the ray theory equations - aancl
TRANSCRIPT
Noise Engineering / Acoustics - 1 -
Chap.4 Ray Theory
The Ray theory equations Plane wave of homogeneous medium
• A plane wave has the distinctive property that its strength and direction of propagation do not vary as it propagates through a homogeneous medium
( ) θθτ sincos, cycxyx +=
( )( )yxtiIep ,τω −=′
p′ vanes in this direction over the scale of one wavelength
Direction of propagation
wavefront : the magnitude of p′ is the same at all points along the wave front
θ x
y
Noise Engineering / Acoustics - 2 -
Ray Theory
The Ray theory equations • On the other hand, a wave propagating through a region with a
slowly varying sound speed will have a slightly curved wave front • Rays are defined to be curves which are always normal to the
wavefront • Rays are useful concept if L≫λ , i.e. high frequency waves. Then
the only effect of variation in the sound speed is that speed of propagation of the wavefront along the ray.
wavefront at time t=τ(x,y)
wavefront at time t+δt=τ(x,y)
ray
Noise Engineering / Acoustics - 3 -
The Ray theory equations
(Note)
• (i) I varies because of gradual changes in the sound speed. • (ii) The variation of I occurs over the length scale over which the
sound speed varies (generally long compared to wavelength) • (iii) The phase is a function of position and accounts for the
variation p’ over the scale of wavelength • (iv) If, L, the scale on which the sound speed varies, is MUCH
greater than the wave length, then I is slowly varying compared to τ and the moving surface, t- τ (x,y) = constant to be wavefronts
Ray Theory
p′τ
( )( )yxtieyxItyxp ,),(),,( τω −=′
Noise Engineering / Acoustics - 4 -
Ray Theory
The Ray theory equations Transmission through the stratified medium
• A continuous variation can be considered as an approximation of N jumps
c
c2
c1
θ
Wave front
c1 c2
c1 + δc
Noise Engineering / Acoustics - 5 -
Ray Theory
The Ray theory equations • When a plane wave propagates from the one medium to another
with a different sound speed, Snell’s law is valid in transmission.
• Snell’s law can be used to determine the rays paths and hence to discover where sound is heard
• The rays bent to the direction of decreasing of ‘c’
constantsin=
cθ
0
0
sinθcc ≤
c increasing
Night
θ
c decreasing
Day
Noise Engineering / Acoustics - 6 -
Ray Theory
The Ray theory equations (Example) the ocean
• Snell's Law :
• where
CCθθ sinsin
0
0 =
)1(sinsin})(1{
sin 00
0222 21 x
CC
dydxdy
dxdydxdy
αθθθ +==+
=+
=
Noise Engineering / Acoustics - 7 -
Ray Theory
The Ray theory equations (Example) the ocean
•
)sin)1)(1)(()( 02222 θαx
dxdy
dxdy
++=∴
21}sin)1(1{
sin)1(
022
0
θαθα
xx
dxdy
+−+
±=
∫ +−+
±=x
xdxxy
00
220
21}sin)1(1{
sin)1(θα
θα
0
00
22
0 sincos}sin)1(1{
sin1
21
θαθθα
θα±+−= x
0
00
2
0 sincos)sin1(
sin1,0 2
1
θαθθ
θα =−== yx
0)sin)1(1(sin1cot
022
022
20 =+−+
− θα
θααθ xy
20
22
20 cos)1(cot
αθ
ααθ ecxy =++
−
Noise Engineering / Acoustics - 8 -
Ray Theory
The Ray theory equations Continuously varying medium
• ① c is constant within each slab • ② :
Noise Engineering / Acoustics - 9 -
Ray Theory
The Ray theory equations Reflection Coefficient of Plane wave at the interface with
sound speed and densities ρ0 , ρ1 The pressure & velocity (normal direction) should be continuous @ interface
θρ
φρ
θρ
φρ
coscos
coscos
0011
0011
cc
cc
IR
+
−=
21
20
212sin1cos
−=
ccθφ
I
θ
θ
R T
φ
00cρ 11cρ
Nc
IR 1~∆=
Noise Engineering / Acoustics - 10 -
Ray Theory
The Ray theory equations compact layer (L≪λ)
• No phase difference • Transmission & Reflection depends only on the net change in
acoustic properties 'across' the layer. (Neglect the change of acoustic properties within the layer)
non - compact layer (L≫λ)
• Large phase difference between incident and reflected waves. • Energy reflected from each interface
2
222
N
II I
R =NI
EiN R
2
~nterfaces−
Noise Engineering / Acoustics - 11 -
Ray Theory
The Ray theory equations • If N→∞
" the actual continuous variation in sound speed and no energy is reflected . " (ER→0 )
• No reflection energy for a high frequency sound ray propagation through a medium in which the sound speed varies continuously .
• Energy flux is constant along the ray tube of sound propagating through a medium in which ρ0and c vary slowly ( L≫λ)
High sound pressure
Converging rays
“Ray theory can be used to determine the level of sound heard”
Noise Engineering / Acoustics - 12 -
Ray Theory
A more rigorous derivation of Ray theory Derivation of Ray theory
• Conservation of mass states that
• Momentum equation of each direction is
• These two equations can be used for substitution
• Ray series is derived from the harmonic functions of time.
( ) ( )vt
uxt 00 ρρρ
∂∂
−∂∂
−=∂∂
xp
tu
∂∂
−=∂∂
0
1ρ y
ptv
∂∂
−=∂∂
0
1ρ
∂′∂
∂∂
+
∂′∂
∂∂
=∂
′∂yp
yxp
xtp
c 00
002
2
2
111ρ
ρρ
ρ
( ) ( )( ) ( ) ( )∑∞
=
−−=′0
, ,,,n
nnyxti yxIietyxp ωτω
Noise Engineering / Acoustics - 13 -
Ray Theory
A more rigorous derivation of Ray theory • Rays are defined to be curves which are everywhere normal to the
wavefront
• From the derivatives of ray series, substitution of the correct form for the wave equation leads to
ττ grad grad, =
dxdY
dsdX
( ) ( )
0 11
111
00
00
00
00
02
222
=
∂∂
∂∂
+
∂∂
∂∂
+
∂∂∂
+∂∂
∂+
∂∂
∂∂
+
∂∂
∂∂
−
−
∂∂
+
∂∂∑
∞
=
−
yI
yxI
xyI
yxI
x
Iyy
Ixx
iIcyx
ii
nnnn
nnn
nn
ρρ
ρρττ
τρ
ρτρ
ρωττωω
Noise Engineering / Acoustics - 14 -
Ray Theory
A more rigorous derivation of Ray theory • First, the equation is to be true for all values of ω
• The coefficient of ω must also vanish,
012
22
=−
∂∂
+
∂∂
cyxττ
011
00
00 =
∂∂∂
+∂∂
∂+
∂∂
∂∂
+
∂∂
∂∂
yI
yxI
xI
yyI
xxnn
nnτττ
ρρτ
ρρ
ray ( ) ( )( )sYsX ,
s
τ grad
wavefront
‘Eikonal Equation’
Noise Engineering / Acoustics - 15 -
Ray Theory
A more rigorous derivation of Ray theory
• Using the chain rules of direction with regard to ‘s’ direction, eikonal equation leads to the Snell’s law
• The intensity, I0 , must satisfy the 2nd equation.
θττττ sin ,1=
∂∂
∂∂
∂∂
+∂∂
∂∂
=∂∂
=
dsdY
yyyxxc
ydsd
dsdY
cdsd
01120
00
02
2
2
2
000 =
∂∂
∂∂
+
∂∂
∂∂
+∂∂
+∂∂
+
∂∂
∂+
∂∂∂
ρτρ
ρτρττττ
yyxxyxI
yI
yxI
x
Noise Engineering / Acoustics - 16 -
A more rigorous derivation of Ray theory The solution to this equation is
Ray Theory
( ) ( )
∂∂
+∇−= ∫s
s scsIsI
0 00
2000
121exp
ρρτ
( ) ( ) ( ) ( )( )
( )( ) ( )
21
000
00000
∆∆
=scs
sAsA
scssIsIρ
ρ
n
n
n
s
s + δs
∆A
grad τ
δs
( ) ( )( ) ( ) ray tube thealongconstant
0
20 =
scssAsI
ρ
Energy Flux Conserved
Noise Engineering / Acoustics - 17 -
Ray Theory
Underwater sound propagation Sound waves of underwaters
• Water transmits sound waves far better than it does optical, radio or magnetic waves, and so sound is used extensively for underwater communication
• The temperature near the surface is warm, and temperature is decreased also to 1000m. It means that speed of sound is also decreases
• Under the 1000m, however, the increase of pressure leads to an increase of sound speed
Sound Speed C
Depth
Noise Engineering / Acoustics - 18 -
Ray Theory
Underwater sound propagation • The rays propagating downwards will eventually reaches the
regions which speed of sound is changed inversely, and the rays are propagating upwards. Finally the rays are trapped within the region. This regions are called ‘sound channel’
Depth
Sound Speed
A Sound Channel
1480
1000m
Noise Engineering / Acoustics - 19 -
Ray Theory
Underwater sound propagation • The refraction of sound rays by anomalies of the sound speed
profile in the ocean can lead to the formation of ‘shadow zones’ • In the case of a sound source in the ocean near a position of
maximum in the sound velocity, rays propagating upwards move into regions where the sound speed decrease and are refracted upwards, and vice versa.
Depth
Sound Speed
Source
Shadow zones
Noise Engineering / Acoustics - 20 -
Ray Theory
Sound propagation in the atmosphere Sound refraction on a day time
• During the daytime the air temperature tends to decrease with height above earth, and hence the sound speed decrease upwards. Rays are then bent up, and most of the sound will pass over the head of a distant observer
c decrease upward
Sound propagation on a typical day
Noise Engineering / Acoustics - 21 -
Ray Theory
Sound propagation in the atmosphere Sound refraction on a night time
• Sometimes on a clear night, the ground cools more quickly than the air, then rays are bent back to the ground are heard more distinctly by an observer
c increase upward
Sound propagation on a clear night following a warm day