The Great Wave Hokusai. LO: Recognize physical principles associated with terms in sonar equation.

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1 Sona Equation: The Wave Equation The Geat Wave Hokusai LO: Recognize hysical inciles associated with tems in sona equation.

2 the Punchline If density too high to esolve individual oganisms, then: E[enegy fom volume] n * enegy fom an individual enegy fom individual σ bs backscatteing coss section enegy fom volume s v volume backscatte coefficient So E[s v ] n σ bs - measue enegy fom volume - assume, measue, o model enegy fom eesentative individual - can calculate density of individuals

3 Definitions Wave Numbe: k π/λ (units ad/m) Angula Fequency: ω π/τπf (units ad/s) Pessue (): foce/aea (units Pascals N/m, 1µPa 10-6 Pa) () ( o ) o / Intensity (I): owe/aea (units W/m ) I() I( o ) ( o /) I

4 Hugen s Pincile Evey oint in a wave field acts as a oint souce Intefeence among souces (e.g. iles in a ond) Constuctive Destuctive + +

5 Poagating Waves Am D 1 (x) D (x) x 1 10 D (x) D 1 (x-10) D 1 (x - ms -1 * 5 s) D(x) D 1 (x-ct) Since c λ/t D(x) D 1 (x-λ/t* t) D 1 (x-λft) Make non-dimensional: divide by λ D(x/λ) D 1 (x/λ ft) seed ms -1

6 Non-dimensional Wave Equation Make non-dimensional: divide by λ D(x/λ) D 1 (x/λ ft) Recall π/λ k; πf ω D(k,x) D1(kx - ωt) +x diection D(k,x) D1(kx + ωt) -x diection

7 Hamonic Motion k m x Foce -kx F ma Theefoe: ma -kx m d x dt kx 0 kx m d x dt Divide by m: 0 d x dt + kx m

8 Hamonic Motion x made u of ats: k k x Acos t 1 m x Bsin t m So: k k x Acos + t Bsin t m m x since ω ( ωt) Bsin( ωt) Acos + k m k Units k N/kg (kg/s )/kg s - Squae oot s -1 ad/s ω Sinusoidal fom with magnitude 1 B A B and hase α tan A C + x C cos t ( ω α ) A,B,C comlex constants

9 Altenate Solution fo Hamonic Motion x made u of ats: So: e t i D x ω 1 e t i E x ω e e t i t i E D x ω ω + Wave Equation fo Acoustics 1 c t δ δ Fo lane waves in 1 diection: δx δ 1 c t x δ δ δ δ LaPlacian oeato

10 Plane Wave Incident Solution ( x, t) A e i ( kx t ) i( kx t ) ω +ω + B e fowad Fowad Taveling Plana Wave: evese (x,t) C cos(kx-ωt), whee C is efeence essue ( o ) Altenate fom: e ( k ωt ) o o i x ange

11 Plana Incident Pessue inc o o e i ( k ωt ) t t o t o e i( k ωt)

12 Sheical Scatteing δ δ δ δ + LaPlacian Wave Equation 1 t c δ δ δ δ δ δ + is distance fom souce cente Solution ( ) ( ) e e t k i o o t k i A ω ω amlitude fequency

13 Diectivity: D D( θ ) kl sin sinθ kl sinc sinθ k sinθ L sinc(sin(x)/x)

14 Tansduce Diectivity: D whee: D I o / I o adiated intensity at acoustic axis mean intensity ove all diections Diectivity Index D i D i 10log(D) fom tansduce D i 10 log(4πa/λ) whee a active tansduce aea fom beam angles D i 10log(.5/sin(β 1 /) sin(β /)) whee β s beam width at 3dB oints

15 Tansduce Beam Patten Shading used to fom main lobe though constuctive and destuctive intefeence * sound atten is indeendent of intensity (analogous to flashlight with low batteies)

16 Don t Foget Tansmission Losses Sheical Seading Δ α 1/ ΔI α 1/ I Absotion α/0 α 1/ o 10 log 10 ( 1 / ) I I α/10 α 1/ o 10 log 10 (I 1 /I ) o unit distance

17 Putting Comonents Togethe Pessue () ( o ) o / 10 - α(- o /0) Intensity I() I( o ) ( o /) 10 - α(- o /10)

18 Sona Equation fo Pessue () ( o ) o / 10 - α(- o /0) Divide by efeence essue o ()/ o ( o ) / o o / 10 - α(- o /0) Take 0 log 10 of both sides: 0 log 10 (()/ o ) 0 log 10 (( o ) / o ) + 0 log 10 ( o /) - α(- o )

19 Sona Equation fo Pessue 0 log 10 (()/ o ) 0 log 10 (( o ) / o ) - 0 log 10 ( o /) - α(- o ) SPL SL - TL - TL Sound Pessue Level Echo Level Souce Level Tansmission Losses (one way)

20 Scatteing fom an Object I i Incident sound on taget I Intensity of eflected sound at 1 mete ρ 1 c 1 ρ c TS 10log10 I I i

21 Point o Sheical Scatteing Scatteed Field scat o t o e i ( k ωt ) e ik s t amlitude fequency incident field eflected field s f f comlex scatteing amlitude (a.k.a. comlex scatteing length) eflectivity, taget size, oientation, geomety of tansmit & eceive

22 Acoustic Imedance Scatteing is caused by an Acoustic Imedance mismatch Reflection (1 diection) Scatteing (all diections) i i LARGE objects (e.g. beakwate) small objects (e.g. iling) R / i What is acoustic imedance (Z)? Z ρ c g ρ /ρ 1 h c /c 1 density sound seed

23 g ρ /ρ 1 h c /c gh gh c c c c c c c c R ρ ρ ρ ρ ρ ρ ρ ρ density contast sound seed contast Echoes: Acoustic Imedance Mismatch Reflection

24 Reflection Coefficient ootion of sound eflected at an inteface Incident R w,fb T T R w,fb fb,w fb,sb R fb,sb R eflected T tansmit w wate fb fish body sb swim bladde sb fb R fb, sb g g fb, sb fb, sb h h fb, sb fb, sb 1 + 1

25 Reflectivity Examles Lead in wate g h 1.49 R 14/16 1 Pefect eflecto Ai wate inteface g h 0. R 0-1/0+1-1 Pessue elease suface Atic kill (Euhasia sueba) g h R 0.06/ Atlantic cod (Gadus mohua) Body Swimbladde g g h h 0.3 R R

26 Backscatte Definition f comlex scatteing amlitude (a.k.a. comlex scatteing length) - if f is big then taget scattes lots of sound - backscatte is most common geomety fo a tansduce σ bs backscatteing coss-sectional aea (units m ) TS taget stength (units db e 1 µpa) σ bs bs * bs f f f bs * comlex conjugate TS 10 log ( ) 10 ( σ ) 10 f bs log10 bs

27 Some Final Definitions EL echo level SL souce level 10log 10 ( ) 1µ Pa 10log i i 10 ( ) 1µ Pa (1m ) TL tansmission loss taget (1m ) 10log10 assumes no absotion TS taget stength 10log σ bs (1m ) 10

28 Two Way Scatteing Equation scat 1 1 ( ) σ o o bs taget souce EL SL TL to taget + TL to souce + TS Reaange: TS EL SL + TL (anything missing?)

29 Sona Equation Schematic Tansmit Receive

30 Active vs Passive Systems SPL SL - TL Passive enegy eceived enegy tansmitted * faction not absobed * faction not sead I I o o 10 α 10 o one-way system souce tansmission loss Active SPL ec SL - TL + TS + noise (Σ themal, electical, anthoogenic) two-way system

31 Active vs Passive Alications Passive - detemine how fa away we can hea animals - detemine how fa away animals can hea us (and coesonding intensity level) - detemine ange of animal communication - census oulations - monito behavio Active - detemine how much owe needed to detect animal - detemine ange of edato detecting ey - census and size oulations - ma distibutions of animals

32 Dolhin Sound Recetion Additional Stuctues: Pathway of sound to the cochlea is via the lowe jaw (cf. Bullock et al. 1968; McComick et al. 1970) Nois 1968

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