Deivation of the Geen s Funtions fo the Helmholtz and Wave Equations Alexande Miles Witten: Deembe 19th, 211 Last Edited: Deembe 19, 211 1 3D Helmholtz Equation A Geen s Funtion fo the 3D Helmholtz equation must satisfy 2 G, + k 2 G, δ, By Fouie tansfoming both sides of this equation, we an show that we may take the Geen s funtion to have the fom G, g and that Fist we take the Fouie tansfom of both sides: sin2ρ g 4π k 2 4π 2 ρ2 dρ F 2 G, + k 2 G, 1 2πi Gρ, ρ + k 2 Gρ, ρ 2 Gρ, ρ [ 2πi + k 2] e 2πiρ 3 G, Gρ, ρ π 2π G, 2π π e 2πiρ 2πi + k 2 4 e 2πiρ 2πi + k 2 e2πiρ os θ sin θdρdθdφ 5 2πi ρ os θ e k 2 4π 2 sin θdρdθ 6 u os θ du sin θdθ 7 1 e 2πi ρu G, 2π 1 k 2 4π 2 ρ2 dρdu 8 1 G, 2π k 2 4π 2 e 2πi ρu dudρ 1 9 G, 2π k 2 4π 2 G, 2π [ ] 1 2πi ρ e 2πi ρ e 2πi ρ dρ 1 [ ] 2i sin 2π ρ k 2 4π 2 dρ 2πi ρ 11 1
G, 2π G, 4π This means that little g has the expeted fom: g 4π k 2 4π 2 [ sin 2π ρ π ρ ] dρ 12 k 2 4π 2 sin 2π ρdρ 13 sin2ρ k 2 4π 2 ρ2 dρ 14 Howeve, this integal passes though a singulaity of the integand. If we use the Cauhy pinipal value to deal with this poblem, we use ontou integation we find that the solution is popotional to a spheial wave fom a monohomati point soue. [ ] e ik Re g 4π sin2ρ k 2 4π 2 ρ2 dρ 4π g 2 sin2πρ 2πρk 2 4π 2 ρ2 dρ 15 sin2πρ k 2 4π 2 ρdρ 16 We eognize that sinx is an even funtion, so we an get the same esult by integating ove infinite limits and halving the esult. g { g Im sin2πρ k 2 4π 2 ρdρ 17 } k 2 4π 2 ρdρ Now let us fous on solving the integal, then we will take the imaginay pat at the end. We eognize two poles at ρ ± k 2π. We must levely e-phase the denominato in ode to lealy anel with ou additional fato. Res k/2π ρ k 2π 4π 2 ρ k 2π ρ + k 2π ρ ρk/2π 4π 2 ρ + k e ik 4π 2 k e ik k k 2π e ik 8π 2 2π + k 2π ρ ρk/2π k 2π 2π 18 19 2
Res k/2π ρ + k 2π 4π 2 ρ k 4π 2 k k 2π e ik k e ik 8π 2 4π 2 ρ k 2π ρ + k 2π ρ ρ k/2π e ik 2π k k 2π 2π 2π ρ ρ k/2π Now we an utilize the esidues, eognizing that sine both lie on the line of integation, they have half the influene: 2 k 2 4π 2 ρdρ 2πi Residue i i 21 1 e ik Residue i 2πi 2 8π 2 + 1 e ik e ik 2 8π 2 i + e ik 22 Residue i i e ik + e ik i osk + i sink + os k + i sin k 23 Residue i i osk + i sink + osk i sink 24 Residue i i i osk osk + osk 25 Finally, we take the imaginay pat in ode to solve ou oiginal integal: g 4π sin2ρ k 2 4π 2 ρ2 dρ osk 26 g osk If we go ahead and plot the esult of ou opeato ating upon this funtion, we do indeed find a delta funtion! 2 3D Wave Equation A Geen s funtion fo the 3D wave equation must satisfy 2 G, t,, t 1 2 2 t 2 G, t,, t δ, δt t We utilize the spae-time Fouie tansfom, defined as: F ρ, ω f, te 2πi ρ ωt dtd 3 R 3 27 3
Using this Fouie tansfom we an show that we may assume the Geens funtion has the fom: G, t,, t g, t t Instead of pefoming the whole Fouie tansfom at one, we instead pefom just the time-dependent tansfom fist. F t 2 G, t,, t 1 2 2 t 2 G, t,, t δ, e 2πitω 28 2 G, t,, t 1 2 4π2 t 2 G, t,, t δ, e 2πitω 29 2 + 4π2 ω 2 2 G, t,, t δ, e 2πitω 3 If we make the substitution k 2πω we see that the wave equation is vey simila to the Helmholtz equation we woked with in the pevious setion. 2 + k 2 G, t,, t δ, e 2πit tω 31 Beause of this similaity, we an utilize the solution all the way up to the point whee we ae invesetansfoming though the ω vaiable in the next potion of the question. We an use ontou integation to show that g, t is a linea ombination of an inoming wave and an outgoing wave δ + t δ t sin2ρ g 4π k 2 4π 2 ρ2 e 2πitω dρdω 32 g g osk e 2πitω dω 33 osk e 2πitω dω 34 g 1 1 e ik + e ik e 2πitω dω 35 2 4
g 1 e ik e 2πitω dω + 8π g 1 e 2πiω e 2πitω dω + 8π g 1 e 2πiω +t dω + 8π g 1 G 1 [ δ [ δ + t + δ t e ik e 2πitω dω e 2πiω e 2πitω dω e 2πiω t dω ] ] + t t + δ t t To get this to esemble the fom alluded to in the question, we must multiply by a fom of one, and following the saling ules fo delta funtions. G [δ + t t + δ t t ] 41 These ae alled inoming and outgoing waves beause the loation of the delta funtion eithe advanes outwad fom the oigin with time oesponding to the t agument o shinks in towad the oigin as time gows oesponding to the + t agument. Also a delta funtion of the adial oodinate looks like a shell, so the name fits quite well. 36 37 38 39 4 5