Acoustic Wave Equation

Acoustic Variables

• Pressure

• Density – Condensation

• Velocity (particle)

• Temperature

op P -Po

o

s

ut

Sound Speed

Bulk modulus Bc

density

Air Water SteelBulk Modulus 1.4(1.01 x 105) Pa 2.2 x 109 Pa ~2.5 x 1011 Pa

Density 1.21 kg/m3 1000 kg/m3 ~104 kg/m3

Speed 343 m/s 1500 m/s 5000 m/s

Please Memorize!!!

Necessary Differential Equations to Obtain a Wave Equation

• Mass Continuity• Equation of State• Force Equation – N2L

Assumptions: homogeneous, isotropic, ideal fluid

Equations of State

Ideal Gasses: k= rTP

o

o

PP

o

o

PP

Real Fluids: o o

22

o o2

1...

2

o

P PP=P

Vp B B Bs

V

o

oB

P

Continuity Equation

u 0t

x xu x x dx

u

dxdy

dzyx z

dmdm dmdM

dt dt dt dt

y zx xx x dx

dm dmdMu dydz u dydz

dt dt dt

yx zx xx x

x

dmu dmdMu dydz u dx dydz

dt x dt dt

yx z

x zy

uu udMdxdydz dxdydz dxdydz

dt x y z

yx z

x zy

uu ud

dt x y z

Force Equation

xP x dx

P

dxdy

dz x x x dxdf dydz dydz

P P

x x xx x

df dydz dx dydz dxdydzx x

P PP P

x zy

ˆ ˆ ˆdf i j k dxdydz dmg dxdydz dmgx y z

P P PP

Fluid Accelerationu

dxdydz g dxdydz dxdydzt

P

u x, y, z, t u x dx, y dy, z dz, t dt

t 0

u x dx, y dy, z dz, t dt u x, y, z, ta lim

t

u u u uu x dx, y dy, z dz, t dt u x, y, z, t dx dy dz dt

x y z t

x y z

u u u uu x, y, z, t u dt u dt u dt dt

x y z t

x y z x y z

u u u u u u ua u u u u u u u u u

x y z t t x y z t t

Lagrangian and Eulerian Variables

• Eulerian – Fixed Moorings• Lagrangian – Drifting Buoys

Du ua u u

Dt t

Material, substantial or Lagrangian Derivative

Eulerian Derivative

ConvectiveTerm

Newton’s Second Law

udxdydz g dxdydz dxdydz u u

t

P

ug u u

t

P

u 0t

p Bso

oB

P

Linear Continuity Equation

o p P P

o os

o oo o

su s u 0

t t

oo

su 0

t

su 0

t

Linear Force Equation

ug u u

t

P

o p P P

o os

o o o o o

up g s p gs

t

P

o

up

t

o

up

t

su 0

t

2o

up

t

2

2

s u0

t t

2 22 o

o 2 2

s pp

t B t

p Bs

22

2 2

1 pp

c t

2

o

Bc

Linear Wave Equation

Velocity Potential

u

o o

up

t t

o op p 0t t

o p 0t

Variation of sound speed with temperature

o

2o oB c

o

PP 2

o

c

oP

k o ko= rT = rT oP P

2 o koo ko

o

rTc rT

2

kc rT

2C Ck k

2o ko

273 T Tc T T1

c T 273 273 273

Co

Tc c 1

273

Speed of sound in water-temperature, pressure, and salinity

2 2 4 3 2 2c t, z,S 1449.2 4.6t 5.5x10 t 2.9x10 t 1.34 10 t S 35 1.6x10 z

with the following limits:

0 t 35 C

0 S 45 p.s.u.

0 z 1000 meters

Sound Speed Variations with Temperature and Salinity (z = 0 m)

13801400142014401460148015001520154015601580

0 5 10 15 20 25 30 35 40

Temperature (C)

So

un

d S

pe

ed

(m

/s)

0

30

35

40

ppt salinity

Class Sound Speed Data

Class Sound Speed in Water Data

y = 0.0004x3 - 0.0807x2 + 6.2061x + 1393.4

1400

1420

1440

1460

1480

1500

1520

0 5 10 15 20 25

Temp (C)

So

un

d S

pee

d (

m/s

)

Series1

Poly. (Series1)

Harmonic 1-D Plane Waves2

22 2

1 pp

c t

j t kx j t kxp Ae Be

o

up

t

j t kx j t kx j t kx j t kxo

uAe Be Ake Bke

t

j t kx j t kx

o

1 uAke Bke

t

j t kx j t kx j t kx j t kx

o o

1 k k 1u A e B e Ae Be

c

Condensation and Velocity Potential

2

p ps

B c

j t kx j t kx

o

1u Ae Be

c

j t kx j t kx

o o o

1 1 1 p jpA e B e

c jk jk j ck

Specific Acoustic Impedancef

Zu

Mechanical Impedance

Z f / A pz

A u u

Pa s

raylm

For a plane wave:

j t kx

oj t kx

o

j t kx

oj t kx

o

Aec

1Ae

cpz

u Bec

1Be

c

In general:r specific acoustic resistance

z r jxx specific acoustic reactance

Sound Speed

Bulk modulus Bc

density

Air Water SteelBulk Modulus 1.4(1.01 x 105) Pa 2.2 x 109 Pa ~2.5 x 1011 Pa

Density 1.21 kg/m3 1000 kg/m3 ~104 kg/m3

Speed 343 m/s 1500 m/s 5000 m/s

Spec. Ac. Imp. 415 Pa-s/m 1.5 x 106 Pa-s/m 5 x 107 Pa-s/m

z cAnalogous to E-M wave impedance

Z

Plane wave in an arbitrary direction

x y zj t k x k y k zp Ae

2

22 2

1 pp

c t

x y z

x y z

j t k x k y k z2

j t k x k y k z22 2

Ae1Ae

c t

x y z x y z

2j t k x k y k z j t k x k y k z2 2 2

x y z 2k k k Ae Ae

c

22 2 2x y z 2

k k kc

Shorthandx y zˆ ˆ ˆk k i k j k k

ˆ ˆ ˆr xi yj zk

x y z j t k rj t k x k y k zp Ae Ae

x

y

z

xkcos

k

ykcos

k

zkcos

k

k

Direction Cosines

k r const

k r k Surfaces (planes) of constant phase

Propagation Vector

2 2 2 2 2 2x y zk k k k k cos cos cos

-1

-0.5

0

0.5

1 -1

-0.5

0

0.5

1

-2

-1

0

1

2

-1

-0.5

0

0.5

1

-1

-0.50

0.51

-1-0.500.51

0

5

10

-1

-0.50

0.51

-1-0.500.51

k in x-y plane

x

z

ky x n2

k

k

ˆ ˆk k cos i k sin j

j t kx cos kysinp Ae

Energy

2 2k o o

1 1E mu V u

2 2

0 0

x V

p

x V

E Fdx pdV

o

2o

pB c

P

V const. dV Vd 0

2

d dpdV V V

c

0

pV 2

p o2 2oV 0

dp 1 pE pdV pV V

c 2 c

Energy Density2 2

2 2k p o o o o o2 2 2

o o

1 1 p 1 pE E E u V V u V

2 2 c 2 c

2 22 2

i o o2 2 2o o o

E 1 p pu u

V 2 c c

E

p P cos t kx o

Pu cos t kx U cos t kx

c

2 2T T 2

i i 2 2To o0 0

P cos t kx1 1 Pdt dt

T T c 2 c

E E E

2o o

o

U U P PU

2 2 c 2c

E

Average Power and Intensity

A

cdt

iT TdE Acdt E

TT

dEAc

dt E

2 2T

oTo

U 1 P 1I c c PU

A 2 2 c 2

E

For plane waves

Effective Average - RMS

T

2e rms

0

1F F f t dt

T

e

UU

2 e

PP

2

22 e

o e e eTo

PI cU P U

c

Intensity of sound• Loudness – intensity of the wave. Energy

transported by a wave per unit time across a unit area perpendicular to the energy flow.

Source Intensity (W/m2) Sound Level

Jet Plane 100 140

Pain Threshold 1 120

Siren 1x10-2 100

Busy Traffic 1x10-5 70

Conversation 3x10-6 65

Whisper 1x10-10 20

Rustle of leaves 1x10-11 10

Hearing Threshold 1x10-12 1

2 2

o o

p PI

c 2 c

Sound Level - Decibel

ref

IIL 10log

I

12

ref 2

WI 1x10

m

• Ears judge loudness on a logarithmic vice

linear scale

• Alexander Graham Bell

• deci =

• 1 bel = 10 decibel

Why the decibel?

1

10

ref

I"bel" log

I

ref

IIL(in dB) 10log

I

Reference Level Conventions

LocationReference Intensity

Reference Pressure

Air 1 x 10-12 W/m2 20 Pa

Water 6.67 x 10-19 W/m2 1 uPa

2ref

refo

pI =

c

Historical Reference

• 1 microbar• 1 bar = 1 x 105 Pa• 1 bar = 1 x 105 Pa

• So to convert from intensity levels referenced to 1 bar to intensity levels referenced to 1 Pa, simply add 100 dB

510 Pa20log 100 dB

1 Pa

Sound Pressure Level

ref

IIL 10log

I

2

rms

ref ref

p pSPL 20log 20log

p p

Mean Squared Quantities:Power, Energy, Intensity

Root Mean Squared Quantities:Voltage, Current, Pressure

“Intensity Level”

“Sound Pressure Level”

Example

• Tube with a piston driver– a=2.5 cm– f = 1 kHz– 154 dB in air

• What are the– rms piston displacement– intensity– power

Spherical Waves2

22 2

1 pp

c t

2 22 2

2 2 2 2 2 2 2

1 p 1 p 1 p 1 pp r sin

r r r r sin r sin c t

n m jm jwt

nn

j krp P cos e e

y kr

Standing waven=0,1,2,3,…m=-n,…,+n

(1)n m jm jwt

n(2)n

h krp Y cos e e

h kr

(1),(2)n n nh j jy

Traveling wave

Spherical Waves For Usjkr

(1),(2)n

eh

r

j t krAp e

r

o

1 1 jkru p

c jkr

2 2j

o 2 2

k r jkrz c ze

1 k r

o 2 2

krz c

1 k r

1

tankr

z

acr

acx

z

acr

acx