Underwater Acoustics and Sonar Signal Processing

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1 Undewate Acoustics and Sona Signal Pocessing Contents 1 Fundamentals of Ocean Acoustics Sound Popagation Modeling 3 Sona Antenna esign 4 Sona Signal Pocessing 5 Aay Pocessing 1 Sound Popagation Modeling Sound popagation in the ocean is mathematically fomulated by the wave equation, whose paametes and bounday conditions ae desciptive of the ocean envionment. As summaied in the figue below, thee ae a vaiety of diffeent techniques available fo solving the wave equation (numeically fo evaluating sound popagation in the sea. Abbeviations FE: Finite Element PE: Paabolic Equation F: Finite iffeence FFP: Fast Field Pogam NM: Nomal Mode RT: Ray Tacing

2 Wave Equation FFP NM Range Independent Coupled FFP Coupled NM Adiabatic NM RT PE F/FE Range ependent 3.1 The Wave Equation The wave equation in an ideal fluid can be deived fom hydodynamics and the adiabatic elation between pessue and density. The following figue is used to deive the wave equation by exploiting the equation fo consevation of mass, the Eule s equation and the adiabatic equation of state. x p g (x,t v(x,t p g (x + dx,t v(x + dx,t A x x + dx 4

3 Fo deiving the following equations we define the total pessue and the total density as follows. p g = p 0 + p and ρ g = ρ 0 + ρ, whee p g, p 0, p, ρ g, ρ 0 and ρ denote the total pessue, static pessue, change in pessue, total density, static density and change in density, espectively. Continuity Equation Employing the figue above the equation fo the consevation of mass can be expessed by ρ g (x + dx,tav(x + dx,t ρ g (x,tav(x,t!####### "####### $ = Adx ρ g % t Resultant mass steam density vaiation!## "## $ Mass vaiation 5 and with ρ g (x + dx,tv(x + dx,t ρ g (x,tv(x,t = (ρ v g dx x efomulated to obtain the so-called continuity equation (ρ g v = ρ g = ρ x t t. Eule s Equation Using the figue above Newton s nd law can be witten as p g (x,ta p g (x + dx,ta!#### "#### $ = ρ Adx dv g % V dt!"$ % and by exploiting Total Foce, F m a 6

4 and dv = v t dt + v x dx, i.e. dv dt = v t + v x p g (x,t p g (x + dx,t = p (x + dx,t p (x,t g g dx dx we obtain Eule s equation p g x = p x = ρ g Adiabatic Equation of State p g = p 0 ρ g ρ 0 κ v t + v v x = p 0 + p g ρ + 1 ρ g ρ 0 dx dt = v t + v v x. p g ρ g = p g x ρ + ρ 0 7 whee κ denotes the adiabatic exponent. Fo convenience we define p g pg p0 c = = κ κ ρ g ρ ad. g ρ0 which tuns out late to be the squaed sound speed in an ideal fluid. Fo p p 0 and ρ ρ 0 the adiabatic equation of state becomes appoximately p g p 0 + c ρ, i.e. p c ρ. Since the time scale of oceanogaphic changes is much longe than the time scale of the acoustical popagation, we suppose that the mateial popeties ρ 0 and c ae independent of time. 8

5 Taking the patial deivative of the continuity equation with espect to t and of Eule s equation with espect to x povides ( ρgv ( ρgv ρ = =, ρ = pc t x x t t ρ g v 1 p = v + ρg = x t x t c t and p v v = ρ + ρ g gv x x t x x espectively. Fo lowe paticle velocities v as well as the tems p p 0 and ρ ρ 0 9 ρ g v v and ρgv x t x x can be neglected. Thus, the fome equations simplify to 1 ρ v p and p ρ v g = = g x t c t x x t and povide by equating the 1dimensional linea wave equation p 1 p = x c t which can be extended by staightfowad agumentation to the 3dimensional case given by Δp = 1 c p t with Δ = x + y +. 10

6 Helmholt Equation Suppose hamonic pessue oscillation, i.e. p(x, y,,t = P(x, y,exp( jω t, we obtain ΔP + k P = 0 with k = ω c = π λ. If P possesses spheical symmety, i.e. P is only depending on R, the Laplacian in spheical coodinates simplifies to Δ = +. R R R Hence, the spheical wave solution of the Helmholt equation is P(R= Aexp( jkr R with R= (x x S +( y y S +( S, 11 whee x S, y S and S ae the coodinates of an omnidiectional point souce (pulsating sphee of small adius. Anothe simple and impotant solution is povided by the plane wave ( x y Pxy (,, = Aexp jkx ( + ky+ k, whee k x, k y and k ae the wave numbes that satisfy k = k T k = k x + k y + k, k = ω c = π λ. The wave vecto k can also be expessed by T ( kx ky k k( ϕ θ ϕ θ θ k =,, = cos cos,sin cos,sin, whee φ and θ denote the aimuth and elevation, espectively. T 1

7 . Homogeneous Waveguide Suppose the medium within infinitely extended boundaies is homogeneous. S = 0 ai R R Receive S Souce wate column c = 1480 m/s ρ = 1 g/cm 3 sediment Fo the given point souce coodinates (0, S the pessue shall be detemined at an abitay eceive location ( R, R Image Souce Appoach Image Suface S 0 The wave field within a homogeneous waveguide can be intepeted as the L 0 θ 0 R Ai R 1 R S S L 01 θ 03 L 03 L 04 θ 04 Sediment R Image Bottom + S supeposition of infinitely many spheical waves that ae eflected at the boundaies. 14

8 As a fist appoximation, the sound pessue in the waveguide can be detemined by supeimposing the fou contibutions indicated in the figue above, i.e. P( R, R,ω = A(ω e j k L 01 L 01 + R (θ 03,ω e j k L 03 + R 1 (θ 0,ω e j k L0 L 03 L R 1 (θ 04,ω R (θ 04,ω e j k L04 L 04 with L 01 = R + ( R S L 0 = R + ( S + R L 03 = R + ( S R L 04 = R + ( + S R and ( ( θ 0 = actan ( S + R / R θ 03 = actan ( S R / R θ 04 = actan (( + S R / R. 15 Continuation of the image souce technique in multiples m = 1,, of goups of fou contibutions povides P( R, R,ω = A(ω m=0 R m 1 (θ m1,ω R m (θ m1,ω e j k L m1 + R m+1 1 (θ m,ω R m (θ m,ω e j k L m + R m L 1 (θ m3,ω R m+1 (θ m3,ω e j k L m3 m + R m+1 1 (θ m4,ω R m+1 (θ m4,ω e j k L m4 L m4 with L = + ( m + L = + ( ( m+ 1 m1 R S R m3 R S R L = + ( m+ + L = + ( ( m+ 1 + m R S R m4 R S R L m1 L m3 16

9 and ( θ m3 =actan (((m+1 S R / R ( θ m4 =actan (((m+1+ S R / R θ m1 =actan (m S + R / R θ m =actan (m+ S + R / R Taking into account that the eflection coefficients at the ocean suface and bottom can be appoximated by R 1 wate-ai-inteface R 1 wate-had bottom-inteface the calculation of the sound pessue simplifies to e i k L P(,,ω = A(ω ( 1 m m1 m=0 L m1 e i k L m L m + e i k Lm3 L m3 e i k Lm4 L m4. 17 Assignment 6: evelop a Matlab pogam fo detemining P(,,ω fo the following paametes. Signal paametes Sinusoidal wavefom Amplitude: A =1, Fequency: f =10 H, 100 H, 1 kh, 10kH and 100 kh Waveguide paametes Wate depth: = 0 m Souce location: S = 0 m, S = 5 m Receive location: ( R, R T [0,500] [0, ] Suface/Bottom Reflection: R 1 = 1 (calm, R = 1 (had bottom Sound speed: c = 1480 m/s epict the pessue distibution P(,,ω in colou coded two dimensional diagams and intepet the esults. 18

10 .. Nomal Mode Appoach Fo cylinde symmetic sound popagation, i.e. p = p(,, t, the wave equation is given by p + 1 p + p = 1 p c t. Let us suppose that p can be expessed by p f( g((. h t Hence, insetion in the wave equation povides (( (( ght (( d f + g h t df + f(( ht d g = f g d h d d d c dt and afte some manipulations f( + d d + = g( d c h( t dt d f df d g d h. 19 Fo hamonic souces with h(t = A e jω t we obtain 1 d f f ( d + 1 df d + 1 d g g( d!### "### $!# " $# -dependent tem -dependent tem = ω c = k. Fo all values of and, the -dependent and -dependent tem ae equal to constants. With the sepaation constant k fo the adial and k fo the vetical tem, the sepaated odinay diffeential equations ae d f 1 df d g + + k 0and 0with. f = + k g= k = k + k d d d The fist equation is a eo-ode Bessel equation. Its solution can be witten in tems of a eo ode Hankel function, i.e. 0

11 whee J 0 (1 H0 ( k = J0( k + jy0( k f( = ( H0 ( k = J0( k jy0( k = eo-ode Bessel function of the 1st kind, (besselj(, Y 0 = eo-ode Bessel function of the nd kind, (bessely(, also known as eo-ode Neumann function N 0, (1 H 0 = eo-ode Hankel function of the 1st kind, (besselh(, ( H 0 = eo-ode Hankel function of the nd kind, (besselh(, both ae also known as eo-ode Bessel function of the 3d kind. The asymptotic fom of the Hankel function fo k is H 0 (1 (k π k e j k π 4 and H 0 ( (k j k π k e π 4, 1 (1 H 0 ( H 0 whee and epesent the conveging and diveging cylindical waves, espectively. The second equation epesents an odinay linea diffeential equation, whee g has to satisfy the bounday conditions g(0 = 0 (R = 1. g( = max (R = 1 Step I: (Elementay Solution λ λ g ( = e g( = λ e g + k g = + k = =± jk ( ( 0 λ 0 λ1, { } = cos(k and Im{ exp( jk } = sin(k With Re exp( jk the set of independent solutions is given by { cos(k,sin(k }.

12 Thus, the geneal solution of the odinay linea diffeential equation can be expessed by g ( = Acos( k + Bsin( k. Step II: (Bounday Conditions g(0 = 0 A = 0 g( = Bsin(k = max sin(k = 1 k =(m 1π /, m=1,, The bounday conditions ae satisfied fo a discete set of values of k. Hence, we obtain π km, = (m 1 (eigenvalues and 3 π gm( = Bmsin (m 1 (eigenfunctions fo m = 1,, The solutions ae called modes because they descibe the natual ways in which the system vibates. 0 Pessue Amplitude [elative units] m = 1 m = m = 3 m = 4 k m, k m, The eigenvalues and ae elated by k = k,m + k,m. 4

13 k m k m Consideing, and, to be the hoiontal and vetical component of k, espectively, we can wite ( ( k ( ω = k( ωcos α ( ω and k = k( ωsin α ( ω. m, m m, m k,1 e α1( ω1 k( ω 1 k e α ( ω k( ω,1 1 k ( ω e,1 1 k ( ω e,1 k, e k( ω 1 k e k( ω, α( ω1 k ( ω e, 1 α( ω k ( ω e, Consequently, we obtain π jkm, 4 m(. πkm, f e 5 Step III: (Initial Conditions The last step is to add the fundamental solutions π g ( = Bmsin( k, m = Bmsin (m 1 m= 1 m= 1 in such a way that the initial condition g ( = φ( is satisfied. Substituting the sum into the initial condition gives φ( = Bmsin( k, m. m= 1 By multiplying each side of this equation with sin(k,n and integating fom 0 to, we obtain φ ( sin( kn, d = Bn sin ( kn, d Bn 0 = 0 6

14 due to the othogonality popety 0 m n sin( km, sin( kn, d =. 0 m= n Hence, the B n ae detemined by B = n ( sin( k, n d n 1,,... φ = 0 which ae the Fouie coefficients of φ( ( uniqueness. Fo a souce function given by φ( = δ( S we have Bn = δ ( Ssin( k, n d = sin( k, ns 0 7 and accodingly gm( = sin( k, mssin( k, m. Finally, fo the bounday and initial condition given above the solution of the wave equation can be expessed by p(,,t = h(t m=1 g m ( f m ( = Ae jωt sin(k sin(k,m S,m m=1 j k,m π k,m e π 4 = A 8 π e j ω t + π 4 sin(k,m S sin(k,m e j k,m. m=1 k,m 8

15 .3 Inhomogeneous Waveguide If the sound speed in the wate column is not constant the medium/waveguide is called inhomogeneous. Howeve, one geneally assumes that the waveguide is cylinde symmetic with egad to the souce location and is eithe ange independent, i.e. the medium is hoiontally statified such that c = c( is only a function of depth o ange dependent, i.e. the sound speed vaies vesus hoiontal ange and depth such that c = c(, is a function of ange and depth. 9 In the following ange independent scenaios ae assumed fo simplicity. S = 0 ai R c (m/s R Receive S Souce wate column ρ = 1 g/cm 3 sediment 30

16 .3.1 Ray-Tacing, eo ode sound speed appoximation The sound speed pofile c( is appoximated by a staicase function c (!c( 1500 c [m/s] c(!c( N = Ai 3 4 S ε 1 ε 1 c 1 Snell s Law: 1 ε ε ε3 c > c 1 sinε n sinε n+1 = c n c n+1 n = 1,, N 3 ε 3 ε 4 ε 4 c 3 > c c 4 < c 3 4 ε 5 ε 5 ε 5 c 5 < c 4 5 = Sediment 3

17 The ay tace can be detemined by applying Snell s law at each bounday laye. sinε 1 = sinε = = sinε N = a = const. c 1 c c N At the n-th bounday laye (0 = suface,, N = bottom holds With we obtain sinε n = ac n. cos x = 1 sin x and tan x = sin x cos x, cosεn 1 acn and tanεn = = ac n 1 ac n 33 such that the hoiontal position and the tavel time of the ay at the l-th bounday laye can be detemined by l l n n 1 la (, = ( n n 1 tanε n and Tla (, = c cosε n= 1 n= 1 n n espectively, whee 0 = S, N = and l = 1,, N..3. Ray-Tacing, fist ode sound speed appoximation Fo any [0, ] Snell s law sin ε( S c( S = sin ε( c( = a povides ε( = acsin c( c( S sinε( S, 34

18 whee sinε( = ac( with a = sinε( S c( S has been exploited. 0 S ε( S c ( ε( = acsin sin ε( S c ( S 35 Using again the identities we obtain and cos = 1 sin and tan = sin cos x x x x x cos ε( = 1 sin ε( = 1 a c( tan ε( = sin ε( cos ε( = ac( 1 a c(. Hoiontal patitioning of the wate column in thin layes of thickness Δ as indicated in the figue on the ight side leads to 0!! Δ 36

19 N( a (, = (, a + Δ tan ε( + nδ. n= 1 Fo Δ 0 the Riemann sum becomes a Riemann integal so that we can wite Coespondingly, the tavel time S (,a = ( S,a + tanε( d = ( S,a + T(, a = T(, a + S N( S ac( d 1 a c(. S n= 1 S S S Δ c ( + nδ cos ε( + nδ 37 esults in T (,a = T ( S,a + = T ( S,a + 1 c( cosε( d fo Δ 0. Now, supposing the velocity pofile can be appoximated piecewise by linea functions, i.e. 0 S S 1 d c( 1 a c( i c( i c i! i+1! i+1 c( = c( + g ( i i i 38

20 the integals a( c( i + gi( i a (, = ( i, a + d 1 ( ( ( i a c i + gi i and d T (,a=t ( i,a+ i c( i + g i ( i i can be solved fo ( 1 a ( c( i + g i ( [ i, i+1 ] analytically. Substitution v= c( + g ( with dv = g i d leads to i i i c( + g ( i i i av 1 a (, = ( i, a + dv= 1 av gi c( i 39 = ( i,a 1 a v ag i c( i c( i +g i ( i ( = ( i,a+ 1 a c( i 1 a c( i + g i ( i ag i The subsequent efomulation shows that ays follow cicula paths in case of linea depth dependent velocity pofiles. (,a ( i,a + 1 a c( i ag i = (,a ( i,a + 1 a c( i ag i 1 a ( c( i + g i ( i ag i = 1 ( a c( i + g i ( i a g i 40

21 i i i i i i i 1 ac ( (( c + g( 1 a (, ( i, a + + = ag g a g 1 ac ( i c ( i 1 i ag i gi a gi a (, ( i, a + + = Moeove, the expession fo the tavel time can be similaly deived by c( i +g i ( i 1 1 T (,a = T ( i,a + dv v 1 a v g c( i i = T ( i,a 1 ln 1+ 1 a v g i av c( i +g( i c( i = ac ( i = T( i, a + ln gi ac( i a ( c i g i ( ( ( 1+ 1 ( + ( ' ln a c i + g i 1 a c( i + gi( i 1+ 1 a c( i = T( i, a + ln. g i ( 1 1 a ( c( i gi( i ac( + + i Since explicit expessions (, a, T(, a ae available fo linea velocity pofiles, computationally efficient and satisfactoy accuate ay-path calculations can be caied out afte piecewise linea appoximation of the actual velocity pofile. ( ( 4

B: Computational Ocean Acoustics, Spinge, 000 [4] Luton, X.

22 Liteatue [1] Bekhovskikh, L.M.; Lysanov, Y.P.: Fundamentals of Ocean Acoustics, Spinge, 003 [] Ette, P.C.: Undewate Acoustic Modeling, Spon Pess, 003 [3] Jensen, F.B: Computational Ocean Acoustics, Spinge, 000 [4] Luton, X.: An Intoduction to Undewate Acoustics, Spinge, 004 [5] Medwin, H.;Clay C.S: Acoustical Oceanogaphy, Academic Pess, 1998 [6] Tolstoy, I.; Clay C.S: Ocean Acoustics, AIP-Pess, 1987 [7] Uick, R.I: Pinciples of Undewate sound, McGaw Hill,

1 Fundamental Solutions to the Wave Equation

1 Fundamental Solutions to the Wave Equation Physical insight in the sound geneation mechanism can be gained by consideing simple analytical solutions to the wave equation. One example is to conside acoustic