P R O B L E M S E T

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1 52 CHAP. 2 Prtil Differentil Equtions (PDEs) The simple proof of this importnt theorem is quite similr to tht of Theorem in Sec. 2. nd is left to the student. Verifiction of solutions in Probs. 2 3 proceeds s for ODEs. Problems 6 23 concern PDEs solvble like ODEs. To help the student with them, we consider two typicl exmples. E X A M P L E 2 Solving u xx u Like n ODE Find solutions u of the PDE u xx u depending on x nd y. Solution. Since no y-derivtives occur, we cn solve this PDE like us u. In Sec. 2.2 we would hve obtined u Ae x Be x with constnt A nd B. Here A nd B my be functions of y, so tht the nswer is with rbitrry functions A nd B. We thus hve gret vriety of solutions. Check the result by differentition. E X A M P L E 3 Solving u xy u x Like n ODE Find solutions u u (x, y) of this PDE. u(x, y) A( y)e x B( y)e x Solution. Setting u we hve p y p, p y >p, ln ƒ p ƒ y c(x), p c (x)e y x p, nd by integrtion with respect to x, u (x, y) f (x)e y g (y) where f (x) c (x) dx, here, f (x) nd g( y) re rbitrry. P R O B L E M S E T 2.. Fundmentl theorem. Prove it for second-order PDEs in two nd three independent vribles. Hint. Prove it by substitution. 2 3 VERIFICATION OF SOLUTIONS Verifiy (by substitution) tht the given function is solution of the PDE. Sketch or grph the solution s surfce in spce. 2 5 Wve Eqution () with suitble c Het Eqution (2) with suitble c Lplce Eqution (3).. u x 2 t 2 u cos t sin 2x u sin kct cos kx u sin t sin bx u e t sin x u e v2 c 2t cos vx u e 9t sin vx u e p2t cos 25x u e x cos y, e x sin y u rctn ( y>x) 2. u cos y sinh x, sin y cosh x 3. u x>(x 2 y 2 ), y>(x 2 y 2 ). TEAM PROJECT. Verifiction of Solutions () Wve eqution. Verify tht u (x, t) v (x ct) w (x ct) with ny twice differentible functions v nd w stisfies (). (b) Poisson eqution. Verify tht ech u stisfies () with f (x, y) s indicted. (c) u y>x f 2y>x 3 u sin xy f (x 2 y 2 ) sin xy u e x2 y 2 f (x 2 y 2 )e x 2 y 2 u > 2x 2 y 2 Lplce eqution. Verify tht u > 2x 2 y 2 z 2 stisfies (6) nd u ln (x 2 y 2 ) stisfies (3). Is u > 2x 2 y 2 solution of (3)? Of wht Poisson eqution? (d) Verify tht u with ny (sufficiently often differentible) v nd w stisfies the given PDE. u v (x) w (y) f (x 2 y 2 ) 3>2 u xy u v (x)w(y) uu xy u x u y u v (x 2t) w (x 2t) u tt u xx 5. Boundry vlue problem. Verify tht the function u (x, y) ln (x 2 y 2 ) b stisfies Lplce s eqution

2 SEC. 2.2 Modeling: Vibrting String, Wve Eqution 53 (3) nd determine nd b so tht u stisfies the boundry conditions u on the circle x 2 y 2 nd u on the circle x 2 y PDEs SOLVABLE AS ODEs This hppens if PDE involves derivtives with respect to one vrible only (or cn be trnsformed to such form), so tht the other vrible(s) cn be treted s prmeter(s). Solve for u u (x, y): 6. u 7. u xx 6p 2 yy u 8. 25u yy u u y y 2 u 2u xx 9u x u 3 cos x 29 sin x u yy 6u y 3u e 3y 22. u 23. x 2 xy u x u xx 2xu x 2u 2. Surfce of revolution. Show tht the solutions z z (x, y) of yz x xz y represent surfces of revolution. Give exmples. Hint. Use polr coordintes r, u nd show tht the eqution becomes z u. 25. System of PDEs. Solve u xx, u yy 2.2 Modeling: Vibrting String, Wve Eqution In this section we model vibrting string, which will led to our first importnt PDE, tht is, eqution (3) which will then be solved in Sec The student should py very close ttention to this delicte modeling process nd detiled derivtion strting from scrtch, s the skills lerned cn be pplied to modeling other phenomen in generl nd in prticulr to modeling vibrting membrne (Sec. 2.7). We wnt to derive the PDE modeling smll trnsverse vibrtions of n elstic string, such s violin string. We plce the string long the x-xis, stretch it to length L, nd fsten it t the ends x nd x L. We then distort the string, nd t some instnt, cll it t, we relese it nd llow it to vibrte. The problem is to determine the vibrtions of the string, tht is, to find its deflection u (x, t) t ny point x nd t ny time t ; see Fig u (x, t) will be the solution of PDE tht is the model of our physicl system to be derived. This PDE should not be too complicted, so tht we cn solve it. Resonble simplifying ssumptions (just s for ODEs modeling vibrtions in Chp. 2) re s follows. Physicl Assumptions. The mss of the string per unit length is constnt ( homogeneous string ). The string is perfectly elstic nd does not offer ny resistnce to bending. 2. The tension cused by stretching the string before fstening it t the ends is so lrge tht the ction of the grvittionl force on the string (trying to pull the string down little) cn be neglected. 3. The string performs smll trnsverse motions in verticl plne; tht is, every prticle of the string moves strictly verticlly nd so tht the deflection nd the slope t every point of the string lwys remin smll in bsolute vlue. Under these ssumptions we my expect solutions u (x, t) relity sufficiently well. tht describe the physicl u T α P Q β T 2 α P Q T 2 β T x x + Δx L Fig Deflected string t fixed time t. Explntion on p. 5

3 SEC. 2.3 Solution by Seprting Vribles. Use of Fourier Series 55 For grphing the solution we my use u (x, ) f (x) nd the bove interprettion of the two functions in the representtion (7). This leds to the grph shown in Fig. 29. f*(x) 2 u(x, ) t = L L L f *(x + ) L f *(x ) t = L/5c 2L f *(x + ) 2L f *(x ) t = 2L/5c L f *(x + ) 2 2 L f *(x ) 2 2 t = L/2c 3L f *(x + ) 3L f *(x ) t = 3L/5c L f *(x + ) L f *(x ) t = L/5c 2 f*(x L) = f*(x + L) 2 t = L/c Fig. 29. Solution u(x, t) in Exmple for vrious vlues of t (right prt of the figure) obtined s the superposition of wve trveling to the right (dshed) nd wve trveling to the left (left prt of the figure) P R O B L E M S E T Frequency. How does the frequency of the fundmentl mode of the vibrting string depend on the length of the string? On the mss per unit length? Wht hppens if we double the tension? Why is contrbss lrger thn violin? 2. Physicl Assumptions. How would the motion of the string chnge if Assumption 3 were violted? Assumption 2? The second prt of Assumption? The first prt? Do we relly need ll these ssumptions? 3. String of length p. Write down the derivtion in this section for length L p, to see the very substntil simplifiction of formuls in this cse tht my show ides more clerly.. CAS PROJECT. Grphing Norml Modes. Write progrm for grphing u with nd c 2 n L p of your choice similrly s in Fig Apply the progrm to u 2, u 3, u. Also grph these solutions s surfces over the xt-plne. Explin the connection between these two kinds of grphs. 5 3 DEFLECTION OF THE STRING Find u (x, t) for the string of length L nd c 2 when the initil velocity is zero nd the initil deflection with smll k (sy,.) is s follows. Sketch or grph u (x, t) s in Fig. 29 in the text. 5. k sin 3px 6. k (sin px 2 sin 2px)

4 552 CHAP. 2 Prtil Differentil Equtions (PDEs) 7. kx ( x) 8. kx 2 ( x) y-xis in the figure, r density, A cross-sectionl 9. re). (Bending of bem under lod is discussed in Sec. 3.3.) Substituting u F (x)g ## (t) into (2), show tht F () >F G>c 2 G b const, x = F (x) A cos bx B sin bx C cosh bx D sinh bx, G (t) cos cb 2 t b sin cb 2 t. x = L x (A) Simply supported x = x = L (B) Clmped t both ends 3. 2x x 2 if x 2, if 2 x 3. Nonzero initil velocity. Find the deflection u(x, t) of the string of length L p nd c 2 for zero initil displcement nd tringulr initil velocity u t (x, ).x if x u t (x, ). (p x) if p 2 p, 2 x p. (Initil conditions with u t (x, ) re hrd to relize experimentlly.) x = Fig x = L Supports of bem (C) Clmped t the left end, free t the right end 6. Simply supported bem in Fig. 293A. Find solutions u n F n (x)g n (t) of (2) corresponding to zero initil velocity nd stisfying the boundry conditions (see Fig. 293A) x = L x u (, t), u (L, t) (ends simply supported for ll times t), u xx (, t), u xx (L, t) (zero moments, hence zero curvture, t the ends). u y Fig Elstic bem 7. Find the solution of (2) tht stisfies the conditions in Prob. 6 s well s the initil condition u (x, ) f (x) x (L x). 5 2 SEPARATION OF A FOURTH-ORDER PDE. VIBRATING BEAM By the principles used in modeling the string it cn be shown tht smll free verticl vibrtions of uniform elstic bem (Fig. 292) re modeled by the fourth-order PDE 2 u (2) (Ref. [C]) t 2 c2 x where c 2 EI>rA ( E Young s modulus of elsticity, I moment of interti of the cross section with respect to the u 8. Compre the results of Probs. 7 nd 7. Wht is the bsic difference between the frequencies of the norml modes of the vibrting string nd the vibrting bem? 9. Clmped bem in Fig. 293B. Wht re the boundry conditions for the clmped bem in Fig. 293B? Show tht F in Prob. 5 stisfies these conditions if bl is solution of the eqution (22) cosh bl cos bl. Determine pproximte solutions of (22), for instnce, grphiclly from the intersections of the curves of cos bl nd >cosh bl.

5 SEC. 2. D Alembert s Solution of the Wve Eqution. Chrcteristics Clmped-free bem in Fig. 293C. If the bem is clmped t the left nd free t the right (Fig. 293C), the boundry conditions re u (, t), u x (, t), u xx (L, t), u xxx (L, t). Show tht F in Prob. 5 stisfies these conditions if is solution of the eqution (23) cosh bl cos bl. Find pproximte solutions of (23). bl 2. D Alembert s Solution of the Wve Eqution. Chrcteristics It is interesting tht the solution (7), Sec. 2.3, of the wve eqution () 2 u 2 u t 2 c2 x, 2 c 2 T r, cn be immeditely obtined by trnsforming () in suitble wy, nmely, by introducing the new independent vribles (2) v x ct, w x ct. Then u becomes function of v nd w. The derivtives in () cn now be expressed in terms of derivtives with respect to v nd w by the use of the chin rule in Sec Denoting prtil derivtives by subscripts, we see from (2) tht v x nd w x. For simplicity let us denote u (x, t), s function of v nd w, by the sme letter u. Then u x u v v x u w w x u v u w. We now pply the chin rule to the right side of this eqution. We ssume tht ll the prtil derivtives involved re continuous, so tht u wv u vw. Since v x nd w x, we obtin u xx (u v u w ) x (u v u w ) v v x (u v u w ) w w x u vv 2u vw u ww. Trnsforming the other derivtive in () by the sme procedure, we find u tt c 2 (u vv 2u vw u ww ). By inserting these two results in () we get (see footnote 2 in App. A3.2) (3) u vw 2 u w v. The point of the present method is tht (3) cn be redily solved by two successive integrtions, first with respect to w nd then with respect to v. This gives u v h (v) nd u h (v) dv c (w).

6 556 CHAP. 2 Prtil Differentil Equtions (PDEs) Type New Vribles Norml Form Hyperbolic v w u vw F Prbolic v x w u ww F 2 Elliptic v w ( ) u vv u ww F 3 2 ( ) 2i Here, (x, y), (x, y), F F (v, w, u, u v, u w ), etc., nd we denote u s function of v, w gin by u, for simplicity. We see tht the norml form of hyperbolic PDE is s in d Alembert s solution. In the prbolic cse we get just one fmily of solutions. In the elliptic cse, i, nd the chrcteristics re complex nd re of minor interest. For derivtion, see Ref. [GenRef3] in App.. E X A M P L E D Alembert s Solution Obtined Systemticlly The theory of chrcteristics gives d Alembert s solution in systemtic fshion. To see this, we write the wve eqution u in the form () by setting By the chin rule, nd u tt c 2 tt c 2 u xx y ct. u t u y y t cu y u yy. Division by c 2 gives u s stted before. Hence the chrcteristic eqution is yr 2 xx u yy, (yr ) (yr ). The two fmilies of solutions (chrcteristics) re (x, y) y x const nd (x, y) y x const. This gives the new vribles v y x ct x nd w y x ct x nd d Alembert s solution u f (x ct) f 2 (x ct). P R O B L E M S E T 2.. Show tht c is the speed of ech of the two wves given by (). 2. Show tht, becuse of the boundry conditions (2), Sec. 2.3, the function f in (3) of this section must be odd nd of period 2L. 3. If steel wire 2 m in length weighs.9 nt (bout.2 lb) nd is stretched by tensile force of 3 nt (bout 67. lb), wht is the corresponding speed of trnsverse wves?. Wht re the frequencies of the eigenfunctions in Prob. 3? 5 8 GRAPHING SOLUTIONS Using (3) sketch or grph figure (similr to Fig. 29 in Sec. 2.3) of the deflection u (x, t) of vibrting string (length L, ends fixed, c ) strting with initil velocity nd initil deflection (k smll, sy, k.). 5. f (x) k sin px f (x) k sin 2px 8. f (x) k ( cos px) f (x) kx ( x) 9 8 NORMAL FORMS Find the type, trnsform to norml form, nd solve. Show your work in detil. 9. u xx u yy. u xx 6u yy. u xx 2u xy u yy u xx 5u xy u yy. 5. xu xx yu xy u xx u xy 5u yy Longitudinl Vibrtions of n Elstic Br or Rod. These vibrtions in the direction of the x-xis re modeled by the wve eqution u tt c 2 u xx, c 2 E>r (see Tolstov [C9], p. 275). If the rod is fstened t one end, x, nd free t the other, x L, we hve u (, t) nd u x (L, t). Show tht the motion corresponding to initil displcement u (x, ) f (x) nd initil velocity zero is u A n 2 L L (2n )p f (x) sin p n x dx, p n. 2L n xu xy yu yy A n sin p n x cos p n ct, u xx 2u xy u yy u xx 2u xy u yy u xx 6u xy 9u yy 2. Tricomi nd Airy equtions. 2 Show tht the Tricomi eqution yu xx u yy is of mixed type. Obtin the Airy eqution Gs yg from the Tricomi eqution by seprtion. (For solutions, see p. 6 of Ref. [GenRef] listed in App..) 2 Sir GEORGE BIDELL AIRY (8 892), English mthemticin, known for his work in elsticity. FRANCESCO TRICOMI ( ), Itlin mthemticin, who worked in integrl equtions nd functionl nlysis.

7 566 CHAP. 2 Prtil Differentil Equtions (PDEs) This shows tht the expressions in the prentheses must be the Fourier coefficients f (x); tht is, by () in Sec..3, b n of b n A* n sinh npb 2 f (x) sin npx dx. From this nd (6) we see tht the solution of our problem is (7) u(x, y) n u n (x, y) n A* n sin npx npy sinh where (8) A* n 2 sinh (npb>) f (x) sin npx dx. We hve obtined this solution formlly, neither considering convergence nor showing tht the series for u, u xx, nd u yy hve the right sums. This cn be proved if one ssumes tht f nd f r re continuous nd f s is piecewise continuous on the intervl x. The proof is somewht involved nd relies on uniform convergence. It cn be found in [C] listed in App.. Unifying Power of Methods. Electrosttics, Elsticity The Lplce eqution () lso governs the electrosttic potentil of electricl chrges in ny region tht is free of these chrges. Thus our stedy-stte het problem cn lso be interpreted s n electrosttic potentil problem. Then (7), (8) is the potentil in the rectngle R when the upper side of R is t potentil f (x) nd the other three sides re grounded. Actully, in the stedy-stte cse, the two-dimensionl wve eqution (to be considered in Secs. 2.8, 2.9) lso reduces to (). Then (7), (8) is the displcement of rectngulr elstic membrne (rubber sheet, drumhed) tht is fixed long its boundry, with three sides lying in the xy-plne nd the fourth side given the displcement f (x). This is nother impressive demonstrtion of the unifying power of mthemtics. It illustrtes tht entirely different physicl systems my hve the sme mthemticl model nd cn thus be treted by the sme mthemticl methods. P R O B L E M S E T Decy. How does the rte of decy of (8) with fixed n depend on the specific het, the density, nd the therml conductivity of the mteril? 2. Decy. If the first eigenfunction (8) of the br decreses to hlf its vlue within 2 sec, wht is the vlue of the diffusivity? 3. Eigenfunctions. Sketch or grph nd compre the first three eigenfunctions (8) with B n, c, nd L p for t,.,.2, Á,... WRITING PROJECT. Wve nd Het Equtions. Compre these PDEs with respect to generl behvior of eigenfunctions nd kind of boundry nd initil

8 SEC. 2.6 Het Eqution: Solution by Fourier Series 567 conditions. Stte the difference between Fig. 29 in Sec. 2.3 nd Fig LATERALLY INSULATED BAR Find the temperture u (x, t) in br of silver of length cm nd constnt cross section of re cm 2 (density.6 g>cm 3, therml conductivity. cl>(cm sec C), specific het.56 cl>(g C) tht is perfectly insulted lterlly, with ends kept t temperture C nd initil temperture f (x) C, where f (x) sin.px f (x).8 ƒ x 5 ƒ f (x) x ( x) 8. Arbitrry tempertures t ends. If the ends x nd x L of the br in the text re kept t constnt tempertures U nd U 2, respectively, wht is the temperture u (x) in the br fter long time (theoreticlly, s t : )? First guess, then clculte. 9. In Prob. 8 find the temperture t ny time.. Chnge of end tempertures. Assume tht the ends of the br in Probs. 5 7 hve been kept t C for long time. Then t some instnt, cll it t, the temperture t x L is suddenly chnged to C nd kept t C, wheres the temperture t x is kept t C. Find the temperture in the middle of the br t t, 2, 3,, 5 sec. First guess, then clculte. BAR UNDER ADIABATIC CONDITIONS Adibtic mens no het exchnge with the neighborhood, becuse the br is completely insulted, lso t the ends. Physicl Informtion: The het flux t the ends is proportionl to the vlue of u>x there.. Show tht for the completely insulted br, u x (, t), u x (L, t), u (x, t) f (x) nd seprtion of vribles gives the following solution, with A n given by (2) in Sec..3. u(x, t) A n A n cos npx L 2 5 Find the temperture in Prob. with L p, c, nd 2. f (x) x 3.. f (x) cos 2x 5. 2 e(cnp>l) t f (x) f (x) x>p 6. A br with het genertion of constnt rte H ( ) is modeled by u t c 2 u xx H. Solve this problem if L p nd the ends of the br re kept t C. Hint. Set u v Hx(x p)>(2c 2 ). 7. Het flux. The het flux of solution u (x, t) cross x is defined by (t) Ku x (, t). Find (t) for the solution (9). Explin the nme. Is it physiclly understndble tht goes to s t :? 8 25 TWO-DIMENSIONAL PROBLEMS 8. Lplce eqution. Find the potentil in the rectngle x 2, y whose upper side is kept t potentil V nd whose other sides re grounded. 9. Find the potentil in the squre x 2, y 2 if the upper side is kept t the potentil sin 2 px nd the other sides re grounded. 2. CAS PROJECT. Isotherms. Find the stedy-stte solutions (tempertures) in the squre plte in Fig. 297 with 2 stisfying the following boundry conditions. Grph isotherms. () u 8 sin px on the upper side, on the others. (b) u on the verticl sides, ssuming tht the other sides re perfectly insulted. (c) Boundry conditions of your choice (such tht the solution is not identiclly zero). y Fig x Squre plte 2. Het flow in plte. The fces of the thin squre plte in Fig. 297 with side 2 re perfectly insulted. The upper side is kept t 25 C nd the other sides re kept t C. Find the stedy-stte temperture u (x, y) in the plte. 22. Find the stedy-stte temperture in the plte in Prob. 2 if the lower side is kept t U C, the upper side t U C, nd the other sides re kept t C. Hint: Split into two problems in which the boundry temperture is on three sides for ech problem. 23. Mixed boundry vlue problem. Find the stedystte temperture in the plte in Prob. 2 with the upper nd lower sides perfectly insulted, the left side kept t C, nd the right side kept t f (y) C. 2. Rdition. Find stedy-stte tempertures in the rectngle in Fig. 296 with the upper nd left sides perfectly insulted nd the right side rditing into medium t C ccording to u x (, y) hu (, y), h constnt. (You will get mny solutions since no condition on the lower side is given.) 25. Find formuls similr to (7), (8) for the temperture in the rectngle R of the text when the lower side of R is kept t temperture f (x) nd the other sides re kept t C.

9 SEC. 2. Lplcin in Polr Coordintes. Circulr Membrne. Fourier Bessel Series 59 P R O B L E M S E T 2. 3 RADIAL SYMMETRY. Why did we introduce polr coordintes in this section? 2. Rdil symmetry reduces (5) to 2 u u rr u r >r. Derive this directly from 2 u u xx u yy. Show tht the only solution of 2 u depending only on r 2x 2 y 2 is u ln r b with rbitrry constnts nd b. 3. Alterntive form of (5). Show tht (5) cn be written 2 u (ru r ) r >r u uu >r 2, form tht is often prcticl. BOUNDARY VALUE PROBLEMS. SERIES. TEAM PROJECT. Series for Dirichlet nd Neumnn Problems () Show tht, Á u n r n cos nu, u n r n sin nu, n,, re solutions of Lplce s eqution 2 u with 2 u given by (5). (Wht would u n be in Crtesin coordintes? Experiment with smll n.) (b) Dirichlet problem (See Sec. 2.6) Assuming tht termwise differentition is permissible, show tht solution of the Lplce eqution in the disk r R stisfying the boundry condition u(r, u) f (u) (R nd f given) is (2) where n, b n Sec..). u(r, u) re the Fourier coefficients of f (see (c) Dirichlet problem. Solve the Dirichlet problem using (2) if R nd the boundry vlues re u (u) volts if p u, u (u) volts if u p. (Sketch this disk, indicte the boundry vlues.) (d) Neumnn problem. Show tht the solution of the Neumnn problem 2 u if r R, u N (R, u) f (u) (where u N u>n is the directionl derivtive in the direction of the outer norml) is u(r, u) A b n r R b nsin nu d n n c n r R b ncos nu r n (A n cos nu B n sin nu) with rbitrry A n B n A nd (e) Comptibility condition. Show tht (9), Sec.., imposes on f (u) in (d) the comptibility condition (f) Neumnn problem. Solve 2 u in the nnulus r 2 if u r (, u) sin u, u r (2, u). 5 8 ELECTROSTATIC POTENTIAL. STEADY-STATE HEAT PROBLEMS The electrosttic potentil stisfies Lplce s eqution 2 u in ny region free of chrges. Also the het eqution u t c 2 2 u (Sec. 2.5) reduces to Lplce s eqution if the temperture u is time-independent ( stedy-stte cse ). Using (2), find the potentil (equivlently: the stedy-stte temperture) in the disk r if the boundry vlues re (sketch them, to see wht is going on). 5. u (, u) 22 if p 2 u p 2 nd otherwise 6. u (, u) cos 3 u 7. u (, u) ƒ u ƒ if p u p 8. u (, u) u if p 2 u p 2 nd otherwise 9. CAS EXPERIMENT. Equipotentil Lines. Guess wht the equipotentil lines u (r, u) const in Probs. 5 nd 7 my look like. Then grph some of them, using prtil sums of the series.. Semidisk. Find the electrosttic potentil in the semidisk r, u p which equls u (p u) on the semicircle r nd on the segment x.. Semidisk. Find the stedy-stte temperture in semicirculr thin plte r, u p with the semicircle r kept t constnt temperture u nd the segment x t. CIRCULAR MEMBRANE pnr n pnr n p f (u) cos nu du, p p f (u) sin nu du. p p f (u) du. p 2. CAS PROJECT. Norml Modes. () Grph the norml modes s in Fig. 36. u, u 5, u 6

10 592 CHAP. 2 Prtil Differentil Equtions (PDEs) (b) Write progrm for clculting the A m s in Exmple nd extend the tble to m 5. Verify (2) F rr r F r r 2 F uu k 2 F. numericlly tht m (m )p nd compute the error for m, Á,. (c) Grph the initil deflection f (r) in Exmple s Show tht the PDE cn now be seprted by substituting F W (r)q(u), giving well s the first three prtil sums of the series. Comment on ccurcy. (25) Qs n 2 Q, (d) Compute the rdii of the nodl lines of u 2, u 3, u when R. How do these vlues compre to those of the nodes of the vibrting string of length? Cn you estblish ny empiricl lws by experimenttion with further? u m 3. Frequency. Wht hppens to the frequency of n eigenfunction of drum if you double the tension?. Size of drum. A smll drum should hve higher fundmentl frequency thn lrge one, tension nd density being the sme. How does this follow from our formuls? 5. Tension. Find formul for the tension required to produce desired fundmentl frequency f of drum. 6. Why is A A 2 Á in Exmple? Compute the first few prtil sums until you get 3-digit ccurcy. Wht does this problem men in the field of music? 7. Nodl lines. Is it possible tht for fixed c nd R two or more u m [see (6)] with different nodl lines correspond to the sme eigenvlue? (Give reson.) 8. Nonzero initil velocity is more of theoreticl interest becuse it is difficult to obtin experimentlly. Show tht for (7) to stisfy (9b) we must hve (26) 2 Periodicity. Show tht Q (u) must be periodic with period 2p nd, therefore, n,, 2, Á in (25) nd (26). Show tht this yields the solutions Q n * sin nu, W n J n (kr), n,, Á Q n cos nu,. 2. Boundry condition. Show tht the boundry condition (27) leds to k k mn mn >R, where s nm is the mth positive zero of J n (s). 22. Solutions depending on both r nd U. Show tht solutions of (22) stisfying (27) re (see Fig. 3) (28) r 2 Ws rwr (k 2 r 2 n 2 )W. u (R, u, t) u nm (A nm cos ck nm t B nm sin ck nm t) J n (k nm r) cos nu u* nm (A* nm cos ck nm t B* nm sin ck nm t) J n (k nm r) sin nu (2) where B m K m R rg (r)j ( m r>r) dr K m 2>(c m R)J 2 ( m ). u u 2 u 32 Fig. 3. Nodl lines of some of the solutions (28) VIBRATIONS OF A CIRCULAR MEMBRANE DEPENDING ON BOTH r AND U 9. (Seprtions) Show tht substitution of into the wve eqution (6), tht is, (22) u tt c 2 u rr r u r r 2 u uu b, gives n ODE nd PDE (23) G # # l 2 G, where l ck, u F (r, u)g (t) 23. Initil condition. Show tht u t (r, u, ) gives B nm, B* nm in (28). 2. Show tht u* m nd u m is identicl with (6) in this section. 25. Semicirculr membrne. Show tht u represents the fundmentl mode of semicirculr membrne nd find the corresponding frequency when c 2 nd R.

11 598 CHAP. 2 Prtil Differentil Equtions (PDEs) (22) A n 55 (2n ) 2 n M m () m (2n 2m)! m!(n m)!(n 2m )!. Tking n, we get A 55 (since! ). For n, 2, 3, Á we get A ! 65!!2! 2, A 2 275! 2! b,!2!3!!!! A !!3!!! b 385!2!2! 8, etc. Hence the potentil (7) inside the sphere is (since P ) (23) u (r, ) (Fig. 3) 2 r P 385 (cos ) 8 r 3 P 3 (cos ) Á with P, P 3, Á given by ( r), Sec Since R, we see from (9) nd (2) in this section tht B n A n, nd (2) thus gives the potentil outside the sphere (2) u (r, ) 55 r 65 2r 2 P (cos ) 385 8r P 3(cos ) Á. Prtil sums of these series cn now be used for computing pproximte vlues of the inner nd outer potentil. Also, it is interesting to see tht fr wy from the sphere the potentil is pproximtely tht of point chrge, nmely, 55>r. (Compre with Theorem 3 in Sec. 9.7.) y π π t 2 Fig. 3. Prtil sums of the first, 6, nd nonzero terms of (23) for r R E X A M P L E 2 Simpler Cses. Help with Problems The techniclities encountered in cses tht re similr to the one shown in Exmple cn often be voided. For instnce, find the potentil inside the sphere S: r R when S is kept t the potentil f () cos 2. (Cn you see the potentil on S? Wht is it t the North Pole? The equtor? The South Pole?) Solution. w cos, cos 2 2 cos 2 2w 2 3P 2 (w) 3 3 (3 2w 2 2 ) 3. potentil in the interior of the sphere is Hence the u 3 r 2 P 2 (w) 3 3r 2 P 2 (cos ) 3 2 3r 2 (3 cos 2 ) 3. P R O B L E M S E T 2.. Sphericl coordintes. Derive (7) from 2 u in 3. Sketch P n (cos u), u 2p, for n,, 2. (Use sphericl coordintes. (r) in Sec. 5.2.) 2. Cylindricl coordintes. Verify (5) by trnsforming. Zero surfces. Find the surfces on which u, u 2, u 3 2 u bck into Crtesin coordintes. in (6) re zero.

12 SEC. 2. Lplce s Eqution in Cylindricl nd Sphericl Coordintes. Potentil CAS PROBLEM. Prtil Sums. In Exmple in the text verify the vlues of nd compute A, Á A, A, A 2, A 3, A. Try to find out grphiclly how well the corresponding prtil sums of (23) pproximte the given boundry function. 6. CAS EXPERIMENT. Gibbs Phenomenon. Study the Gibbs phenomenon in Exmple (Fig. 3) grphiclly. 7. Verify tht nd u* n in (6) re solutions of (8). u n 8 5 POTENTIALS DEPENDING ONLY ON r 8. Dimension 3. Verify tht the potentil u c>r, r 2x 2 y 2 z 2 stisfies Lplce s eqution in sphericl coordintes. 9. Sphericl symmetry. Show tht the only solution of Lplce s eqution depending only on r 2x 2 y 2 z 2 is u c>r k with constnt c nd k.. Cylindricl symmetry. Show tht the only solution of Lplce s eqution depending only on r 2x 2 y 2 is u c ln r k.. Verifiction. Substituting u (r) with r s in Prob. 9 into u xx u yy u zz, verify tht us 2ur>r, in greement with (7). 2. Dirichlet problem. Find the electrosttic potentil between coxil cylinders of rdii r 2 cm nd r 2 cm kept t the potentils U 22 V nd U 2 V, respectively. 3. Dirichlet problem. Find the electrosttic potentil between two concentric spheres of rdii r 2 cm nd r 2 cm kept t the potentils U 22 V nd U 2 V, respectively. Sketch nd compre the equipotentil lines in Probs. 2 nd 3. Comment.. Het problem. If the surfce of the bll r 2 x 2 y 2 z 2 R 2 is kept t temperture zero nd the initil temperture in the bll is f (r), show tht the temperture u (r, t) in the bll is solution of u t c 2 (u rr 2u r >r) stisfying the conditions u (R, t), u (r, ) f (r). Show tht setting v ru gives v t c 2 v rr, v (R, t), v (r, ) rf (r). Include the condition v (, t) (which holds becuse u must be bounded t r ), nd solve the resulting problem by seprting vribles. 5. Wht re the nlogs of Probs. 2 nd 3 in het conduction? 6 2 BOUNDARY VALUE PROBLEMS IN SPHERICAL COORDINATES r, U, Find the potentil in the interior of the sphere r R if the interior is free of chrges nd the potentil on the sphere is 6. f () cos f () cos 2 9. f () f () cos 2 2. f () cos 3 3 cos 2 5 cos 2. Point chrge. Show tht in Prob. 7 the potentil exterior to the sphere is the sme s tht of point chrge t the origin. 22. Exterior potentil. Find the potentils exterior to the sphere in Probs. 6 nd Plne intersections. Sketch the intersections of the equipotentil surfces in Prob. 6 with xz-plne. 2. TEAM PROJECT. Trnsmission Line nd Relted PDEs. Consider long cble or telephone wire (Fig. 35) tht is imperfectly insulted, so tht leks occur long the entire length of the cble. The source S of the current i (x, t) in the cble is t x, the receiving end T t x l. The current flows from S to T nd through the lod, nd returns to the ground. Let the constnts R, L, C, nd G denote the resistnce, inductnce, cpcitnce to ground, nd conductnce to ground, respectively, of the cble per unit length. S x = Fig. 35. Trnsmission line T x = l Lod () Show tht ( first trnsmission line eqution ) where u (x, t) is the potentil in the cble. Hint: Apply Kirchhoff s voltge lw to smll portion of the cble between x nd x x (difference of the potentils t x nd x x resistive drop inductive drop). (b) Show tht for the cble in () ( second trnsmission line eqution ), u x i x Ri L i t Gu C u t. Hint: Use Kirchhoff s current lw (difference of the currents t x nd x x loss due to lekge to ground cpcitive loss). (c) Second-order PDEs. Show tht elimintion of i or u from the trnsmission line equtions leds to u xx LCu tt (RC GL)u t RGu, i xx LCi tt (RC GL)i t RGi. (d) Telegrph equtions. For submrine cble, G is negligible nd the frequencies re low. Show tht this leds to the so-clled submrine cble equtions or telegrph equtions u xx RCu t, i xx RCi t.

13 6 CHAP. 2 Prtil Differentil Equtions (PDEs) Find the potentil in submrine cble with ends (x, x l) grounded nd initil voltge distribution U const. (e) High-frequency line equtions. Show tht in the cse of lternting currents of high frequencies the equtions in (c) cn be pproximted by the so-clled high-frequency line equtions u xx LCu tt, i xx LCi tt. Solve the first of them, ssuming tht the initil potentil is U sin (px>l), nd u t (x, ) nd u t the ends x nd x l for ll t. 25. Reflection in sphere. Let r, u, be sphericl coordintes. If u (r, u, ) stisfies 2 u, show tht v (r, u, ) u (>r, u, )>r stisfies 2 v. 2.2 Solution of PDEs by Lplce Trnsforms Reders fmilir with Chp. 6 my wonder whether Lplce trnsforms cn lso be used for solving prtil differentil equtions. The nswer is yes, prticulrly if one of the independent vribles rnges over the positive xis. The steps to obtin solution re similr to those in Chp. 6. For PDE in two vribles they re s follows.. Tke the Lplce trnsform with respect to one of the two vribles, usully t. This gives n ODE for the trnsform of the unknown function. This is so since the derivtives of this function with respect to the other vrible slip into the trnsformed eqution. The ltter lso incorportes the given boundry nd initil conditions. 2. Solving tht ODE, obtin the trnsform of the unknown function. 3. Tking the inverse trnsform, obtin the solution of the given problem. If the coefficients of the given eqution do not depend on t, the use of Lplce trnsforms will simplify the problem. We explin the method in terms of typicl exmple. E X A M P L E Semi-Infinite String Find the displcement w (x, t) of n elstic string subject to the following conditions. (We write w since we need u to denote the unit step function.) (i) The string is initilly t rest on the x-xis from x to ( semi-infinite string ). (ii) For t the left end of the string (x ) is moved in given fshion, nmely, ccording to single sine wve w (, t) f (t) e sin t if t 2p otherwise (Fig. 36). (iii) Furthermore, lim w (x, t) for t. x: f(t) Fig. 36. π 2ππ t Motion of the left end of the string in Exmple s function of time t