Classical wave experiments on Chaotic Scattering

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Classical Wave Scattering Part III O. Rath Spivack April 18, 2007 This is just a first draft of the material covered in this course. I should very much appreciate being told of any corrections or possible improvements Comments, please, to O.Rath-Spivackdamtp.cam.ac.uk. 1 Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.CONTENTS Contents 1 Governing equations for acoustic and electromagnetic waves 3 1.1 Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 The Kirchoff-Helmholtz equation . . . . . . . . . . . . . . . . 18 2 Canonical cases 21 2.1 Scattering from a flat surface . . . . . . . . . . . . . . . . . . 21 2.2 Scattering from a semi-infinite plane . . . . . . . . . . . . . . 24 2.3 Scattering from a wedge . . . . . . . . . . . . . . . . . . . . . 32 3 Approximations 37 3.1 Born Approximation . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Rytov Approximation . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 WKB Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Parabolic Equation . . . . . . . . . . . . . . . . . . . . . . . . 46 4 Scattering from randomly rough surfaces 50 4.1 Rayleigh criterion . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Surface Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3 Properties and Approximate Solutions of Scattering Equations 56 5 Wave Propagation through Random Media 65 5.1 Propagation beyond a thin phase screen . . . . . . . . . . . . 65 5.2 Propagation in an extended random medium . . . . . . . . . . 70 6 Electromagnetic scattering in layered media 86 6.1 Reflection from a layered medium . . . . . . . . . . . . . . . . 91 7 The inverse scattering problem 95 7.1 Tikhonov regularisation . . . . . . . . . . . . . . . . . . . . . 96 8 Methods for solving the inverse scattering problem 98 8.1 The method of Imbriale and Mittra . . . . . . . . . . . . . . . 98 8.2 Optimization method . . . . . . . . . . . . . . . . . . . . . . . 106 8.3 Inverse scattering in the Born approximation . . . . . . . . . . 109 9 References and further reading 113 Part III - Classical Wave Scattering 2 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1 Governing equations for acoustic and elec- tromagnetic waves 1.1 Acoustic Waves Acoustic waves are mechanical disturbances that propagate in a medium, for example air, water, or a solid structure such as, for example, the shell of a ship. The governing equations that describe how acoustical waves propagate in a mediumarederivedfromthebasicconservationlawsforfluidsandthelawsof thermodynamics. The detailed derivation can be found in many books (see, for example, Pierce 8, LD Landau and EM Lifschitz, Fluid Mechanics). Here we shall just give the main steps. For a fluid with density ρ and velocity v, the conservation of mass in the non-relativistic case is expressed as: ∂ρ +∇·(ρv) = 0 (1.1) ∂t In the presence of a mass source, the r.h.s. will be non-zero. If we have an ideal fluid, i.e. with zero viscosity, the surface force F is directed normally s into thesurface, soF =−np, wherep is the pressure. With this assumption s and neglecting gravity and any other external forces, the conservation of momentum is expressed as: µ ¶ ∂v ρ +v·∇v =−∇p (1.2) ∂t Ingeneral, forasysteminlocalthermodynamicequilibrium, anequationof state will hold, whereby a function of state can be expressed in terms of any other two. We shall use the equation of state that relates the pressure to the entropyS andthedensityρofthesystem: p =p(ρ,S),wheretheentropyisa measure of the disorder of a system and is such that, for a reversible process, the differential change in entropy equals the differential change in absorbed heat divided by the system temperature: dS =dQ/T. If one can ignore any heat flow (so no temperature gradient imposed externally, gradient of the fluid small), then the fluid motion is adiabatic and the entropy is constant, S (isentropic process). In this case 0 p =p(ρ,S ) (1.3) 0 and depends only on the density. Part III - Classical Wave Scattering 3 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.1 Acoustic Waves If sound, i.e. acoustic disturbance, is regarded as a small-amplitude pertur- bation of an ambient state (ρ ,p ,v ), then when the disturbance is present 0 0 0 one has ′ ′ ′ p =p +p; ρ =ρ +ρ; v =v +v, 0 0 0 which also satisfy the conservation laws: ′ ∂(ρ +ρ) 0 ′ ′ +∇·((ρ +ρ)v) = 0 (1.4) 0 ∂t µ ¶ ′ ∂v ′ ′ ′ ′ (ρ +ρ) +v ·∇v =−∇(p +p) (1.5) 0 0 ∂t and ′ ′ p +p =p(ρ +ρ,S ) (1.6) 0 0 0 Herev = 0, so this derivation applies in the absence of mean flow. We shall 0 also take the fluid to be homogeneous so p and ρ are constants related by 0 0 p =p(ρ ,S ). 0 0 0 ′ ′ If p is expanded in a Taylor series in ρ: µ ¶ µ ¶ 2 ∂p 1 ∂ p ′ ′ ′ 2 p = ρ + (ρ) +... , 2 ∂ρ 2 ∂ρ 0 0 by using this expansion in the above equations and truncating to first order we obtain the linear acoustic equations ′ ∂ρ ′ +ρ ∇·v = 0 (1.7) 0 ∂t ′ ∂v ′ ρ =−∇p (1.8) 0 ∂t µ ¶ ∂p ′ ′ 2 ′ p = ρ =c ρ. (1.9) ∂ρ 0 The quantity c introduced in (1.9) is denoted speed of sound for reasons that will become clear once we obtain the solution to the equation governing the time evolution of the acoustic pressure p. The factor of proportionality ρc, equal to the ratio between pressure and velocity, is called characteristic impedance of the medium. Let’s now use (1.9) in (1.7): ′ 1 ∂p ′ +ρ ∇·v = 0 (1.10) 0 2 c ∂t Part III - Classical Wave Scattering 4 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.1 Acoustic Waves If we then take the partial time derivative of (1.10) and use (1.8), we obtain the wave equation for the acoustic pressure: 2 1 ∂ p 2 ∇ p− = 0, (1.11) 2 2 c ∂t where we have now dropped the primes. The wave equation can be formulated alternatively in terms of a velocity potential. Ifwetakethecurlof (1.8)andusethevectoridentity∇×(∇ϕ) = 0, valid∀ϕ, it follows that (again dropping all primes) ∂(∇×v) = 0, ∂t i.e. thevorticity (∇×v) is constant in time. Therefore the velocity field is irrotational (∇×v = 0) if it is irrotational initially, and we can introduce a velocity potential φ by writing v =∇φ. (1.12) Note that v =∇φ+v will apply if the fluid is initial moving with velocity 0 v . Substituting (1.12) in (1.8), we obtain 0 ∂φ p =ρ . (1.13) 0 ∂t Now, using (1.12), (1.13) and (1.9) in (1.8) gives 2 1 ∂ φ 2 ∇ φ− = 0, (1.14) 2 2 c ∂t which is the wave equation in terms of the velocity potential. A general solution of (1.11) is µ ¶ µ ¶ ξ ξ p =f t− +g t+ , (1.15) c c where f and g are arbitrary functions which will be determined by initial and boundary conditions, and ξ is the coordinate along which the acoustic pressure varies, i.e. the direction along which the acoustic disturbance trav- els. This solution is the sum of two waves travelling at speedc in the +ξ and −ξ direction respectively. In an arbitrarily oriented coordinate frame, if n is the unit vector in the direction of increasing ξ, then at a point x we can write ξ = n·x. If one assumes, as is usually appropriate from physical considerations, that there Part III - Classical Wave Scattering 5 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.1 Acoustic Waves exists a time t in the past before which the wave hasn’t arrived and all field 0 quantities are zero (causality), then the solution reduces to waves travelling in the positive direction: ³ ´ n·x p =f t− . (1.16) c Foranacousticdisturbanceofconstantfrequency, thefieldvariablesoscillate sinusoidally with time, so (iϕ−iωt) p =Acos(ωt−ϕ) = ReAe , (1.17) where ω = angular frequency, ϕ = phase, 2π and we have T = = period, ω ω f = = frequency. 2π If a sinusoidal wave p =Acos(ωt) travels in the n direction, then we must have p =f(t−n·x/c), and consequently h i n·x n·x −iω(t− ) ik·x−ωt c p =Acos ω(t− ) = Ree = Ree , (1.18) c ω where we have used the wavevectork = n. The above rightmost expression c is the one usually and most conveniently used in practical calculations. NOTE: Even though the physical quantity is given by the real part only, full complex waves are normally used in calculations, and the real part is subsequently taken as appropriate. Consequently, if the acoustic field is expressed in terms of a complex velocity potential ψ, we should be careful to take p = Reiωρψexp(−iωt) v = Re∇ψexp(−iωt) (1.19) when dealing with real physical quantities. Any acoustic disturbance p(x,t) can be written as a superposition of time- harmonic waves 1.17. This can be done using a Fourier transform (as long 2 asp(x,t) andp(x,t)∈L ): Z ∞ 1 p(x,t) = p(x,ω)exp(−iωt)dω (1.20) 2π −∞ Part III - Classical Wave Scattering 6 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.1 Acoustic Waves where Z ∞ 1 p(x,ω) = p(x,t)exp(iωt)dt (1.21) 2π −∞ ik·x−ωt If we substitute a harmonic wave p = e in the wave equation (1.11) ∂ (noting that Re· and · commute), we obtain ∂t 2 ω 2 p+∇ p = 0 , 2 c ω or, by using the wavenumber k = c 2 2 ∇ p+k p = 0 . (1.22) This form of the wave equation, suitable for time-harmonic waves, is usually called the Helmholtz equation, or reduced wave equation. When considering time-harmonic problems then, it is usual (and obviously very convenient) to drop the time-dependent part of the wave altogether. This is possible, at least for part of the calculations, in the case of a non- monochromatic wave, by decomposing it into monochromatic waves using Fourier analysis. Since the wave equation is linear, each Fourier component obeys the Helmholtz equation, and the total field can be reconstracted after solvingthescatteringproblemforwhateverboundaryconditionsonanyfinite surfacesareappropriate. Inthiscase, though, itisnotpossibletoexpressthe causality condition in the same way as before. Causality then is expressed by the integrability condition implicit in assuming that a Fourier representation ofthewaveexists. Whatwasintroducedasaconditionintime(initialvalue), and cannot in that form be readily applied to a superposition of stationary waves, is equivalent to a condition in space (boundary condition at infinity): −1/2 p(x) =O(x ) (1.23) or, more usually: µ ¶ ∂p(x) x −ikp(x) → 0 (1.24) ∂x uniformly as x→ ∞. This is the Sommerfeld radiation condition, and it expresses the requirement that the field should contain no incoming wavesasx→∞. Ingeneral, integrability, hencecausality, willalsoresultin restrictions imposed on thecontour chosen for the integration in the complex plane. Part III - Classical Wave Scattering 7 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.2 Electromagnetic waves 1.2 Electromagnetic waves Inthis section thewaveequation obeyedbyelectromagnetic wavesis derived, and we introduce the general scattering problem for electromagnetic waves. We shall begin with Maxwell’s equations for an electromagnetic field in a generic medium with permittivity ǫ and permeability μ, in SI units (also sometimes called MKS): ∂B ∇×E = − (1.25) ∂t ∇·B = 0 (1.26) ∂D ∇×H = +J (1.27) ∂t ∇·D = ρ , (1.28) Here E is the electric field intensity, B is the magnetic induction, H is the magnetic field intensity, D is the so-called electric displacement, J is the current density, and ρ is the electric charge density. These quantities are related by D = ǫE+P (1.29) B = μH+M , (1.30) where P is the electric polarization and M the magnetization. In free space, we haveP =0 andM =0, and Maxwell’s equations reduce to ∂H ∇×E = −μ (1.31) 0 ∂t ∇·H = 0 (1.32) ∂E ∇×H = ǫ (1.33) 0 ∂t ρ ∇·E = , (1.34) ǫ 0 where ǫ and μ are the permittivity and permeability of free space respec- 0 0 tively. It is straightforward to see from the Maxwell equations that there exist scalarandvectorpotentialsfortheelectromagneticfield. Since∇·B = 0, ∃ a vector field A such that B =∇×A . (1.35) Using this in the first of Maxwell’s equations shows that E must satisfy: ∂A E =−∇V − , (1.36) ∂t Part III - Classical Wave Scattering 8 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.2 Electromagnetic waves where V is a scalar field. A and V are not unique. It is always possible to find an arbitrary scalar Φ such that the vector A =A−∇Φ 0 also satisfies (1.35) giving the same B, and the scalar ∂Φ V =V + 0 ∂t gives the same E. This is a gauge transformation, and any particular choice of Φ is a choice of gauge. We shall see that the electric field E and the magnetic field B obey a wave equation equivalent to that derived in section 1.1 for acoustic waves. Let us derive the wave equation first in free space, i.e. and in the case when there are no charges nor currents: ρ = 0,J =0. We shall start with equation (1.27), which in this case becomes: ∂E ∇×B =μ ǫ (1.37) 0 0 ∂t ∂ ∂ Noting that ∇×· and · commute, if we now apply to (1.37), and ∂t ∂t use equation (1.25), we obtain 2 ∂ E ∇×(∇×E) =μ ǫ (1.38) 0 0 2 ∂t −2 and, since ∇·E = 0 in this case, and μ ǫ = c , where c is the speed of 0 0 light, we arrive at the wave equation for E 2 1 ∂ E 2 ∇ E− = 0 . (1.39) 2 2 c ∂t It is straightforward to derive a wave equation of the same form for the magnetic field B. A wave equation for E can be similarly derived in the more general case were charges and currents are present, and has the form: 2 1 ∂ E ∂J 2 −1 ∇ E− =ǫ ∇ρ+μ , (1.40) 0 0 2 2 c ∂t ∂t where the r.h.s. represents source terms due to charges and currents. A similar equations for B also applies. We notice here that the vector product E× H has the dimensions of an energy flux. It is indeed taken as the energy flow at a point (even though it is not unique), and is called Poynting vector: 1 S =E×H = E×B (1.41) μ Part III - Classical Wave Scattering 9 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.2 Electromagnetic waves ThePoyntingvectorgivesthedirectionoftheenergyflow. Similarlytoacous- tic plane waves, an electromagnetic plane wave shall be written E(r,t) = ik·r E (t)e , from which we can see that for plane waves the energy flow is 0 perpendicular to the wavefront, and the energy travels in the direction of the wavevector k. Note that, even though the functional form of an electromag- netic plane wave is the same as that of an acoustic plane wave, electromag- netic waves are vector waves, so all the equations are vector equations. −iωt For a time-harmonic field E(r,t) = ReE(r)e we can derive, as in the case of acoustic waves, a reduced wave equation: the Helmholtz equation for electromagnetic waves 2 2 ∇ E(r)−k E(r) = 0 , (1.42) 2 2 where k =ω μǫ. The radiation condition for electromagnetic waves can be expressed (as before) in terms of the scalar and vector potentials, but is usually more conveniently expressed in terms of the field components: rEK , rHK ˆ r(E+Z i ×H) → 0 , asr→∞ , (1.43) 0 r ˆ r(H−i ×E/Z ) → 0 , asr→∞ , (1.44) r 0 p where Z = μ/ǫ = impedance of the medium. 0 Polarized waves Plane waves solutions of (1.40) or (1.39) and their equivalents for the mag- netic field are again fundamental in practical applications, as in the case of acoustic waves, either because only far-field solutions are of interest, or because any wave can be represented as a superposition of plane waves. Ofparticularinterestareplanewaveswhicharelinearlypolarized. Twokinds of linear polarizations are possible. Let’s take Cartesian coordinates and a plane wave with direction of propagation k in the (x,y)-plane. Then, either the electric vector E is parallel to the z-coordinate: E =ˆzE , E-polarization (1.45) z or TM wave or: H =ˆzH , H-polarization (1.46) z or TE wave . Part III - Classical Wave Scattering 10 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.2 Electromagnetic waves When talking of ”direction of polarization”, one normally refers to the di- rection of E (but note that the opposite convention is sometime found in the literature). It is immediately apparent that in many scattering problems withlinearlypolarizedwaves, thevectorwaveequationwillreducetoascalar equation for either E or H . z z For example, if a TM wave is incident on a surface that can be described by S =f(ρ,φ) in cylindrical polar coordinates, independently ofz, then for this scattering problem the incident field is given by µ ¶ inc inc i ∂E ∂E inc inc inc z z E =ˆzE , H =− xˆ− yˆ , (1.47) z kZ ∂y ∂x p where Z = μ/ǫ is the surface impedence, and depends on the properties of the two media and the surface, and usually varies with the incoming field at each point. In general, Z is also a function of frequency and angle of incidence. Since the boundary conditions are independent of z, then the scattered field must also be E-polarized, and of the form µ ¶ sc sc i ∂E ∂E sc sc sc z z E =ˆzE , H =− xˆ− yˆ , (1.48) z kZ ∂y ∂x sc therefore the scattering problem reduces to finding the scalar function E , z and is analogous to the problem of an acoustic field scattered by a soft sur- face. Similarly, the case ofH-polarization is analogous to that of an acoustic field scattered by a hard surfaces. All problems where the scatterer is ax- isymmetricandtheincidentelectromagneticfieldispolarizedinthedirection parallel to the axis of symmetry therefore reduce to a scalar problem. Part III - Classical Wave Scattering 11 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.3 Boundary conditions 1.3 Boundary conditions The constraints imposed on the solutions of the wave equation at a surface mustreflectthenatureofthesolidobjectdefinedbythesurfaceor, ifthesur- face in question is an interface between two fluids, the different characteristic properties of the two fluids. Ifthesurfaceisperfectlyreflecting, (foracousticwaves)orperfectlyconduct- ing (for electromagnetic waves), i.e. the tangential component of the total electric field at the surface is zero: E−(E·n)n = 0 , (1.49) then two cases are possible: Neumann condition, when the normal derivative of the potential field is given at the boundary, i.e., ifn is the unit normal pointing outward from the surface: ∂ψ(r) = 0, r on S. (1.50) ∂n Foracousticwaves, thiscorrespondstoanacoustically hardsurface, orinthe caseofelectromagneticwavesin2D,toavertically polarizedelectromagnetic wave on a perfectly conducting surface. Dirichlet condition, when the value of the potential field is given at the boundary: ψ(r) = 0, r on S. (1.51) which, for acoustic waves, corresponds to a pressure-release or acoustically soft surface, and in the case of electromagnetic waves corresponds to a hor- izontally polarized electromagnetic wave in 2D on a perfectly conducting surface. In most real cases the surfaces is neither perfectly reflecting, nor perfectly conducting, and the boundary condition is of mixed type: Cauchy condition (also called Robin, or impedance boundary condi- tion). In this case both the potential and its normal derivative are different from zero at the boundary, and the boundary condition is then expressed as an equation relating these two quantities: ∂ψ (r) =iZ(r,ω,θ,...)ψ(r) r on S. (1.52) ∂n Forelectromagneticwavestheboundaryconditionrelatesthetangentialcom- ponent of the electric field at the surface to the normal component of the magnetic field at the surface: E−(E·n)n =Z(r,ω,θ,...)n×H , (1.53) Part III - Classical Wave Scattering 12 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.3 Boundary conditions Here Z depends on the properties of the two media and usually varies with the incoming field at each point. In general, Z is also a function of frequency and angle of incidence. The impedance boundary condition can also be expressed as n×∇×E =iZn×(E×n) T in a form similar to the one for scalar waves. Exact boundary conditions at an interface between two media are given by the jump (continuity) conditions: ρ ψ = ρ ψ 1 1 2 2 (2) ∂ψ ∂ψ ∂ψ 1 2 = (1.54) ∂n ∂n ∂n where the subscripts 1 and 2 refer to the two media, and we take n as the normal directed into medium 1. For electromagnetic waves, the boundary conditions at an interface are con- tinuity of the normal component of B and the tangential component of E: (B −B )·n = 0 (1.55) 2 1 n×(E −E ) = 0 , (1.56) 2 1 plus (D −D )·n = ρ (1.57) 2 1 s n×(H −H ) = J , (1.58) 2 1 S where ρ is surface charge and J surface current. s S Part III - Classical Wave Scattering 13 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.4 Green’s functions 1.4 Green’s functions In most problems of practical interest in acoustics, there will be one or more sources of sound, and the space where the problem needs to be solved will include one or more surfaces. Consequently, the differential equation to be solved will be an inhomogeneous version of (1.11) or (1.22), and the solutions will be subject to other boundary conditions in addition to (1.24). In general the problem in question will then be defined by a differential equation 2 2 ∇ p(x,t)+k p(x,t) =f(x,t) , (1.59) together with boundary conditions on one or more surfaces and the Sommer- feld conditions. It is usually not easy to find solutions for such boundary value problems, but the task is greatly facilitated by the use of an auxil- iary function associated with the differential equation, known as Green’s function. InordertoillustratetheconceptofaGreen’sfunction,andprovidethemeans ofconstructingGreen’sfunctionsfordifferentproblems,let’sfirstwrite(1.59) in operator form as Lp(ξ) =f(ξ) , (1.60) where L is a linear operator, p the unknown function, and f is a known function determined by the source. The variable ξ denotes a point in an n-dimensional space which can include time as one of the coordinates. The solution of (1.60) can be sought in principle by finding the inverse of the operator L, −1 p(ξ) =L f(ξ) , (1.61) but this is so far not particularly useful in practice. Since L is a differential −1 operator, if L exists, it can be reasonably assumed to be an integral oper- −1 ator. If we assume that L is an integral operator with kernel K, i.e. such that Z −1 L f(ξ) = K(ξ,η)f(η)dη for any functions f defined in the same domain as p, then we can write Z −1 p(ξ) =LL p(ξ) =L K(ξ,η)p(η)dη , Since L is a differential operator with respect to the variable ξ, we can for- mally write Z p(ξ) = LK(ξ,η)p(η)dη . Part III - Classical Wave Scattering 14 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.4 Green’s functions This can be true only if LK(ξ,η) =δ(η−ξ) , (1.62) in which case we can write the solution to (1.60) as Z p(ξ) = K(ξ,η)f(η)dη (1.63) −1 The kernel K of the operator L is called the Green’s function for the problem and will therefter be denoted by G(ξ,η). We can see from (1.63) that its knowledge allows us to find the solution of the wave equation for any known source f(ξ), at least in principle. Equation (1.62) shows that the Green’s function is the field generated by a delta-function inhomogeneity, i.e. the solution of the inhomogeneous wave equation (1.60) withthe source term f =δ(η−ξ). Due to the symmetric property of G: ∗ G(ξ,η) =G (η,ξ) ′ This reciprocity relation means that G(x,y,t,t) can equivalently represent thefieldatapointxduetoa’disturbance’aty,orthefieldatyduetoa’dis- turbance’ at x. In other words, the Green’s function is unchanged if source and receiver are interchanged. We note that, with regard to the time coor- dinate, the reciprocity implies time reversal: G(x,y,t,0) =G(y,x,0,−t), so causality is satisfied. The Green’s function defined above is not unique: it is always possible to add to it a solution of the homogeneous wave equation, and the result will of course still satisfy (1.62). The particular solution for the Green’s function which is independent of any boundary conditions is called the free space Green’sfunction,andshallusuallybedenotedbyG (ξ,η). AnyotherGreen’s 0 function can be written as G(ξ,η) =G (ξ,η)+G (ξ,η) , (1.64) 0 H where G (ξ,η) is a solution of H L(ξ)G(ξ,η) = 0 . (1.65) When G (ξ,η) is chosen to satisfy the boundary conditions for the problem, H then G(ξ,η) is the exact Green’s function for the problem. We shall derive here the free space Green’s function for time-dependent wave equation in 1D, i.e. the function G satisfying: 2 2 ∂ Gx,t ∂ G(x,t) 2 −c =δ(x−y)δ(t−τ) (1.66) 2 2 ∂t ∂x Part III - Classical Wave Scattering 15 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.4 Green’s functions If we Fourier transform (1.66) in both space and time, it becomes 2 2 2 iky −iωτ ˆ ˆ −ω G(k,ω)+c k G(k,ω) =e e , (1.67) so the transform of the required Green’s function is given by iky −iωτ 1 e e ˆ G(k,ω) = , (1.68) 2 2 2 2 c k −ω /c and G(x,t) can be obtained by transfoming back: Z Z ∞ ∞ −ik(x−y) iω(t−τ) 1 e e G(x,t) = dkdω (1.69) 2 2 2 2 2 4π c k −ω /c −∞ −∞ The integral in (1.69) must be calculated taking care that the contour of integration is chosen in a way that satisfies the causality condition. As dis- cussed in section 1.1, this means requiring that the time-Fourier transformed function G(x,ω) must be analytic in Im(ω)≤ 0. Therefore, when integrat- ing in the complex k-plane, we need to take the limit from below at the pole k = ω/c, and the liimit from above at the pole k =−ω/c. In the first case the contour will have a small indentation above the pole, in the second case, a small indentation below. With these contraints then, if we first carry out the inverse in k-space we obtain: Z x−y ∞ −ik(x−y) −iω c 1 e e G(x,ω) = dk = . (1.70) 2 2 2 2 2 4π c k −ω /c 4πiωc −∞ The inverse transform in time then gives: µ ¶ Z x−y ∞ iω(t−τ− ) c 1 e 1 x−y G(x,t) = dω = H t−τ− . (1.71) 4πic ω 2c c −∞ x−y The time (t−τ − ) is called retarded time, and is the time at which c the disturbance observed at (x,t) has been emitted by the source at (y). In 3 dimensions, the free space Green’s function for the time-dependent wave equation is 1 G(x,t) = δ(t−τ−r/c) , (1.72) 2 4πc r where r =x−y, and the free space Green’s function for the Helmholtz equation is ikr e G(x,t) = . (1.73) 4πr Part III - Classical Wave Scattering 16 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.4 Green’s functions The above represents a spherically symmetric wave, and can be derived as the wave generated by a source consisting of an oscillating sphere, in the limiting case where the radius tends to zero. Such source is called a point source, or monopole. In the case of electromagnetic waves, a point source is equivalent to a charge. It is instructive to consider a source Q(r), uniformly distributed within a sphere. The Helmholtz equation for the wave field is then 2 2 ∇ p(x,ω)+k p(x,ω) =Q(x) . (1.74) This can now be written, using (1.63), as: Z ikr 1 e p(x,ω) = Q(y)dy (1.75) 4π r ′ ′ If the radius of the sphere r is very small, so r ≪ r, then we can expand ′ ikr−r ′ (e /(r−r ) in a power series: ′ ikr−r ) ′ (e /(r−r ) = µ ¶ µ ¶ ikr ikr ikr e e 1 e ′ ′ 2 −r ·∇ + (r ·∇) +... r r 2 r If we substitute this expansion in (1.75), we obtain: ikr ikr ikr e e e p =Q +Q +Q +... (1.76) 0 i ij 2 3 r r r The coefficients Q , Q and Q (obtained by integrating over the volume 0 i ij of the sphere containing the sources), are called respectively monopole, dipole and quadrupole strength, and the series just obtained multipole expansion. Part III - Classical Wave Scattering 17 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.5 The Kirchoff-Helmholtz equation 1.5 The Kirchoff-Helmholtz equation By using the Green’s function it is possible to derive an integral form of the Helmholtz equation, which facilitates calculations of sound propagation and scattering, and allows sources and boundary conditions to be treated in a simple and convenient way. In order to derive this integral equation, we shall first recall the following vector identities. Given any two function f and g, we have: 2 ∇·(f∇g) =f∇ g+(∇f)·(∇g) . (V1) If f∇g is a vector field continuously differentiable to first order, which we shall denote by F = f∇g, then we can apply to it the following theorem, which transforms a volume integral into a surface integral: n Gauss theorem If V is a subset ofR , compact and with piecewise smooth boundary S, and F is a continuously differentiable vector field defined on v, then Z Z ∇·FdV = F·ndS , (V2) V S where n is the outward-pointing unit normal to the boundary S. 3 InR , for an F =f∇g and an F =g∇f, we have, using V2 and V1: 1 2 Z Z £ ¤ 2 f∇ g+(∇f)·(∇g) dV = f∇g·ndS , (1.77) V ∂V Z Z £ ¤ 2 g∇ f +(∇g)·(∇f) dV = g∇f·ndS , (1.78) V ∂V and subtracting (1.78) from (1.77) we obtain: Z Z ¡ ¢ 2 2 f∇ g−g∇ f dV = (f∇g−g∇f)·ndS . (1.79) V ∂V This result can be used can be used to solve a general scattering problem, in- volving oneormoresources andwritethesolutionin termsofthe(unknown) field and its normal derivative along the boundary. The integral equations obtained can in principle be solved to find these unknown surface field val- ues. This approach applies whether the problem involves an interface with a vacuum or with a second medium. ConsiderfirstafiniteregionV containedbetweentwosmoothclosedsurfaces S and S , and containing a source Q(r). 0 1 Part III - Classical Wave Scattering 18 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.5 The Kirchoff-Helmholtz equation . Let G be the free space Green’s function, and ψ the solutions to the inho- mogeneous equation 2 2 ∇ ψ+k ψ =Q(r) . (1.80) Using the vector identities introduced above, we can write: Z Z µ ¶ ¡ ¢ ∂G ∂ψ 2 2 ψ∇ G−∇ ψG dV = ψ − G ds , (1.81) ∂n ∂n V S +S 0 1 where we have usedd/dn =n·∇. If we let theouter surfaceS go to infinity, 1 then, provided ψ obeys the Sommerfeld boundary condition at infinity, then the integral over S vanishes. 1 2 2 Substituting in (1.81) the expressions for ∇ ψ and ∇ G obtained by the appropriate wave equations, i.e. 2 ′ 2 ∇ G = δ(r−r)−k G 2 2 ∇ ψ = Q(r)−k ψ we obtain Z Z µ ¶ ∂G ∂ψ ′ ′ ′ ′ ′ ψ(r)δ(r−r)− Q(r)G(r,r)dr = ψ − G ds . (1.82) ∂n ∂n V S +S 0 1 But Z ′ ′ ′ ψ (r) = Q(r)G(r,r)dr. (1.83) i V is the incident field ψ inside the volume V. Using this result, then, we can i write (1.81) as Z · ¸ ∂G(r,r ) ∂ψ 0 ψ(r) =ψ (r)+ ψ(r ) − (r )G(r,r ) dr . (1.84) i 0 0 0 0 ∂n ∂n S 0 This is the Kirchoff-Helmholtz equation, an integral (implicit) form of theHelmholtzequation, whichisofgreatpracticaluseincalculatingthefield induced by sources scattered by finite boundaries. Part III - Classical Wave Scattering 19 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.1.5 The Kirchoff-Helmholtz equation Part III - Classical Wave Scattering 20 O.Rath-Spivackdamtp.cam.ac.uk Copyright © 2007 University of Cambridge. Not to be quoted or reproduced without permission.

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