PH2130 Problem Set 5 1. a. u(x,y, +)=x(x)Y(y)T(+) dx2 dc=T"xx d'u=y"Tx dc=x"Ty df2 . " X" YT + Y"XT = " T"XY = + xx x V 1-11 -( V2 T . => x"= >>X and Y"= x, y . Ax + xy = {/2=" => T " = \'T ( 1+ + 1}) b. 1x = 0; x = 0 (trivial) My =0; y =0 (trivial) Soly: (x) =A+++B2 + => x(0) = A + B : . A, B = 0 x(a) = A& Ha2+ Be-Ha= 0 .. A, B = 0 .. x= 0 (trivial ) and some result for by 1 .< o; Ax =- kx= = > x".kx2x=0 Soly: X(x) = de +++13=4xx => x(x) = Acosk2 x + Bsink, X X(0) = 0 ; x(a) = 0 .. A = 0, Brin (, x) = 0 KXKx=WII .A =- (Ic)' => Xx(x) = Bm sin (NICA) a ly soi la =- hj- = >y"they=0 both non-trivial dy :- (MIT)2 :1 /2(y) =(msin(wicy) C. Already written in (b) xx= - (2) whena box = H I dy : - (s )' where kg = we
d. w(x,y,+)=x(x) x(y)T(+) ; du = 0 at X(x) = Sin (UTTX) X(y )= sin (MIX ) T" = V2 (2x+14)T I" = - k - kg 2 TV2 (+)= xcos(2+kyiv+)+[sin(k="+kyiv+) T'(0) =0. 3=0 Ke = nTe Ky = mit . . Tum (+) = cos/Tv+ +m 2 let wym = TEV /M2 +m la 52 .: Tm(+) = ccs (want) => u (x, y, +) = sin (MTX) des / mTy ) ccs ( want ) 5 e. 14(r.y+) em (x,y,+ dxdy = CHEx6cm a Sin VTEX| Sin (ktex) = {9/2; K=u a 10 ; k&n ?. wily = 53/8 ; m=1 Jo sin - sin 10 ; mal Unm ( x, y, +) = cos (want) sin (IR) sin (MIX) 10 => [[" Ere Enn dedy = cs(+) cos (care +) 2 m &m : (+) =ab cos(want) cs(wat) + Il tve Com dady= CHO m 60
a= 1, b=2 f2.3 (x,y) = A sin/2TEX) s/n /3Rs) 5 +3,2(x,y) = Asin (3TLx) sin /2Ry Wam = C3 . (5)2 a=1, 6 === >(nm=(UTC)+(MIT)2 W213 = 4Th2+/3/25) = /4TC=+ 9 TL2 = 4 = 3/2TC 4 13 TEZ W3,2 = 1 9K=+TC= = JIOUT= = JTOTT .. W312 > W23 => w32 has higher pitch g. a =1, 5 =1 wam= => a=1, 5 =1 = TC Jom 2 + 42 w .. = (1.70) +(1.TC)= = > wAm = TC JT2+12 = VET Wil TI 52 mitnz 2