Sonar Equation: The Wave Equation - University of Washington
Transcript of Sonar Equation: The Wave Equation - University of Washington
Sonar Equation: The Wave Equation
The Great Wave Hokusai
LO: Recognize physical principles associated with terms in sonar equation.
…the Punchline If density too high to resolve individual organisms, then:
E[energy from volume] = n * energy from an individual
energy from individual = σbs= backscattering cross section
energy from volume = sv = volume backscatter coefficient
E[sv] = n σbs So… - measure energy from volume - assume, measure, or model energy from representative individual - can calculate density of individuals
Definitions
Wave Number: k = 2π/λ (units rad/m)
Angular Frequency: ω = 2π/Τ=2πf (units rad/s)
Pressure (p): force/area (units Pascals = N/m2, 1µPa = 10-6 Pa)
p(r) = p(ro) ro/r
Intensity (I): power/area (units W/m2)
I(r) = I(ro) (ro/r)2 I ∝ p2
Hugen’s Principle
Every point in a wave field acts as a point source
Interference among sources (e.g. ripples in a pond)
+ = + =
Constructive Destructive
Propagating Waves
1 10 x Amp
D1(x) D2(x)
D2(x) = D1(x-10) = D1(x - 2 ms-1* 5 s)
D(x) = D1(x-ct)
Since c = λ/T
D(x) = D1(x-λ/T* t) = D1(x-λft)
speed 2 ms-1
Make non-dimensional: divide by λ
D(x/λ) = D1(x/λ – ft)
Non-dimensional Wave Equation
Make non-dimensional: divide by λ
D(x/λ) = D1(x/λ – ft)
Recall 2π/λ = k; 2πf = ω
D(k,x) = D1(kx - ωt) +x direction
D(k,x) = D1(kx + ωt) -x direction
Harmonic Motion
x m
k Force = -kx F = ma Therefore: ma = -kx
kxdt
xdm −=2
2
2
2
0dt
xdmkx −−=
Divide by –m:
mkx
dtxd
+= 2
2
0
Harmonic Motion
x made up of 2 parts:
= t
mkAx cos1
= t
mkBx sin2
So:
+
= t
mkBt
mkAx sincos
mk
=ωsince
( ) ( )tBtAx ωω sincos +=
( )αω −= tCx cosA,B,C complex constants
22 BAC +=
= −
AB1tanαSinusoidal form with magnitude and phase
k Units k = N/kg = (kg/s2)/kg = s-2 Square root = s-1
= rad/s = ω
Alternate Solution for Harmonic Motion
x made up of 2 parts:
So:
e tiDx ω=1 e tiEx ω−=2
ee titi EDx ωω −+=
Wave Equation for Acoustics
222
22 1
ctpp ∗=∇
δδ For plane waves in 1 direction:
2
22
xδδ
=∇
222
2
2
2 1ct
px
p∗=
δδ
δδ
LaPlacian operator
Plane Wave Incident Solution
Forward Traveling Planar Wave:
p(x,t)= C cos(kx-ωt), where C is reference pressure (po)
( ) ( ) ( )ee tkxitkxi BAtxp ωω +− +=,forward reverse
Alternate form: ( )e tkrioo
rrpp ω−= r = x = range
Spherical Scattering
rrr δδ
δδ 2
2
22 +=∇LaPlacian
Wave Equation 2
2
22
2 12tp
crp
rrp
δδ
δδ
δδ
=+ r is distance from source center
Solution ( ) ( )ee tkriootkri
rrp
rAp ωω −− ==
amplitude frequency
Transducer Directivity: D
Di = 10log(D) Directivity Index Di
Di = 10 log(4πa/λ) where a = active transducer area
Di = 10log(2.5/sin(β1/2) sin(β2/2))
where β’s = beam width at –3dB points
from transducer
from beam angles
where: Io = radiated intensity at acoustic axis = mean intensity over all directions
D = Io/
Transducer Beam Pattern
* sound pattern is independent of intensity (analogous to flashlight with low batteries)
Shading used to form main lobe through constructive and destructive interference
Don’t Forget Transmission Losses
Spherical Spreading Δp α 1/r
ΔI α 1/r2
Absorption p2 = p1 10- αr/20
α = 1/ro 10 log10(p12/p2
2)
I2 = I1 10- αr/10 α = 1/ro 10 log10(I1/I2)
I ∝ p2
ro = unit distance
Putting Components Together
Pressure p(r) = p(ro) ro/r 10- α(r-ro/20)
Intensity I(r) = I(ro) (ro/r)2 10- α(r-ro/10)
Sonar Equation for Pressure
p(r) = p(ro) ro/r 10- α(r-ro/20)
Divide by reference pressure po
p(r)/po = p(ro) /po ro/r 10- α(r-ro/20)
Take 20 log10 of both sides:
20 log10(p(r)/po) = 20 log10 (p(ro) /po) + 20 log10(ro/r) - α(r-ro)
Sonar Equation for Pressure
20 log10(p(r)/po) = 20 log10 (p(ro) /po) - 20 log10(ro/r) - α(r-ro) SPL
Sound Pressure Level
Echo Level
= SL
Source Level
- TL TL -
Transmission Losses
(one way)
Scattering from an Object
Ii
Incident sound on target
Ir
Intensity of reflected sound at 1 meter
ρ1c1
ρ2c2
=
i
r
IITS 10log10
Point or Spherical Scattering
( ) fr
eerrpp
s
ikrtkri
t
ooscat
st ××= −ω f = complex scattering amplitude
(a.k.a. complex scattering length)
reflectivity, target size, orientation, geometry of transmit & receive
Scattered Field
amplitude frequency
incident field reflected field
Acoustic Impedance Scattering is caused by an Acoustic Impedance mismatch
What is acoustic impedance (Z)? Z = ρ c
Reflection (1 direction) Scattering (all directions)
small objects (e.g. piling) LARGE objects (e.g. breakwater)
pr pi
pi
R = pr/pi
g = ρ2/ρ1 h = c2/c1
density sound speed
g = ρ2/ρ1 h = c2/c1
11
1
1
11
22
11
22
1122
1122
+−
=+
−=
+−
=ghgh
cccc
ccccR
ρρρρ
ρρρρ
density contrast sound speed contrast
Echoes: Acoustic Impedance Mismatch
Reflection
proportion of sound reflected at an interface
11
R,,
,,, +
−=
sbfbsbfb
sbfbsbfbsbfb hg
hg
Incident
R fb,sb
R w,fb T T Rw,fb fb,w fb,sb
fb
sb
Reflection Coefficient
R = reflected T = transmit w = water fb = fish body sb = swim bladder
Reflectivity Examples
Lead in water g = 11.35 h =1.49 R =14/16 ≅ 1
Air – water interface g = 0.0012 h =0.22 R = 0-1/0+1= -1
Artic krill (Euphasia superba) g = 1.0357 h = 1.0279 R = 0.06/2.06 ≅ 0.03
Atlantic cod (Gadus morhua) Body Swimbladder g = 1.039 g = 0.0012 h =1.054 h = 0.232 R = 0.045 R = - 0.99
Perfect reflector Pressure release surface
Backscatter Definition f = complex scattering amplitude (a.k.a. complex scattering length)
- if f is big then target scatters lots of sound - backscatter is most common geometry for a transducer
σbs – backscattering cross-sectional area (units m2)
TS – target strength (units dB re 1 µPa)
2*bsbsbsbs fff =×=σ
( ) ( )bsbsfTS σ102
10 log10log10 ==
* complex conjugate
Some Final Definitions
EL = echo level ( )
= 2
2
10 1log10
Papr
µ
SL = source level ( )
= 22
22
10 )1(1log10
mParp ii
µ
TS = target strength
= 210 )1(
log10m
bsσ
TL = transmission loss
= 2
2target
10 )1(log10
mr
assumes no absorption
Two Way Scattering Equation
( ) bssource
ooscat rrrpp σ×××= 22
target
22 11
EL = SL – TLto target + TLto source + TS Rearrange:
TS = EL – SL + 2 TL (anything missing?)
Active vs Passive Systems SPL = SL - TL
Passive energy received = energy transmitted * fraction not absorbed * fraction not spread
−
−
= 10
2
10orr
oor r
rIIα
source transmission loss
Active SPLrec = SL - 2TL + TS
+ noise (Σ thermal, electrical, anthropogenic)
one-way system
two-way system
Active vs Passive Applications Passive
- determine how far away we can hear animals - determine how far away animals can hear us (and corresponding intensity level) - determine range of animal communication - census populations - monitor behavior
Active - determine how much power needed to detect animal - determine range of predator detecting prey - census and size populations - map distributions of animals