Project 10.5D Spherical Harmoic Waves I problems ivolvig regios that ejoy spherical symmetry about the origi i space, it is appropriate to use spherical coordiates. The 3-dimesioal Laplacia for a fuctio u( ρ, φ, θ ) expressed i spherical coordiates is give by u =! ρ + + 1 1 1 ρ ρ φ ρ siφ φ φ si φ θ u u u si " $ #. (1) Note that ρ = OP deotes the distace of the poit P from the origi O, φ is the agle dow from the vertical z-axis to OP, ad θ is the ordiary polar coordiate agle i the horizotal xy-plae (though some texts reverse the roles of φ ad θ). If u is idepedet of either ρ, φ, or θ the the correspodig secod-derivative term is missig 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 displacemet at time t of the poit ( φ, θ ) of the surface ρ = c of the plaet, the the wave equatio u a u takes the form tt = u t = b u () φθ where b = a/c ad φθu = + φ 1 u 1 u si. (3) siφ φ φ si φ θ Alteratively, Equatio () models 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 displacemet (from equilibrium) of the water surface at the poit ( φ, θ ) at time 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) T + b λ T = 0, (5) Y + λ Y = 0 (6) φθ Project 10.5D 313
where λ is the usual separatio costat. Next, show that the substitutio i (6) yields the equatios Y( φ, θ) = Φ( φ) Θ ( θ) (7) Θ + µ Θ = 0, (8) (si φ) Φ + (si φ cos φ) Φ + ( λ si φ µ ) = 0 (9) where µ is a secod separatio costat. I order that Θ( θ ) be periodic with period π, it follows from (8) that µ = m, the square of a o-egative iteger, i which case a typical solutio of (8) is Θ m (θ ) = cos mθ. (10) Now show that with µ = m x the substitutio = cos φ (11) i (9) yields the ordiary differetial equatio 7 Φ Φ = d 1 x x d m + λ Φ 0. (1) dx dx 1 x Note that if m = 0 the (1) is a Legedre equatio with depedet variable Φ ad idepedet variable x. Accordig to Sectio 8. i the text, this equatio has a solutio Φ( x ) that is cotiuous for 1 x 1 provided that the parameter λ = ( + 1 ) where is a o-egative iteger. I this case the cotiuous solutio Φ( x ) is a costat multiple of the th degree Legedre polyomial P ( x ). Equatio (1) is a associated Legedre equatio, ad it likewise has a solutio Φ( x ) that is cotiuous for 1 x 1 provided that the parameter λ = ( + 1 ) with beig a o-egative iteger. I this case the cotiuous solutio Φ( x ) is a costat multiple of the associated Legedre polyomial m m m/ ( m) P ( x) = ( 1) ( 1 x ) P ( x), (13) where the mth derivative of the ordiary Legedre polyomial P ( x) appears o the right. For istace, writig Pm for P m ( x ), Mathematica gives P00 = LegedreP[0,x] 1 314 Chapter 10
P01 = LegedreP[1,x] P11 = Sqrt[1-x^] D[LegedreP[1,x],x] x 1 x P0 = LegedreP[,x] P1 = -Sqrt[1-x^] D[LegedreP[,x],x] P = (1-x^) D[LegedreP[,x],{x,}] 3x 1 3x 1 x 31 ( x ) Actually, the associated Legedre fuctios are built ito Mathematica, with LegedreP[,m,x] deotig P m ( x). Usig Maple, you must first load the orthogoal polyomials package. Oly the ordiary Legedre fuctios are immediately available, so you must implemet the defiitio i (13). with(orthopoly): p4 := P(4,x); p4 := 35 8 x4 15 4 x + 3 8 p4 := expad((1-x^)*diff(p4,x$)); p4 := 60 x 15 105 x4 If x is a row vector with k elemets the the MATLAB commad legedre(,x) yields a ( + 1) k matrix whose mth row cotais the values of m 1 P at the elemets of x. Thus, the computatio legedre(3, 0:1/3:1) as = 0-0.4074-0.593 1.0000 1.5000 0.685-1.3665 0 0 4.4444 5.5556 0-15.0000-1.5708-6.113 0 shows that P 3 ( 1/ 3) 4.4444 ad P 3 ( / 3) 5. 5556. (It would be istructive for you to deduce from (13) the exact values 40 P 3 ( 1/ 3) ad 50 P 3 ( / 3). 9 9 Project 10.5D 315
At ay rate, give o-egative itegers m ad with m, substitutio of x = cos φ i the cotiuous solutio (13) of Eq. (1) with λ = ( +1) yields the solutio m m ( m) Φ m ( φ) = P (cos φ) = (si φ) P (cos φ) (14) of Eq. (9) with µ = m ad λ = ( +1). Substitutio of λ = ( +1) i (5) yields the typical solutio with frequecy Puttig it all together, we get fially the eigefuctio T () t = cosω t (15) ω = b ( + 1 ). (16) m u ( φθ,, t) = P (cos φ) cos mθcos ω t (17) m (0 m = 1,, 3,... ) of the wave equatio i (). The remaiig eigefuctios are obtaied by (idepedetly) replacig cos mθ with si mθ ad cos ω t with si ω t. Big Waves o a Small Plaet With all this preparatio, your task is to ivestigate graphically the way i which water waves slosh about o the surface of a small plaet. Let us take a sphere of radius c = 5 with b = 1 i (), ad (somewhat urealistically) cosider waves of amplitude h =. The followig MATLAB fuctio spharm(,m) costructs a φθ-grid o the surface of the sphere ad calculates the correspodig matrix Y of values of the surface spherical m harmoic fuctio Y ( φθ, ) = P (cos φ) cos mθ. m fuctio [Y,phi,theta] = spharm(,m) phi = 0 : pi/40 : pi; % co-latitude theta = 0 : pi/0 : *pi; % polar agle [theta,phi] = meshgrid(theta,phi); Theta = cos(m*theta); Phi = legedre(, cos(phi(:,1))); Phi = Phi(m + 1,:)'; pp = Phi; for k = : size(phi,1) Phi = [Phi pp]; ed; Y = Phi.*Theta; m = max(max(abs(y))); Y = Y/m; 316 Chapter 10
The fuctio spshape(,m) the displays the correspodig iitial graph ρ = c+ hy m ( φ, θ). The ext figure shows such plots for = 4, 5, 6, 7 ad m = 0, 1,, 3 (respectively) geerated by commads such as spshape(6,), which geerates the figure which appears (i livig color) o the cover of the text. fuctio spshape(,m) c = 5; h = ; [Y,phi,theta] = spharm(,m); rho = c + h*y; r = rho.*si(phi); x = r.*cos(theta); y = r.*si(theta); z = rho.*cos(phi); mesh(x,y,z) axis([-6 6-6 6-6 6]) axis square, axis off view(40,30) colormap([0 0 0]) m = 0, = 4 m = 1, = 5 m =, = 6 m = 3, = 7 Project 10.5D 317
The followig fuctio spmovie(,m) costructs ad shows a movie that displays i motio the surface water waves correspodig to oe of our spherical surface eigefuctios. For istace, the commad spmovie(6,) sets i motio the figure o the surface of the text. Warig: With k = 0 frames, the matrix Mslosh storig the movie may occupies from 5 to 15 megabytes of RAM (depedig upo the size of your figure widow). fuctio Mslosh = spmovie(,m) c = 5; h = ; w = sqrt(*(+1)); k = 0; % steps per cycle dt = *pi/(k*w); % (time) step size [Y,phi,theta] = spharm(,m); % Costruct the movie: Mslosh = moviei(k); for j = 0 : k-1 t = j*dt; rho = c + h*y*cos(w*t); r = rho.*si(phi); x = r.*cos(theta); y = r.*si(theta); z = rho.*cos(phi); surf(x,y,z) light lightig phog axis square, axis off axis([-6 6-6 6-6 6]) view(40,30) colormap(jet) Mslosh(:,j+1) = getframe; ed % Show it: movie(mslosh,5) Costruct some movies of your ow. If you're ambitious you ca ivestigate liear combiatios of differet spherical surface harmoics. For istace, the spherical wave motio with iitial positio fuctio 1 3 u( φ, θ, 0) = 5+ P (cos φ) cos θ + P (cos φ) cos 3θ. may remid you (at least vaguely) of a throbbig, beatig huma heart. 4 Maple- ad Mathematica-based ivestigatios correspodig to the MATLAB expositio here ca dowloaded from the DE computig projects web page at the site www.prehall.com/edwards. 6 318 Chapter 10