solutions_exam2

# solutions_exam2 - Physics 604 Midterm#2 Fall ‘11 Dr Drake...

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Unformatted text preview: Physics 604 Midterm #2 Fall ‘11 Dr. Drake 1. (35 points) The Coulomb Wave equation is given by (£210 ppm—2mm =0 (1) where n is real and p > 0. This equation describes a wave propagating towards a turning point and an adjacent singularity. (a) Classify the singular points of this equation (don’t worry about inﬁnity). (1)) Find the behavior of the solutions for ,0 very close to zero. Show that the solutions are linearly independent. (c) Plot the square of the WKB wavevector k2(p) versus p. Calculate the behavior of the solutions for )9 very iarge. 2. (35 points) A wave in a one-dimensional medium is driven by a 5 — function source at :r = 0 that is turned on at t = O as follows: __. 2 273 — 0238—3322;. = Here) (2) where y x 0 for t < 0 and H (t) is zero for t < O and one for t > 0. Solve for the space/time dependence of y(\$, t) by completing Laplace and Fourier transforms of the equation and then completing the subsequent inverse Laplace transform. Finally complete the inverse Fourier transform to obtain y(:r, t). Plot the spatial dependence of y at a ﬁxed time. Interpret the result on the basis of causality. Hint: To reduce the required algebra evaluate y(m, t) explicitly only :1; > O and use the symmetry of the problem to sketch the solution for a: < 0. 3. (30 points) An integral representation for the Couiomb wave equation in (1) can be written as we) : f0 simmers (3) (a) Derive a differential equation for Y(k) by inserting (3) into (1). Explicitiy give the conditions on the contour C in order that the integral solution satisfy (1). Show that the solution of your equation for YUc) is given by YUC) = (1 + klilimﬂ — WW”- (4) (b) Choose two contours in the complex k plane which satisfy the constraint given in part (a) and therefore give two linearly independent solutions to Draw them. Hint: You must deﬁne cuts for the singularities of YUC) at k = ii. The contours wrap around the cuts. Where do they end? Recall that p > 0. Had-116mm #2 Sv/wams é) % @ €W€€+C€’-l”?)uﬂ:0 4) anjué'w/Qoivﬂc (L74 6 1"— O raga/at" €/S‘r&€~»€ ‘ J €2Wee + {62—21%} w = 0 W ‘ ahalx/‘Hg 47L {22—0 1:1) Raf. {mg/>71 47‘ 6:0 3 Wk Siv/io'ﬁfam //£ﬂj€ 6 *5: € —;’Z :C‘ {478’ ‘6!) (U M e: i :ere \$67M?) 2: a 1:56 2 e: 164’: >3 __ any” __ f E. gm {:6 guy???“ ~—w c3 :15 140 (46:47?“ 5%71/5’976 1455’: (ab/15K Pond? QMC-é y :0 42% X: 1%? mm/ (if 24:0 w , m c1075 SM €05] QM C ‘W 0 ("gaff ‘f’C‘ f a: e { 1 -___., (it’d o [W ﬂﬂff me>@ 50 5.546%?qu WA f: W léT h v - JR ( 3’“ £7ch L— EW w 1%ch “*“ ’ if “’2‘ _L [16,75 :3; .5— .ﬂchie W :LU { , L“ c: 2‘ @“ﬁxww’ﬁcj 54\$ 7% we , é “4&4 {W r“7L/€g' 1:06“ 7925-9 6/6956 maﬁm l”; 5/793 ,4 {SQ/rdCéwC/S LZVVLMG. (67n%m[,éM/7éaq 74449;? Sfmf'ﬁ «fwd/é /S Q'fma, bAKSm/orrf Q0544 %6/Jd‘t [Ow/ZQ“, (,1 [KCT -» 754+ '~— .% K _, ,3 r C r I g E: ./ + __ , lug, +- _{.— £40746 ﬂew/7‘ mtg gthéi—c/étwl7z/ ﬂ 1’), 2.. \$30 Sic/16\$ (fig k5?" .V [—- iéf 2, . t L‘éjf : k (X'C‘g) { :2 Jr _ ﬂﬂ/ ( J— _ J 7/ :2chng E1 @“26 *ib C; H; I“... I] 4» EL 4’ [omggcgé’w 5V1 X 7?? 7:5 :5 +3ch 1EKVLC7LIO¢4§ 64X :5 6/056 m ai+® mo (fwd/555% {DO/Cg 5k x—¢% I ‘2; w #26:”; if, ( > {I 3’ 1:1 153‘ W :53 Z: L E: @1143 - .C‘L">g 34:12 1C2” jEL;: <3 xxgﬁﬁ>cwf 2C)— J 2:; i __l__ ‘ L Mia-H27“ “BE 21.)- Slcﬁ: #:2’ a a «gem a“ K56 (Jag: M diff}; 21% M0746 ﬂaﬂ’ 7:20 ‘gbw X >67.“ :5 wave fﬂyatjcﬂﬁ‘fg V’jﬂ’aclé’ C; 36 me (AWN/f3 défyOMC/e (x/ = as 6} ’7‘: 2‘ _ L? [-K 5‘ iii (h V :1 “L? {(L‘ H: “V? - Q‘K ) 3 ( [4—K “I? y (wk; l--KX (hwy/V4sz .4 7:6 are? / 2 ...
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## This note was uploaded on 12/30/2011 for the course PHYSICS 604 taught by Professor Drake during the Fall '11 term at Maryland.

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solutions_exam2 - Physics 604 Midterm#2 Fall ‘11 Dr Drake...

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