 ,, kkkk




Spherical wave

• A wave in which all the flow parameters are functions of the radial

distance, r, and time, t, only.

• The pressure gradient polar coordinates is given as follow.

• Since p′ is independent of θ and φ, the wave equation reduces to

2

2

222

2

2

2

sin

1sin

sin

11

p

r

p

rr

pr

rrp

r

pr

rrt

p

c

2

22

2

2

11




Spherical wave (Alternative derivation)

• Conservation of mass

• Conservation of momentum of radial component

• Eliminate u and ρ' from above equations

2

0

2

integral

2

0

2 4'

4 urrt

rurt

rr

r

p

t

u

0

r

pr

rt

p

c

r 2

2

2

2

2

r tru ,

r




Spherical wave

• Consequently, a little algebra shows that this equation is precisely

the same as

• This satisfies one-dimensional wave equation :

• So that,

011

2

2

22

2

2

pr

rrpr

tc

ctrgctrftrrp ,'

r

ctrg

r

ctrftrp

,




Causality condition (Sommerfeld radiation condition)

• In open space, information about the current source activity should

not be contained in past waves. (The sound must not anticipates its

cause.)

• The solution of the wave equation in open space consists entirely of

outward travelling waves.

• Specifically, Sommerfeld required that the limit

0lim

r

pc

t

pr

t




Spherical wave

• This is a test for their physical reasonableness. Only the outward

wave component is admitted.

• For example, for an outward propagating spherically symmetric

wave of frequency ω,

where A is a complex constant.

r

Aep

crti

r

ctrftrp

,'




Spherical wave

• From conservation of the radial component momentum,

where φ, the phase angle, is

trpc

rc

c

rcee

r

A

ercr

i

i

Atru

i

crti

crti

,'/1

/1

1,

0

2

0

2

/

/

2

0

rc /tan 1




Spherical wave (Note : special features)

1. Pressure & velocity perturbation are not in phase

2. Impedance is less then that of the plane wave

3. For small r (for NEAR field)

4. For large r (for FAR field)

The pressure and velocity are nearly in phase and the spherical

wave behaves like a plane wave.

cc

rcu

pz 00

2/1

1'

t

up

r

p

c

i

re

r

A

r

p

t

ucrti

~'

'1'0

trpc

trur

c

rr,'

1,0tanlimlim

0

1



❖Some examples of 3-D waves

The sound scattered by a small bubble

• If the fluid is in unsteady motion, the pressure in the vicinity of the

bubble will vary, and the bubble will respond by pulsating and

thereby producing a secondary centered sound wave.

• In practice, it is occurred from metallic casting, bubble in the fluid,

cavitations, and boiling coolant, etc.

• Assumption:

- High frequency approximation

- The case which is the bubble pulsate with

large amplitude and at very low frequency

will be consider later more thoroughly.





• The surface of the void is unable to sustain any pressure variation

the sum of the incident and scattered pressures must vanish at the

boundary surface r=a.

where

p'i(t) : environmental pressure fluctuation

p'(r, t) : centered outgoing sound wave

0,'' taptp i

arr

ctrftrp

,,





• From the boundary condition,

• Let X=a-ct,

tpa

ctaftap i','

c

XaapXf i '





• Hence,

• The scattered pressure wave travels

radially outwards with its amplitude

inversely proportional to the distance

travelled.

• The pressure time history at the void is

faithfully mimicked at a time (r-a)/c later

by the sound pressure at r.

c

artp

r

a

r

ctrftrp i,




The sound generated by a pulsating sphere

• Suppose that the radial velocity at radius a on the surface of the

spherical radiator is specified to be ua(t).

• Pressure field generated by this vibration

• The radial component of the momentum balance is

c

artap

r

a

c

artp

r

atrp i ,'','

c

arta

t

p

rc

a

c

artap

r

a

r

p

t

u

,,

20





• Consider a time harmonic wave with angular frequency ω. Let

• Then, the momentum equation becomes

• At r=a, u=ua, so that

ti

aa

titi eutuerutruerptrp ˆ,)(ˆ,,)(ˆ,'

apr

a

c

rieui cari ˆ1ˆ

20

c

ai

uaiap a

1

ˆˆ 0





• the complete field induced by the pulsating spherical boundary is

cari

aeuc

c

aic

ai

r

arp /

0ˆ

1

ˆ





• The (specific) radiation impedance, Z

- The radiation impedance, Z

- This is often presented as a fraction of a plane’s acoustic impedance,

ρ0c. That is termed the specific radiation impedance

- When ωa/c is large the specific radiation impedance is purely real and

equal to unity, but when ωa/c is small the specific radiation

impedance is purely imaginary.

u

pZ

'

2

01

1

1

Z

a

c

a

ci

c

aic

ai

c





• Helmholtz number (Compactness ratio)

• Case I : ωa/c≫1

the circumstance of sphere is larger than the acoustic wave length (the

sphere is large enough to be called non-compact)

The entire disturbance field set up by the pulsations of the sphere is

simple and wave-like.

Z becomes purely real :

a

c

a 2

1 , , 0

c

a

c

artcu

r

atrp a

1Z





• Case II : ωa/c≪1

the circumstance of sphere is smaller than the acoustic wave length

(the sphere is small enough to be called compact)

the vicinity of the vibrating surface is not wave-like.

It's virtually independent of the compressibility.

Z becomes purely imaginary :

1 , , 0

2

c

a

c

art

t

u

r

atrp a

c

aiZ





• By comparing two cases,

• This implies that

1. Small acoustic radiator are very inefficient and their response

increase with frequency.

2. Large acoustic radiator are efficient and their response remain same

over all frequency range.

com pactnoncom pact rpc

airp

ˆˆ




Resonant scatterer

• The sound generated by an isolated gas bubble with large amplitude

and at low frequency

where

: mean radius

: complex amplitudes of pressure and velocity

• Total surface pressure on the bubble’s external surface is

where p0 is the mean pressure in the external fluid.

tia e

c

ai

uiatap

0

00

1

ˆ,ˆ

0a

aup ˆ,ˆ

tappp i ,ˆ'0




Resonant scatterer

• Pressure inside the bubble, pg is

• Assume that the gas inside the bubble is perfect gas and in an

adiabatic state, then

• Differentiating above equation w.r.t. time,

a

Ttapppp ig

2,ˆ'0

3

0

3 apap gg

033 13

00

3

0

13

00

3

0

ag

g

g

guapa

t

p

t

aapa

t

p

a

g

g ua

pp

t 0

03




Resonant scatterer

• The bubble surface velocity can be determined in terms of the

incident pressure field.

1

2

0

0

2

00

1

3

00

2

00

0

0

2

00

1

3

00

2

00

0

0

2

00

/1ˆ

213

3

/1ˆ

23

/1ˆˆ

caip

a

i

a

T

a

p

caip

a

i

a

T

a

p

caip

a

iu

i

i

g

ia




Resonant scatterer

• For air and vapor bubbles in water,

so that the bubble response is seen to be that of a lightly damped

oscillator which resonates ate the frequency ω0.

• Surface tension

- Acts to raise the resonance frequency but not usually by very much

- Can be negligible for bubbles resonant below 30kHz

10 c

a

3

00

2

00

02

0

213

3

a

T

a

p



❖A more elaborate 3-D wave field

A whole variety of solution

One such pressure field can be obtained

where θ is the angle between the 1-axis and the position

vector x, cosθ=x1/r.

00 22

2

222

2

2

ii xc

xtc

t

r

ctrf

rr

ctrf

xtp cos ,

1

x




The sound of an oscillating rigid sphere

• Consider a rigid sphere whose center moves along the x1 axis in a

small amplitude harmonic motion about the origin of coordinates.

• The sound field induced by the

sphere’s vibration is actually the

field of the form expressed with

harmonic time dependence;

r

e

rAtxp

crti /

cos,'

ti

r eUu cos

a

tiUe





• The complex amplitude A can be determined by applying the radial

momentum equation at the surface of the sphere

• So that

r

e

rA

r

pui

t

crti /

2

2

00 cos

crtiAerccr

i

riu /

2

2

23

0

22cos

ti

r eUu cos

a

tiUe





• On the surface of sphere,

• There is the parameter ωa/c, the compactness ratio.

1. Compact sphere

1. Near-field

2. Far-field

2. Non-compact sphere

222

/3

0

//12 cacai

eUaiA

cai

r

e

rcacai

Uaitrp

carti /

222

3

0

//12,,





• Compact sphere (ωa/c«1)

- Radiates sound very ineffectively

• Far field

- The pressure field is in phase with the velocity field

- Radiates to infinity carrying energy away from the sphere

• Near field

- The motion is hardly influenced by the speed of sound, or, therefore,

by the material’s compressibility

r

ice

r

acU

c

a

r

e

rUaitrp

crti

crti

1cos

cos,,'

/

021

2

22

/3

021





• On the surface of the compact sphere

• The reaction of the material surrounding the vibrating sphere

produces the drag. Its magnitude is equal to the force required to

overcome the inertia of half the mass of fluid displaced by the

sphere.

Total drag

In-phase drag

1 ; 22

11cos

2

1,,

32

0

c

a

c

ai

c

aeaUitap ti

32

3

00 22

11

3

2cos,,sin2

c

ai

c

aUeiaadtapa ti

4

2

0

2

3

1

c

aUca





• Non-compact sphere (ωa/c»1)

- radiates sound very effectively

- the surface pressure is in-phase with surface velocity.

• The drag on the non-compact sphere is

Total drag

In-phase drag

cartier

acUtrp /

0 cos,,

dcUeaadtapa ti

0

2

0

2

0cossin2cos,,sin2

2

0

2

3

4Uca



❖Two dimensional sound waves

The pressure in the water

• Varies due only to hydrostatic effects

• The vertical acceleration being negligible compared to gravity

Mass conservation

Combining two equations

t

v

x

p

xg

zhgpp

0

tx

vh

02

22

2

2

xc

t



2-D wave equation

• valid only when λ»h, an analogue of 2-D sound wave.

• what happens if λ«h?

- va is a function of frequency

- groups of waves at different frequency disperse.

General solutions

• 1-D wave

• 3-D wave

• 2-D wave :

Huygen's Principle does not work

in Two Dimensional wave field.


ctxf 1

ctrfr

1



A simple 2-D wave solution

In general,

is singular when


ctr

ctrrtc

txp

,0

,1

,'222

crt

drtc

ftxp

/

222,'

rtcrtcrtc 222

rtc



Approximately,

Far field form of the integral is

where


rtcr

rtcrtcrtc

2

222

r

ctrFtxp

~,'

crt

dcrt

f

cctrF

/

/2

1



Illustration of simple wave fields


One dimensional

: Pulse propagates with speed c at constant

amplitude and shape.

Two dimensional

: Singular wave front propagates with

constant speed c. Succeeding ‘wake’

becomes weaker as it travels.

Three dimensional

: Pulse propagates with constant shape and

speed but amplitude is inversely

proportional to distance travelled.

r

1

r

1




Simple harmonic wave

• Consider the sound field has frequency ω,

• Changing variable s=c(t-τ)/r, then we have

• This integral cannot be evaluated explicitly. Instead it is given a

name and tabulated in Mathematical Tables.

→ Hankel function!

crt

ti

drtc

eAtrp

/

222,'

1 2/12

/

1,' ds

s

e

c

Aetrp

csriti





• Hankel function: Zeroth order

• Using a zeroth order Hankel function, the pressure field is

simplified as

1 2/12

2

0

1

2ds

s

eiXH

iXs

c

rHe

ic

Atrp ti 2

02

,'





• Far field approximation

• Near field approximation (for small r)

• The pressure perturbation and hence the velocity potential have a

logarithmic singularity at the origin. This is the characteristic form

for the velocity potential due to two-dimensional sources in steady

aerodynamics.

4//

2~,'

icrti

rAe

rctrp

c

re

c

Atrp e

ti log~,'

Three-dimensional wave equation The one-dimensional wave...

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