TheWaveandHelmholtzEquations

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1 TheWaveandHelmholtzEquations Ramani Duaiswami The Univesity of Mayland, College Pak Febuay 3, 2006 Abstact CMSC828D notes (adapted fom mateial witten with Nail Gumeov). Wok in pogess 1 Acoustic Waves 1.1 Baotopic Fluids The usual assumptions fo acoustic poblems ae that acoustic waves ae petubations of the medium (fluid) density ρ (,t), pessuep (,t), and mass velocity, v (,t), wheet is time. It is also assumed that the medium is inviscid, and that petubations ae small, so that ρ = ρ 0 + ρ 0, p = p 0 + p 0, ρ 0 ρ 0, p 0 p 0, v 0 p0 c. (1) ρ 0 Hee the petubations ae nea an initial spatially unifom state (ρ 0,p 0 ) of the fluid at est (v 0 = 0) and ae denoted by pimes. The latte equation states that the mass velocity of the fluid is much smalle than the speed of sound c in that medium. In this case the lineaized continuity (mass consevation) and momentum consevation equations can be witten as ρ 0 t + (ρ 0v 0 v 0 )=0, ρ 0 t + p0 =0, (2) whee = i x x + i y y + i z z, (3) is the invaiant nabla opeato, epesented by fomula (3) in Catesian coodinates, whee (i x, i y, i z ) ae the Catesian basis vectos. Diffeentiating the fome equation with espect to t and excluding fom the obtained expession v 0 / t due to the latte equation, we obtain 2 ρ 0 t 2 = 2 p 0. (4) Note now that system (2) is not closed since the numbe of vaiables (thee components of velocity, pessue, and density) is lage than the numbe of equations. The elation needed to close the system is equation of state, which elates petubations of the pessue and density. The simplest fom of this elation is povided by baotopic fluids, whee the pessue is a function of density only: p = p(ρ). (5) We can expand this in seies nea the unpetubed state p = p(ρ 0 )+ dp (ρ ρ 0 )+O ³(ρ 2 ρ 0 ). (6) dρ ρ=ρ0 1

2 Taking into account that p(ρ 0 )=p 0, we obtain neglecting the second-ode nonlinea tem: p 0 = c 2 ρ 0, c 2 = dp, (7) dρ ρ=ρ0 wheeweuseddefinition of the speed of sound in the unpetubed fluid, which is a eal positive constant (popety of the fluid). Substitution of expession (7) into elation (4) yields the wave equation fo pessue petubations 1 2 p 0 c 2 t 2 = 2 p 0. (8) Obviously, the density petubations satisfy the same equation. The velocity is a vecto and satisfies the vecto wave equation: 1 2 v 0 c 2 t 2 = 2 v 0. (9) This also means that each of the components of the velocity v 0 = vx,v 0 y,vz 0 0 satisfies the scala wave equation (8). Note that these components ae not independent. The momentum equation (2) shows that thee exists some scala function φ 0, which is called the velocity potential, such that v µ φ = φ, c 2 t 2 = φ 2 φ ρ 0 t = p0. (10) So the poblem can be solved fo the potential and then the velocity field can be found as the gadient of this scala field. 1.2 Isentopic Fluids The analysis that was done in class in Lectue 2, coesponds to the case of an isentopic gas. In this case as was noted in class, the equations that ae finally deived ae the same, except that the sound velocity is γp0 c =, ρ 0 with γ the atio of specific heats fo the gas. 1.3 Fouie Tansfom The wave equation deived above is linea and obey paticula solutions peiodic in time. Paticulaly, if the time dependence is a hamonic function of cicula fequency ω we can wite φ (,t) =Re e iωt ψ (), i 2 = 1, (11) whee ψ () is some complex valued scala function and the eal pat is taken, since φ (,t) is eal. Substituting expession (11) into the wave equation, (10), we can see that the latte is satisfied, if ψ () is a solution of the Helmholtz equation 2 ψ ()+k 2 ψ () =0, k = ω c. (12) The constant k is called the wavenumbe and is eal fo eal ω. The name is elated to the case of plane wave popagating in the fluid, whee the wavelength is λ =2π/k, and so k is the numbe of waves pe 2π. The Helmholtz equation stands theefoe fo monochomatic waves, o waves of some given fequency ω. Fo polychomatic waves, o sums of waves of diffeent fequencies, we can sum up solutions with diffeent ω. Moe geneally, we can pefom the invese Fouie tansfom of potential φ (,t) with espect to the tempoal vaiable: ψ (,ω) = e iωt φ (,t) dt. (13) 2

3 In this case ψ (, ω) satisfies the Helmholtz equation (12). Solving this equation we can detemine solution ofthewaveequationusingthefowad Fouie tansfom: φ (,t) = 1 e iωt ψ (,ω) dω, ω = ck. (14) 2π We note that in the Fouie tansfom the fequency ω can be eithe negative o positive. This esults in eithe negative o positive values of the wavenumbe. Howeve the Helmholtz equation depends on k 2 and is invaiant with espect to change of the sign of k. This phenomenon, in fact, has a deep physical and mathematical oigin, and appeas fom the popety of the wave equation to be a two wave equation. It descibes solutions which ae a supeposition of two waves popagating with the same velocity in opposite diections. We will conside this popety and ules fo selection of pope signs late in this chapte, in the section dedicated to bounday conditions. In the case of Fouie tansfom we can state that the Helmholtz equation is the wave equation in the fequency domain. Since methods fo fast Fouie tansfom ae widely available, convesion fom time to fequency domain and back ae computationally efficient, and so the poblem of solution of the wave equation can be educed to solution of the Helmholtz equation, which is an equation of lowe dimensionality (3 instead of 4) than the wave equation. 2 Bounday Conditions The Helmholtz equation is an equation of the elliptic type, fo which it is usual to conside bounday value poblems. Bounday conditions follow fom paticula physical laws (consevation equations) fomulated on the boundaies of the domain in which solution is equied. This domain can be finite (intenal poblems) o infinite (extenal poblems). Fo infinite domains solutions should satisfy some conditions at the infinity. These conditions also have a physical oigin. Fo the Helmholtz equation that aises as a tansfom of the wave equation into the fequency domain the bounday conditions should be undestood in the context of the oiginal wave equation. 2.1 Conditions at Infinity Spheically Symmetical Solutions To undestand conditions which should be imposed fo solutions of the Helmholtz equation in infinite domains we stat with the consideation of spheically symmetical solutions of scala wave equation. In this case the dependence on of a function φ, which satisfies the wave equation (10), is the dependence on the distance = only. It is well known that solution of this equation can be witten in the following fom φ (, t) = 1 [f(t + /c)+g (t /c)], (15) whee f and g ae two abitay diffeentiable functions. The fome function descibes incoming waves to the cente =0and the latte function descibes outgoing waves fom the cente =0. Indeed the incoming wave phase can be chaacteized by some constant value of f, which is ealized at = ct+const, and so the wavefonts convege to the cente as t is gowing. Invesely, the outgoing wave phase is chaacteized by some constant value of g, which is ealized at = ct+const and so the wavefonts fo the outgoing waves divege fom the cente as inceasing t. Theefoe spheically symmetical solution of the scala wave equation can be chaacteized by specification of two functions of time f(t) and g(t). Assume that these functions satisfy necessay conditions to pefom the Fouie tansfom. Then, in the fequency domain we have images of these functions accoding (10): bf (ω) = e iωt f(t)dt, bg (ω) = 3 e iωt g(t)dt. (16)

4 With these definitions and solution (15) we can detemine the image, o phaso ψ (, ω), ofφ (, t) in the fequency domain as ψ (, ω) = e iωt φ (, t) dt = 1 e iωt f(t + /c)dt + e iωt g(t /c)dt (17) = 1 e iω(t0 /c) f(t 0 )dt 0 + e iω(t0 +/c) g(t 0 )dt = 1 b f (ω) e ik + 1 bg (ω) eik, k = ω c. Hee we defined k = ω/c and so this quantity is negative fo ω < 0 and positive fo ω > 0. The function ψ (, ω) is a solution of the spheically symmetical Helmholtz equation (12). It is seen that solutions coesponding to the incoming waves ae popotional to e ik, while solutions coesponding to the outgoing waves ae popotional to e ik. It is not difficult to see also that at lage we have ikψ = 2ikf b 1 (ω) e ik + O, (18) 1 + ikψ =2ikbg (ω) e ik + O. (19) This means that if the condition lim ikψ =0 (20) holds then f b (ω) 0. This esults in f(t) 0 and in this case φ (, t) consists only of outgoing waves. Similaly, in the case if condition lim + ikψ =0 (21) holds then solution consists only of incoming waves and g(t) Sommefeld Radiation Condition The poblems, which ae usually consideed in elation with the wave equation in thee dimensional unbounded domains ae scatteing poblems. In this case the wave function is specified as φ (,t)=φ in (,t)+φ scat (,t), (22) whee both functions φ in (,t) and φ scat (,t) satisfy the wave equation. Function φ in (,t) is the potential of the incident field, while φ scat (,t) is the potential of the scatteed field, which aises due to the pesence of one o seveal scattees. In the absence of scattees φ (,t)=φ in (,t) is some given function (e.g. the potential of a plane wave popagating along the z diection, φ in (,t)=f (t z/c)). To undestand the scatteed field we may tun ou attention to the Huygens pinciple, which epesents wave popagation as emission of seconday wave fom the points located on the cuent wavefont. When the pimay wave descibed by φ in (,t) eaches the scattee bounday the seconday waves ae geneated fom the bounday points located at the intesection of the bounday and the wavefont. Due to finite speed of wave popagation spatial points fa fom the bounday do not know about these seconday waves, so these waves can be thought as waves outgoing fom the bounday points. Fo each point then we can wite the seconday wave potential in the fom (15), whee f 0 and, theefoe, in the fequency domain condition (20) holds. Since the total scatteed field, φ scat (,t), can be seen now as a supeposition of outgoing waves, coesponding potential in the fequency domain should satisfy condition lim µ scat ikψ scat =0. (23) 4

5 This condition is called Sommefeld adiation condition o just adiation condition. It states that the scatteed field consists of outgoing waves only. Solutions of the Helmholtz equation which satisfy the adiation condition ae called adiating solutions o adiating functions. In some wave poblems consideed in infinite domains all the wave souces and scattees can be enclosed inside some sphee. Since in the absence of the wave souces solution of the wave equation is tivial, φ (,t) 0, then all petubations fo points located outside the sphee come only fom some events inside the sphee. This means that in this case the total field in the fequency domain, ψ (, ω), is a adiating function. We emphasize that the adiation condition (23) deived fom consideation of point souces is applied to a set of souces, i.e. to the case ψ scat = ψ scat (,k). Geneally, the fa field asymptotics of ψ scat is ψ scat 1 Ψ (θ, ϕ) eik, (24) whee Ψ (θ, ϕ) is the angula dependence on spheical angles θ and ϕ, and so condition (23) holds. Indeed, fom a vey emote point a set of souces o scattees is seen as one point (like we see galaxies consisting of many stas as one sta ). While fo diffeent angles thee will be diffeent values of of ψ scat (so it is not spheically symmetical), fo a given, o fixed, angles θ and ϕ thee is no diffeence between the asymptotic behavio of a set of souces and an equivalent single souce. 2.2 Tansmission Conditions Real systems can be consideed as a unity of domains occupied by elatively homogeneous media. While the physical popeties of diffeent substances can diffe substantially (say ai and igid paticles) one should keep in mind that waves of diffeent natue can popagate in any substance (e.g. acoustic waves) and theefoe wave-type equations can be used fo thei desciption. Due to the diffeence in the popeties the speed of popagation of petubations is diffeent fo diffeent media. Theefoe fo desciptions of waves in each domain we can apply the wave equation with the speed of sound coesponding to the medium that occupies that domain. The poblem then is to povide sufficient conditions on the domain boundaies, that enable to match solutions in diffeent domains and build solutions fo the coesponding wave equation. These conditions ae known as tansmission conditions, which also can be intepeted as jump conditions o conditions fo discontinuities, since the wave function and/o its deivatives jump on the contact boundaies. In geneal, the jump conditions can be deived fom the same consevation equations that lead to the govening equations. The fom of these consevation laws should be witten in integal fom to allow discontinuities and then the conditions aise afte shinking the domain to the contact sufaces. In acoustics we usually conside poblems, when the boundaies of the domains ae eithe immovable o move with velocities much smalle than the speed of sound. We also conside the case when the amplitude of pessue petubations is small and petubations of the mass velocity ae small as well. In the linea appoximation, this esults in the following two conditions on a contact suface S with nomal n sepaating two media maked as 1 and 2: v1 0 n = v 0 2 n, p 0 1 = p 0 2. (25) The fist condition states that the nomal velocities to the suface ae the same. In fluid mechanics this is known as kinematic condition. In fact, it follows fom the mass consevation equation in assumption that thee is no mass tansfe though the suface S. The second condition, sometimes called dynamic conditions, follows fom the momentum consevation equation and is valid if thee ae no suface foces. As follows fom this desciption these conditions should be modified if mass is tansfeed though the suface (say due to phase tansitions), and if thee ae some appeciable suface foces (fo example, suface tension). These conditions ae sufficient to match solutions of the wave o Helmholtz equation in two domains. Depending on the poblem to be solved (wave equation fo pessue o fo the velocity potential) conditions (25) can be witten in tems of pessue o velocity potential and thei deivatives only. Conside fist the pessue equations. As follows fom the momentum consevation equation (2) witten in the fequency 5

6 domain, the phasos of pessue and velocity petubations, bp 0 and bv 0, satisfy equations iωρ 1 bv bp 0 1 = 0, iωρ 2 bv bp 0 2 = 0. (26) Taking the scala poduct of these equations with nomal n, denoting the nomal deivative / n = n, and noticing that elations (25) also holds in the fequency domain (emembe ou assumption that the speed of the bounday is much smalle than the speed of sound!), we obtain the following tansmission conditions fo pessue petubations applicable to matching solutions of the Helmholtz equation in domains 1 and 2: 1 bp 0 1 ρ 1 n = 1 bp 0 2 ρ 2 n, bp0 1 = bp 0 2. (27) Hee ρ 1 and ρ 2 ae the espective medium densities. Now conside the poblem fomulation fo the Helmholtz equation witten in tems of the velocity potential (10). The integal of momentum equation (expession in the paentheses in equation (10)) can be witten in the phaso space, whee we use notation ψ fo the phaso of φ (see (13)) as Hence, elations (25) lead to the following tansmission conditions: iωρ 1 ψ 1 = p 0 1, iωρ 2 ψ 2 = p 0 2. (28) 1 n = 2 n, ρ 1ψ 1 = ρ 2 ψ 2. (29) Compaing these conditions with elation (27) we can see that in case of pessue fomulation the function which satisfies Helmholtz equations in two diffeent domains is continuous, while its nomal deivatives have a discontinuity. The opposite situation, when the nomal deivative is continuous, while the wave function itself has a jump on the bounday is the case fo fomulation of the same acoustic poblem in tems of the velocity potential. Note that fo acoustic waves in complex media (dispesion, dissipation, elaxation) conditions (27) and (29) should be modified accoding to the model of the media. Pope tansmission conditions in this case can be obtained fom geneal mass and momentum consevation elations (25) and specific equations of state fo the medium, such as (??), witten in the fequency domain. 2.3 Conditions on the Boundaies Conditions on the boundaies of the domain 1 ae used when eithe the popeties of the bounday mateial (medium 2) ae vey diffeent fom the popeties of the medium 1 o can be modelled o assumed. In the fome case the tansmission conditions can be simplified and povide sufficient conditions fo solution of the Helmholtz equation. In the latte case simplification of geneal poblem usually follow fom consideation of some model poblem by applying the esults to moe geneal case. Since such modeling is out of scope of this book, we mention hee the following basic types of bounday conditions fo scala wave equation and Maxwell equations. Hee we assume that the domain of consideation is medium 1 (we also call it as host, caie, o just a medium with no index), and the mateial of the bounday has popeties of medium 2 (we will dop the indexing if it is clea fom the context). The nomal deivative eveywhee is taken inwad the domain of the caie medium (diection fom medium 2 to medium 1). The Diichlet bounday condition: ψ S =0. (30) This condition appeas, e.g. fo complex amplitude pessue in acoustics, when the mateial of the suface has vey low acoustic impedance compaed totheacousticimpedanceofthecaiemedium (ρ 2 c 2 ρ 1 c 1 ). In this case the suface is called sound soft. 6

7 The Neumann bounday condition: =0. (31) n S In acoustics this condition holds fo complex amplitude ofpessue,whenthesufacemateialhas much highe acoustic impedance than the acoustic impedance of the host medium (ρ 2 c 2 À ρ 1 c 1 ). In this case the suface is called sound had. The Robin (o mixed, o impedance) bounday condition: µ n + iσψ S =0. (32) In acoustics this condition is used to model finite acoustic impedance of the bounday. In this case σ is the admittance of the suface. Solutions of the Helmholtz equation with the Robin bounday condition in limiting cases σ 0 and σ tuns into solutions of the same equationwiththeneumannand Diichlet bounday conditions, espectively. The bounday value poblems with those conditions ae called the Diichlet, Neumann, and the Robin poblems, espectively. 3 Solutions of 1-D poblems Let us conside the one dimensional wave equation 7