Question

all parts, please

For this problem, we'll solve the 3D wave equation in a box. The Laplacian in 3 dimensions is a2 vu= a2 a2 ou + ay? U= and the wave equation is 22 a2 at24 = 1 (a) (3 Points) Use separation of variables with ur,y,z,t) = S(x, y, z)T(t) to get a spatial PDE and a temporal ODE for this problem, call the separation constant A. Show all your work! (b) (3 Points) The spatial equation should now look like AS(x, y, z) = V2 S(r, y, z) Use separation of variables again with S(x, y, z) = X(2)Y(y)Z(z) to get X, Y, and Z ODEs for this problem. (c) (5 Points) Suppose we constrain ourselves to a box with 0<<a 0<y<b 0 <2<c and fix the solution to be zero on all the sides of this box. That is, X(0) = X(a) = 0 Y(0) = Y(b) = 0 Z(0) = 2(C) = 0 Find the general solution Umn/(x, y, z) = Xm (*)Y(Y)2(2) for this setup. (d) (2 Points) What are the allowed values of X? (e) (5 Points) Solve the temporal ODE you found in part a for the time solution Tmne(t). (f) (2 Points) Finally, combine all your results. What are the eigenfunc- tions Umner, y, z, t) for this problem?

Answer 1

ПА e-au ӘE 2 ети әл- Ч e 22. u = 22. h t ut и . 2y 22. (4) и (1 ч 2 €) 24 - S (ич 2) т/t) as д 24 — T se С. е ne T 1 че ид h hte he ( as ту T, T бе zhe пе PS Sul T те «а. T д ze 2.8. 82 - s d2 me S olT alt h? 1. 1 rete di 2 & Tie he Т. :: c- S. s.d? ва) 2)T di2 ze the 2d²I v- ( — Separalion const. / df a d27 aler r = sec s amd сѣ The spatial equation А5 - (x) z (6) 3 С) х = (zьк)ѕ Ч 2. Х. ді se Х Y Z 2 Вк - ху! Z he Z,,KXo 2 5 е, he 54, 24 е ful xyz' 2 XYZ" «te

11 Х T Х Y . x + * * *447140 IX = ( ( 1,7 // Z =8 = Z'=dZ :: X Y Z = XYZ + X Y Z + X Y Z Y" Z" Z Az = M (say, separation Y => M =) x"=Q-H) X =u Y =) Y" =(M-6) Y у X Contrat =) žk 11 y! Z t Z Y 2 ration Constal Z - S separation _^) Z y" -M 9-= 11 Z mo Z Z are Thus three ODES Z z" =dz y"=(M-6) Y *" --H) X

Z(O)= Z (C) =0 for nonzero solution, sso, Z (2) A Ces FS Z + B Sih Ff 2 Zlo o A FO zle) 20 Seh rage 20 J-8 n=1, 2, na -f= nale z (2) - Bn Sin (2) n=1,2,- Ylo) y (6=0 Mrd To ie d-myo for non zero solution Y c Ces room yt D Sin room y. Y./9 =) c=0 Sin room b =0 Ylb) 121,2-- J S-M b - nn rom nr nr d-M = n32 br 6 Y CY) = Do Sen (27 y nal, 2, x0=0 -X ( solution For nouzeo x-m.co.ie Mod yo x (2) = E les ruryx & F Sin Punya X (o E ,لا في y ca) so son rmaga=o Thay a Enh

» Turda nn a X (n) = En Sen en na1,2,.. 11, Ze(z) = Be sen (2) Yo (3) = Dn Sen they ņ=1,2,- Xm(") 11 Son Man ) ma1,2,- $ (442) = GSen (wan) Smary) Sim (eredze ) where Gmine a Conszent 2 nn M-) = in 22 92 d). 2 -S 2 Mod= = b2, 2 CL. n2n2 en2 2) u 62 ca 5 lit? m2 a a2 The temporal ral ODE. Т. = X T. aka ca 72 m2 n2 d27 afa M С. 62 x TCE = H cos ( A m2 12 t 727 22 tk Sin (「芸芸感り : 'Tmnelt) = Homne cos (17 m 2 + 1 2 + z 2 t) + k mne +) + Kimne Sin / m2 92722 62 the H

. The Combine all S and T. we get u (ny, 24 = El Hom Cos/17 12 t = ) anne 12 92 mni f mni 22 62 c2 + Kime Same 52 xez 92 X 92 62 Sin Main). sin chay Singers The eigen functions are tt kmne Sin/eim som reg te vay Cos I m2 (2 mnl C2 5 92732 finanz Ĉ sin mnn a Sin niny l, m, n Eh!? O +