Sona Equaion: The Wave Equaion The Gea Wave Hokusai LO: Recognize hysical inciles associaed wih ems in sona equaion.
he Punchline If densiy oo high o esolve individual oganisms, hen: E[enegy fom volume] n * enegy fom an individual enegy fom individual s bs backscaeing coss secion enegy fom volume s v volume backscae coefficien E[s v ] n s bs So - measue enegy fom volume - assume, measue, o model enegy fom eesenaive individual - can calculae densiy of individuals
Definiions Wave Numbe: k π/λ (unis ad/m) Angula Fequency: w π/tπf (unis ad/s) Pessue (): foce/aea (unis Pascals N/m, 1mPa 10-6 Pa) () ( o ) o / Inensiy (I): owe/aea (unis W/m ) I() I( o ) ( o /) I
Hugen s Pincile Evey oin in a wave field acs as a oin souce Inefeence among souces (e.g. iles in a ond) Consucive Desucive + +
Poagaing Waves Am D 1 (x) D (x) 1 10 x seed ms -1 D (x) D 1 (x-10) D 1 (x - ms -1 * 5 s) D(x) D 1 (x-c) Since c λ/t T eiod T ime D(x) D 1 (x-λ/t* ) D 1 (x-λf) Make non-dimensional: divide by λ D(x/λ) D 1 (x/λ f)
Non-dimensional Wave Equaion Make non-dimensional: divide by λ D(x/λ) D 1 (x/λ f) Recall π/λ k; πf ω D(k,x) D1(kx - ω) D(k,x) D1(kx + ω) +x diecion -x diecion
Hamonic Moion k m x Foce -kx F ma Theefoe: ma -kx m d x d kx 0 kx m d x d Divide by m: 0 d x d + kx m
Hamonic Moion x made u of as: k k x Acos 1 m x Bsin m So: k k x Acos + Bsin m m x since ω ( ω) Bsin( ω) Acos + k m k Unis k N/kg (kg/s )/kg s - Squae oo s -1 ad/s ω 1 B an A B A Sinusoidal fom wih magniude C + and hase α x C cos ( ω α ) A,B,C comlex consans
Alenae Soluion fo Hamonic Moion x made u of as: So: e i D x ω 1 e i E x ω e e i i E D x ω ω + Wave Equaion fo Acousics 1 c δ δ Fo lane waves in 1 diecion: δx δ 1 c x δ δ δ δ LaPlacian oeao
Plane Wave Inciden Soluion ( x, ) A e i ( kx ) i( kx ) ω +ω + B e fowad Fowad Taveling Plana Wave: evese (x,) C cos(kx-ω), whee C is efeence essue ( o ) Alenae fom: e ( k ω ) o o i x ange
Plana Inciden Pessue inc o o e i ( k ω ) o o e i( k ω)
Sheical Scaeing δ δ δ δ + LaPlacian Wave Equaion 1 c δ δ δ δ δ δ + is disance fom souce cene Soluion ( ) ( ) e e k i o o k i A ω ω amliude fequency
Tansduce Dieciviy D( θ ) kl sin sinθ kl sinc sinθ k sinθ L sinc(sin(x)/x) Dieciviy Index D i 10log(D) 10log(I o / ) whee: I o adiaed inensiy a acousic axis mean inensiy ove all diecions Calculae fom ansduce D i 10 log(4πa/λ) whee a acive ansduce aea Calculae fom beam angles D i 10log(.5/sin(β 1 /) sin(β /)) whee β s beam widh a 3dB oins
Don Foge Tansmission Losses Sheical Seading Δ α 1/ ΔI α 1/ I Absoion 1 10 - α/0 α 1/ o 10 log 10 ( 1 / ) I I 1 10 - α/10 α 1/ o 10 log 10 (I 1 /I ) o uni disance
Puing Comonens Togehe Pessue () ( o ) o / 10 - α(- o /0) Inensiy I() I( o ) ( o /) 10 - α(- o /10)
Sona Equaion 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 boh sides: 0 log 10 (()/ o ) 0 log 10 (( o ) / o ) + 0 log 10 ( o /) - α(- o )
Sona Equaion 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)
Scaeing fom an Objec I i Inciden sound on age I Inensiy of efleced sound a 1 mee ρ 1 c 1 ρ c TS 10log10 I I i
Poin o Sheical Scaeing Scaeed Field sca o o e i ( k ω ) e ik s amliude fequency inciden field efleced field s f f comlex scaeing amliude (a.k.a. comlex scaeing lengh) efleciviy, age size, oienaion, geomey of ansmi & eceive
Acousic Imedance Scaeing is caused by an Acousic Imedance mismach Reflecion (1 diecion) Scaeing (all diecions) i i LARGE objecs (e.g. beakwae) small objecs (e.g. iling) R / i Wha is acousic imedance (Z)? Z ρ c g ρ /ρ 1 h c /c 1 densiy sound seed
g ρ /ρ 1 h c /c 1 1 1 1 1 1 1 1 1 1 1 1 1 + + + gh gh c c c c c c c c R ρ ρ ρ ρ ρ ρ ρ ρ densiy conas sound seed conas Echoes: Acousic Imedance Mismach Reflecion
Reflecion Coefficien ooion of sound efleced a an ineface Inciden R w,fb T T R w,fb fb,w fb,sb R efleced T ansmi w wae fb fish body sb swim bladde R fb,sb sb fb R fb, sb g g fb, sb fb, sb h h fb, sb fb, sb 1 + 1
Refleciviy Examles Lead in wae g 11.35 h 1.49 R 14/16 1 Pefec efleco Ai wae ineface g 0.001 h 0. R 0-1/0+1-1 Pessue elease suface Aic kill (Euhasia sueba) g 1.0357 h 1.079 R 0.06/.06 0.03 Alanic cod (Gadus mohua) Body Swimbladde g 1.039 g 0.001 h 1.054 h 0.3 R 0.045 R - 0.99
Backscae Definiion f comlex scaeing amliude (a.k.a. comlex scaeing lengh) - if f is big hen age scaes los of sound - backscae is mos common geomey fo a ansduce σ bs backscaeing coss-secional aea (unis m ) TS age sengh (unis db e 1 µpa) σ bs bs * bs f f f bs * comlex conjugae TS 10 log ( ) 10 ( σ ) 10 f bs log10 bs
Some Final Definiions EL echo level SL souce level 10log 10 ( ) 1µ Pa 10log i i 10 ( ) 1µ Pa (1m ) TL ansmission loss age (1m ) 10log10 assumes no absoion TS age sengh 10log σ bs (1m ) 10
Two Way Scaeing Equaion sca 1 1 ( ) σ o o bs age souce EL SL TL o age + TL o souce + TS Reaange: TS EL SL + TL (anyhing missing?)
Sona Equaion Schemaic Tansmi Receive
Acive vs Passive Sysems SPL SL - TL Passive enegy eceived enegy ansmied * facion no absobed * facion no sead I I o o 10 α 10 o one-way sysem souce ansmission loss Acive SPL ec SL - TL + TS + noise (Σ hemal, elecical, anhoogenic) wo-way sysem
Acive vs Passive Alicaions Passive - deemine how fa away we can hea animals - deemine how fa away animals can hea us (and coesonding inensiy level) - deemine ange of animal communicaion - census oulaions - monio behavio Acive - deemine how much owe needed o deec animal - deemine ange of edao deecing ey - census and size oulaions - ma disibuions of animals
Dolhin Sound Receion Addiional Sucues: Pahway of sound o he cochlea is via he lowe jaw (cf. Bullock e al. 1968; McComick e al. 1970) Nois 1968