Application 10.5D Spherical Harmonic Waves

Transcription

1 Applicatio 10.5D Spherical Haroic Waves I probles ivolvig regios that ejoy spherical syetry about the origi i space, it is appropriate to use spherical coordiates. The 3-diesioal Laplacia for a fuctio u( ρφθ,, ) epressed i spherical coordiates is give by 1 u 1 u 1 u u = ρ siφ ρ ρ ρ + siφ φ + φ. (1) si φ θ Note that ρ = OP deotes the distace of the poit P fro the origi O, φ is the agle dow fro the vertical z-ais to OP, ad θ is the ordiary polar coordiate agle i the horizotal y-plae (though soe tets reverse the roles of φ ad θ). If u is idepedet of either ρ, φ, or θ the the correspodig secod-derivative ter is issig o the right-had side i (1). For istace, cosider radial vibratios of the surface of a spherical plaet of radius c. If u( φ, θ, t) deotes the radial displaceet at tie t of the poit ( φ, θ ) of the surface ρ = c of the plaet, the the wave equatio u = a u takes the for tt where b = a/c ad u = b u φθ () t u 1 u 1 u φθ = siφ siφ φ φ +. (3) si φ θ Alteratively, Equatio () odels the oscillatios of tidal waves o the surface of a water-covered spherical plaet with radius c. I this case, u( φ, θ, t) deotes the vertical displaceet (fro equilibriu) of the water surface at the poit ( φ, θ ) at tie t ad b = gh/c, where h is the average depth of the water ad g deotes gravitatioal acceleratio at the surface of the plaet. To solve equatio () by separatio of variables, show first that the substitutio yields the equatios u( φ, θ, t) = Y( φ, θ ) T( t) (4) + = 0, (5) T b λt Y + λ Y = 0 (6) φθ Applicatio 10.5D 315

2 where λ is the usual separatio costat. Net, show that the substitutio i (6) yields the equatios Y ( φ, θ ) =Φ( φ) Θ ( θ ) (7) Θ + µ Θ = 0, (8) Φ + Φ + = (9) (si φ) (si φ cos φ) ( λ si φ µ ) 0 where µ is a secod separatio costat. I order that Θ ( θ ) be periodic with period π, it follows fro (8) that the square of a o-egative iteger, i which case a typical solutio of (8) is µ =, Θ ( θ) = cosθ. (10) Now show that with µ = the substitutio = cos φ (11) i (9) yields the ordiary differetial equatio 1 d Φ d Φ + λ 0 d d Φ = 1. (1) ( ) Note that if = 0 the (1) is a Legedre equatio with depedet variable Φ ad idepedet variable. Accordig to Sectio 8. i the tet, this equatio has a solutio Φ ( ) that is cotiuous for 1 1 provided that the paraeter λ = ( + 1) where is a o-egative iteger. I this case the cotiuous solutio Φ ( ) is a costat ultiple of the th degree Legedre polyoial P ( ). Equatio (1) is a associated Legedre equatio, ad it likewise has a solutio Φ ( ) that is cotiuous for 1 1 provided that the paraeter λ = ( + 1) with beig a o-egative iteger. I this case the cotiuous solutio Φ ( ) is a costat ultiple of the associated Legedre polyoial P = P, (13) / ( ) ( ) ( 1) (1 ) ( ) where the th derivative of the ordiary Legedre polyoial P ( ) appears o the right. For istace, writig P for P ( ), Matheatica gives 316 Chapter 10

3 P00 = LegedreP[0,] 1 P01 = LegedreP[1,] P11 = Sqrt[1-^] D[LegedreP[1,],] 1 P0 = LegedreP[,] P1 = -Sqrt[1-^] D[LegedreP[,],] P = (1-^) D[LegedreP[,],{,}] (1 ) Actually, the associated Legedre fuctios are built ito Matheatica, with LegedreP[,,] deotig P ( ). Usig Maple, you ust first load the orthogoal polyoials package. Oly the ordiary Legedre fuctios are iediately available, so you ust ipleet the defiitio i (13). with(orthopoly): p4 := P(4,); p 4 4 : = p4 := epad((1-^)*diff(p4,$)); p4 : = 60 4 If is a row vector with k eleets the the MATLAB coad legedre(,) yields a ( + 1) k atri whose th row cotais the values of 1 P at the eleets of. Thus, the coputatio legedre(3, 0:1/3:1) as = Applicatio 10.5D 317

4 shows that P 3 (1/ 3) ad P3 ( / 3) (It would be istructive for you to deduce fro (13) the eact values P (1/ 3) ad P (/3) At ay rate, give o-egative itegers ad with, substitutio of = cos φ i the cotiuous solutio (13) of Eq. (1) with λ = ( + 1) yields the solutio ( ) Φ ( φ) = P (cos φ) = (si φ) P (cos φ) (14) of Eq. (9) with µ = ad λ = ( + 1). Substitutio of λ = ( + 1) i (5) yields the typical solutio T( t) = cosωt (15) with frequecy ω = b ( + 1). (16) Puttig it all together, we get fially the eigefuctio u ( φ, θ, t) = P (cos φ)cosθ cosω t (17) (0 = 1,, 3,... ) of the wave equatio i (). The reaiig eigefuctios are obtaied by (idepedetly) replacig cos θ with si θ ad cos ω t with si ω. t Big Waves o a Sall Plaet With all this preparatio, your task is to ivestigate graphically the way i which water waves slosh about o the surface of a sall plaet. Let us take a sphere of radius c = 5 with b = 1 i (), ad (soewhat urealistically) cosider waves of aplitude h =. The followig MATLAB fuctio sphar(,) costructs a φθ-grid o the surface of the sphere ad calculates the correspodig atri Y of values of the surface spherical haroic fuctio Y ( φ, θ ) = P (cos φ) cos θ. fuctio [Y,phi,theta] = sphar(,) phi = 0 : pi/40 : pi; % co-latitude theta = 0 : pi/0 : *pi; % polar agle [theta,phi] = eshgrid(theta,phi); Theta = cos(*theta); Phi = legedre(, cos(phi(:,1))); Phi = Phi( + 1,:)'; pp = Phi; for k = : size(phi,1) Phi = [Phi pp]; ed; 318 Chapter 10

5 Y = Phi.*Theta; = a(a(abs(y))); Y = Y/; The fuctio spshape(,) the displays the correspodig iitial graph ρ = c+ hy ( φ, θ ). The et figure shows such plots for = 4, 5, 6, 7 ad = 0, 1,, 3 (respectively) geerated by coads such as spshape(6,), which geerates the figure which appears (i livig color) o the cover of the tet. fuctio spshape(,) c = 5; h = ; [Y,phi,theta] = sphar(,); rho = c + h*y; r = rho.*si(phi); = r.*cos(theta); y = r.*si(theta); z = rho.*cos(phi); esh(,y,z) ais([ ]) ais square, ais off view(40,30) colorap([0 0 0]) Applicatio 10.5D 319

6 The followig fuctio spovie(,) costructs ad shows a ovie that displays i otio the surface water waves correspodig to oe of our spherical surface eigefuctios. For istace, the coad spovie(6,) sets i otio the figure o the surface of the tet. Warig: With k = 0 fraes, the atri Mslosh storig the ovie ay occupies fro 5 to 15 egabytes of RAM (depedig upo the size of your figure widow). fuctio Mslosh = spovie(,) c = 5; h = ; w = sqrt(*(+1)); k = 0; % steps per cycle dt = *pi/(k*w); % (tie) step size [Y,phi,theta] = sphar(,); % Costruct the ovie: Mslosh = oviei(k); for j = 0 : k-1 t = j*dt; rho = c + h*y*cos(w*t); r = rho.*si(phi); = r.*cos(theta); y = r.*si(theta); z = rho.*cos(phi); surf(,y,z) light lightig phog ais square, ais off ais([ ]) view(40,30) colorap(jet) Mslosh(:,j+1) = getfrae; ed % Show it: ovie(mslosh,5) Costruct soe ovies of your ow. If you're abitious you ca ivestigate liear cobiatios of differet spherical surface haroics. For istace, the spherical wave otio with iitial positio fuctio u( φ, θ,0) = 5 + P (cos φ) cos θ + P (cos φ) cos 3θ ay reid you (at least vaguely) of a throbbig, beatig hua heart. Maple- ad Matheatica-based ivestigatios correspodig to the MATLAB epositio here ca dowloaded fro the DE coputig projects web page at the site 30 Chapter 10