SECTION PROBLEMS IN POLAR, CYLINDRICAL AND SPHERICAL COORDINATES

CHAPTER 9 SECTION 9. 353 PROBLEMS IN POLAR, CYLINDRICAL AND SPHERICAL COORDINATES 9. Homogeneous Problems in Polr, Cylindricl, nd Sphericl Coordintes In Section 6.3, seprtion of vribles ws used to solve homogeneous boundry vlue problems expressed in polr coordintes. With the results of Chpter 8, we re in position to tckle boundry vlue problems in cylindricl nd sphericl coordintes nd initil boundry vlue problems in ll three coordinte systems. Homogeneous problems re discussed in this section; nonhomogeneous problems re discussed in Section 9.. We begin with the following het conduction problem. Exmple 9. An infinitely long cylinder of rdius is initilly t temperture fr) = r, nd for time t>, the boundry r = is insulted. Find the temperture in the cylinder for t>. Solution With the initil temperture function of r nd the surfce of the cylinder insulted, temperture in the cylinder is function Ur, t) ofr nd t only. It stisfies the initil boundry vlue problem t = k U r r ), <r<, t>, 9.) + r, t) =, r t >, 9.b) Ur, ) = r, <r<. 9.c) When function Ur, t) = Rr)T t) with vribles seprted is substituted into the PDE, nd the eqution is divided by krt, the result is T kt = R R + R = α = constnt independent of r nd t. rr This eqution nd boundry condition 9.b yield the Sturm-Liouville system rr ) αrr =, <r<, 9.) R ) =. 9.b) This singulr system ws discussed in Section 8.4 see Tble 8. with ν = ). If we set α = λ, eigenvlues re defined by the eqution J λ) =, nd normlized eigenfunctions re R n r) = J λ n r), n. 9.3) J λ n ) For simplicity of nottion, we hve dropped the zero subscript on R n nd λ n.) The differentil eqution hs generl solution T + kλ nt = 9.4)

354 SECTION 9. T t) =Ce kλ n t. 9.5) In order to stisfy initil condition 9.c, we superpose seprted functions nd tke Ur, t) = C n e kλ n t R n r), 9.6) n= where the C n re constnts. Condition 9.c requires these constnts to stisfy r = C n R n r), <r<. 9.7) n= Thus, the C n re coefficients in the Fourier Bessel series of r, nd, ccording to eqution 8.6 in Section 8.4, C n = r r )R n r) dr = r r )J λ n r) dr. J λ n ) To evlute this integrl when n>, we set u = λ n r, in which cse λn ) u C n = u3 J λ n ) λ n λ 3 J u) du n λ n λn = λ 4 n J λ n λ n ) u u3 )J u) du. For the term involving u 3, we use the reduction formul in Exercise 9 of Section 8.3, [ λn { } C n = λ 4 λ n uj u) du u 3 λn J u) nj λ n ) { } u λn ] λn J u) +4 uj u) du. If we recll the eigenvlue eqution J λ) =, nd eqution 8.4 in Section 8.3 with ν =, we my write [ ] λn C n = λ 4 n J λ n λ n ) J λ n )+ λ n +4) d du [uj u)] du [ { } ] = λ 4 n J λ λ n ) n J λ n )+ λ λn n +4) uj u) = λ. n When n =, the eigenfunction is R r) = /, nd { C = r r r )R r) dr = The solution of problem 9. is therefore } r4 3 =. 4 4

Exmple 9. SECTION 9. 355 ) 3 Ur, t) = + 4 λ e kλ n t J λ n r) n J λ n ) = 4 n= e kλ n t λ n n= J λ n r) J λ n ). 9.8) Notice tht for lrge t, the limit of this solution is /, nd this is the verge vlue of r over the circle r. In the following het conduction problem, we dd ngulr dependence to the temperture function. An infinitely long rod with semicirculr cross section is initilly t = ) t constnt nonzero temperture throughout. For t>, its flt side is held t temperture C while its round side is insulted. Find temperture in the rod for t>. z Solution Temperture in tht hlf of the rod for which x< in Figure 9. is identicl to tht in the hlf for which x ; no het crosses the x = plne. As result, r the temperture function Ur,θ,t) nd it r is independent of z) must stisfy the initil boundry vlue problem x y Figure 9. t = k U r + r r + ) U, <r<, <θ<, t >, 9.9) r θ Ur,,t)=, <r<, t>, 9.9b) U θ r,,t ) =, <r<, t>, 9.9c) U r,θ,t)=, <θ<, t >, 9.9d) Ur, θ, ) = U, <r<, <θ<. 9.9e) In Exercise 4, the problem is solved for <θ<with the condition Ur,,t)= in plce of 9.9c.) When function with vribles seprted, Ur,θ,t) = Rr)Hθ)T t), is substituted into the PDE, or, RHT = kr HT + r R HT + r RH T ) H H = r R R + rr R r T = α = constnt independent of r, θ, nd t. kt When boundry conditions 9.9b,c re imposed on the seprted function, Sturm- Liouville system in Hθ) results, H + αh =, <θ</, 9.) H) = = H /). 9.b)

356 SECTION 9. This system ws discussed in Section 5.. If we set α = ν, then ccording to Tble 5., eigenvlues re ν m =m ) m =,,...), with orthonorml eigenfunctions H m θ) = sin m )θ. 9.) Continued seprtion of the eqution in Rr) nd T t) gives R + r R ν m R r = T = β = constnt independent of r nd t. kt Boundry condition 9.9d leds to the Sturm-Liouville system ] rr ) m ) + [ βr R =, r <r<, 9.) R ) =. 9.b) This is singulr Sturm-Liouville system 8.46 of Section 8.4. If we set β = λ, eigenvlues λ mn re defined by the eqution with corresponding eigenfunctions J m λ) = 9.3) R mn r) = N J m λ mn r), 9.4) where N = [ ) ] m [J m λ mn )]. 9.4b) λ mn The differentil eqution T = kλ mn T 9.5) hs generl solution T t) =Ce kλ mn t. 9.6) To stisfy initil condition 9.9e, we superpose seprted functions nd tke Ur,θ,t)= C mn e kλ mn t R mn r)h m θ), 9.7) m= n= where C mn re constnts. The initil condition requires these constnts to stisfy U = C mn R mn r)h m θ), <r<, <θ</. 9.8) m= n= If we multiply this eqution by H i θ) nd integrte with respect to θ from θ =to θ = /, orthogonlity of the eigenfunctions H m θ) gives

SECTION 9. 357 C in R in r) = n= / = U { i U H i θ) dθ = U / } / cos i )θ = sin i )θdθ U i ). But this eqution implies tht the C in re Fourier Bessel coefficients for the function U /[i ) ]; tht is, U C in = i ) rr inr) dr. Thus, the solution of problem 9.9 for θ / is 9.7, where U C mn = m ) rr mn r) dr. 9.9) For n ngle θ between / nd, we should evlute Ur, θ, t). Since H m θ) = sin m ) θ) = sin m )θ, it follows tht Ur, θ, t) =Ur,θ,t). Hence, solution 9.7 is vlid for θ. Our next exmple is vibrtion problem. Exmple 9.3 Solve the initil boundry vlue problem z t = z c r + z r r + ) z, r θ <r<, <θ, t >, 9.) z,θ,t)=, <θ, t >, 9.b) zr, θ, ) = fr, θ), <r<, <θ, 9.c) z t r, θ, ) =, <r<, <θ. 9.d) Described is membrne stretched over the circle r tht hs n initil displcement fr, θ) nd zero initil velocity. Boundry condition 9.b sttes tht the edge of the membrne is fixed on the xy-plne. Solution When seprted function zr,θ,t) = Rr)Hθ)T t), is substituted into the PDE, RHT = c R HT + r R HT + r RH T ) or, H R H = + r R r T ) = α = constnt independent of r, θ, nd t. R c T Since the solution nd its first derivtive with respect to θ must be -periodic in θ, it follows tht Hθ) must stisfy the periodic Sturm-Liouville system H + αh =, <θ, 9.) H ) =H), 9.b) H ) =H ). 9.c)

358 SECTION 9. This system ws discussed in Chpter 5 Exmple 5. nd eqution 5.). Eigenvlues re α = m, m nonnegtive integer, with orthonorml eigenfunctions, sin mθ, cos mθ. 9.) Continued seprtion of the eqution in Rr) nd T t) gives R + r R m R r = T = β = constnt independent of r nd t. c T When boundry condition 9.b is imposed on the seprted function, Sturm- Liouville system in Rr) results, ) rr ) + βr m R =, r <r<, 9.3) R) =. 9.3b) This is, once gin, singulr system 8.46 in Section 8.4. If we set β = λ, eigenvlues λ mn re defined by with corresponding orthonorml eigenfunctions J m λ) =, 9.4) R mn r) = Jm λ mn r) J m+ λ mn ) 9.5) see Tble 8.). The differentil eqution hs generl solution T + c λ mnt = 9.6) T t) =d cos cλ mn t + b sin cλ mn t, 9.7) where d nd b re constnts. Initil condition 9.d implies tht b =, nd hence T t) =d cos cλ mn t. 9.8) In order to stisfy the finl initil condition 9.c, we superpose seprted functions nd tke R n r) zr,θ,t)= d n cos cλ n t n= ) cos mθ sin mθ + R mn r) d mn + f mn cos cλ mn t, 9.9) m= n= where d mn nd f mn re constnts. Condition 9.c requires these constnts to stisfy fr, θ) = n= d n R n r) + m= n= ) cos mθ sin mθ R mn r) d mn + f mn 9.3)

SECTION 9. 359 Exmple 9.4 for <r<, <θ. If we multiply this eqution by / ) cos iθ nd integrte with respect to θ from θ = to θ =, orthogonlity of the eigenfunctions in θ gives cos iθ fr, θ) dθ = d in R in r). Multipliction of this eqution by rr ij r) nd integrtion with respect to r from r =tor = yields becuse of orthogonlity of the R ij for fixed i) cos iθ rfr, θ)r ij dθ dr = d ij ; tht is Similrly, nd d mn = f mn = d n = n= rr mn cos mθ fr, θ) dr dθ. 9.3) rr mn sin mθ fr, θ) dr dθ, 9.3b) rr n fr, θ) dr dθ. 9.3c) The solution of problem 9. is therefore 9.9, where d mn nd f mn re defined by 9.3. Coefficients d mn nd f mn in this exmple were clculted by first using orthogonlity of the trigonometric eigenfunctions nd then using orthogonlity of the R mn r). An lterntive procedure is to determine the multi-dimensionl eigenfunctions for problem 9.. This pproch is discussed in Exercise 7. Our finl exmple on seprtion is potentil problem. Find the potentil interior to sphere when the potentil is fφ, θ) on the sphere. Solution The boundry vlue problem for the potentil V r,φ,θ)is V r + V r r + r sin φ V ) V + sin φ φ φ r sin φ θ =, <r<, <φ<, <θ, 9.3) V, φ, θ) =fφ, θ), φ, <θ. 9.3b) When function with vribles seprted, V r,φ,θ) = Rr)Φφ)Hθ), is substituted into PDE 9.3, or, R ΦH + r R ΦH + r sin φ φ sin φrφ H)+ RΦH r sin φ = [ ] R r sin φ R + R rr + d r sin φ Φ dφ sin φ Φ ) = H H = α = constnt independent of r, φ, nd θ.

36 SECTION 9. Becuse V r,φ,θ) must be -periodic in θ, s must its first derivtive with respect to θ, it follows tht Hθ) must stisfy the periodic Sturm-Liouville system H + αh =, <θ, 9.33) H ) =H), 9.33b) H ) =H ). 9.33c) This is Sturm-Liouville system 9. with eigenvlues α = m nd orthonorml eigenfunctions, cos mθ, sin mθ. Continued seprtion of the eqution in Rr) nd Φφ) gives r R R + rr R = m sin φ d Φ sin φ dφ sin φ Φ )=β = constnt independent of r nd φ. Thus, Φφ) must stisfy the singulr Sturm-Liouville system d sin φ dφ ) ) + β sin φ m Φ=, <φ<. 9.34) dφ dφ sin φ According to the results of Section 8.6, eigenvlues re β = nn + ), where n m is n integer, with orthonorml eigenfunctions n + )n m)! Φ mn φ) = P mn cos φ). 9.35) n + m)! The remining differentil eqution r R +rr nn +)R = 9.36) is Cuchy-Euler eqution tht cn be solved by setting Rr) =r s, s n unknown constnt. This results in the generl solution Rr) = C r n+ + Arn. 9.37) For Rr) to remin bounded s r pproches zero, we must set C =. Superposition of seprted functions now yields V r,φ,θ)= A n r n Φ n φ) n= + m= n=m r n cos mθ sin mθ Φ mn φ) A mn + B mn ), 9.38) where A mn nd B mn re constnts. Boundry condition 9.3b requires these constnts to stisfy fφ, θ) = A n n Φ n φ) n= + m= n=m ) n cos mθ sin mθ Φ mn φ) A mn + B mn 9.39)

SECTION 9. 36 for φ, <θ. Becuse of orthogonlity of eigenfunctions in φ nd θ, multipliction by / ) sin φ Φ j φ) nd integrtion with respect to φ nd θ give A j = j fφ, θ) sin φ Φ j φ) dφ dθ. 9.4) Similrly, A mn = cos mθ fφ, θ) sin φ Φ n mn φ) dφ dθ, 9.4b) B mn = sin mθ n fφ, θ) sin φ Φ mn φ) dφ dθ. 9.4c) Notice tht the potentil t the centre of the sphere is V,φ,θ)= A Φ φ) = [ ] fφ, θ) sin φ Φ φ) dφ dθ Φ φ). Since Φ φ) =/, V,φ,θ)= 4 = 4 fφ, θ) sin φ dφ dθ fφ, θ) sin φ dφ dθ, nd this is the verge vlue of fφ, θ) over the surfce of the sphere. We cn develop n integrl formul for the solution nlogous to Poisson s integrl formul for circle, eqution 6.34. We chnge vribles of integrtion for the coefficients to α nd β, substitute the coefficients into summtion 9.38 nd interchnge orders of integrtion nd summtion [ r ) n V r,φ,θ)= fα, β) sin α Φn φ)φ n α) n= ] + r ) nfα, β) sin α Φmn φ)φ mn α)cos mθ cos mβ + sin mθ sin mβ) dβ dα m= n= = [ r ) n fα, β) sin α Φn φ)φ n α) n= ] r ) n + Φmn φ)φ mn α) cos mθ β) dβ dα. m= n= Let us define Sr,φ,θ)= r ) n Φn φ)φ n α)+ n= m= n= r ) n Φmn φ)φ mn α) cos mθ β). Consider the potentil t point inside the sphere nd on the z-xis with sphericl coordintes r,,θ), where θ is rbitrry nd <r<. For such point, Sr,,θ)= r ) n r ) n Φn )Φ n α)+ Φmn )Φ mn α) cos mθ β). n= m= n=

36 SECTION 9. Since n + n + n + )n m)! Φ n ) = P n ) =, Φ mn ) = P mn ) =, n + m)! Sr,,θ)= r ) ) n n + P n cos α) = 4 n= n= r ) n n +) Pn cos α). To find closed vlue for this summtion, we differentite the generting function 8.7 for Legendre polynomils = P n x)t n xt + t with respect to t, n= x t xt + t ) = np 3/ n x)t n. If we multiply this by t nd dd it to the generting function, we obtin n +)P n x)t n tx t) = xt + t ) + = t 3/ xt + t xt + t ). 3/ n= It follows tht Thus, Sr,,θ)= 4 r r cos α + r n= ) 3/ = r ) 4 r cos α + r ). 3/ V r,,θ)= r ) fα, β) sin α dβ dα 4 r cos α + r ) 3/ = r ) fα, β) sin α dβ dα. 4 r cos α + r ) 3/ This is the potentil t point r,,θ) on the z-xis. The distnce between this point nd point, α, β) on the sphere is sin α cos β) + sin α sin β) + cos α r) = r + r cos α. The denomintor in the bove integrl is therefore the cube of the distnce from points on the sphere to the point t which the potentil is clculted. Since the xes could lwys be rotted so tht the observtion point is on the z-xis, it follows tht to find the potentil t ny point with sphericl coordintes r,φ,θ) inside the sphere, we need only replce r + r cos α with the distnce from r,φ,θ) to, α, β), nmely, r sin φ cos θ sin α cos β) +r sin φ sin θ sin α sin β) +r cos φ cos α) = r + r[sin φ sin α cos θ β) + cos φ cos α].

Thus, V r,φ,θ)= r ) 4 SECTION 9. 363 fα, β) sin α dβ dα. {r + r[sin φ sin α cos θ β) + cos φ cos α]} 3/ This is clled Poisson s integrl formul for sphere. 9.4) EXERCISES 9. Prt A Het Conduction. ) The initil temperture of n infinitely long cylinder of rdius is fr). If, for time t>, the outer surfce is held t C, find the temperture in the cylinder. b) Simplify the solution in prt ) when fr) is constnt U. c) Find the solution when fr) = r.. An infinitely long cylinder of rdius is initilly t temperture fr) nd, for time t>, the boundry r = is insulted. ) Find the temperture Ur, t) in the cylinder. b) Wht is the limit of Ur, t) for lrge t? 3. A thin circulr plte of rdius is insulted top nd bottom. At time t = its temperture is fr, θ). If the temperture of its edge is held t C for t>, find its interior temperture for t>. 4. Solve Exmple 9. using the boundry condition Ur,,t) = in plce of r, /,t)/ θ =. 5. An infinitely long cylinder is bounded by the surfces r =, θ =, nd θ = /. At time t =, its temperture is fr, θ), nd for t>, ll surfces re held t temperture zero. Find temperture in the cylinder. 6. Repet Exercise 5 if the flt sides re insulted. 7. Repet Exercise 5 if the curved side is insulted. 8. Repet Exercise 5 if ll sides re insulted. Show tht the limit of the temperture s t is the verge of fr, θ) over the cylinder. 9. A flt plte in the form of sector of circle of rdius nd ngle α is insulted top nd bottom. At time t =, the temperture of the plte increses linerly from Ctr =to constnt vlue Ū Ctr = nd is therefore independent of θ). If, for t>, the rounded edge is insulted nd the stright edges re held t temperture C, find the temperture in the plte for t>. Prove tht het never crosses the line θ = α/.. Find the temperture in the plte of Exercise 9 if the initil temperture is fr), the stright sides re insulted, nd the curved edge is held t temperture C.. Repet Exercise if the initil temperture is function of r nd θ, nmely, fr, θ).. A cylinder occupies the region r, z L. It hs temperture fr, z) t time t =. For t>, its end z = is insulted, nd the remining two surfces re held t temperture C. Find the temperture in the cylinder.

364 SECTION 9. 3. Solve Exercise ),b) if het is trnsferred t r = ccording to Newton s lw of cooling to n environment t temperture zero. 4. ) A sphere of rdius is initilly t temperture fr) nd, for time t>, the boundry r = is held t temperture zero. Find the temperture in the sphere for t>. You will need the results of Exercise 8 in Section 8.4). Compre the solution to tht in Exercise of Section 4.. b) Simplify the solution when fr) =U, constnt. c) Suppose the sphere hs rdius cm nd is mde of steel with k =.4 6. Find the temperture t the centre of the sphere fter minutes when fr) =U s in prt b). d) Repet prt c) if the sphere is sbestos with k =.47 6. 5. Repet prts ) nd b) of Exercise 4 if the surfce of the sphere is insulted. See Exercise 8 in Section 8.4.) Wht is the temperture for lrge t? 6. Repet prts ) nd b) of Exercise 4 if the surfce trnsfers het to n environment t temperture zero ccording to Newton s lw of cooling; tht is, tke s boundry condition, t) κ + µu, t) =, t >. r Assume tht µ < κ nd see Exercise 8 in Section 8.4.) 7. Repet Exercise 4) if the initil temperture is lso function of φ. You will need the results of Exercise 9 in Section 8.4.) 8. ) Repet Exercise 4) if the initil temperture is lso function of φ nd the surfce of the sphere is insulted. You will need the results of Exercise 9 in Section 8.4.) b) Wht is the limit of the solution for lrge t? 9. The result of this exercise is nlogous to tht in Exercise 9 of Section 6.4. Show tht the solution of the homogeneous het conduction problem t = k U r r + U z + r l z + h U =, z =, <r<, t>, l z + h U =, z = L, <r<, t>, l 3 r + h 3U =, r =, <z<l, t>, Ur, z, ) = fr)gz), <r<, <z<l, ), <r<, <z<l, t>, where the initil temperture is the product of function of r nd function of z, is the product of the solutions of the problems t = k U r + ), <r<, t>, r r, t) l 3 + h 3 U, t) =, t >, r Ur, ) = fr), <r<; nd

SECTION 9. 365,t) l z L, t) l z. Solve the het conduction problem U r t = k U, <z<l, t>, z + h U,t)=, t >, + h UL, t) =, t >, Uz,) = gz), t = k + r U z r,,t)=, <r<, t>, <z<l. ) r + U z, <r<, <z<l, t>, Ur,L,t)=, <r<, t>, U r,z,t)=, <z<l, t>, Ur, z, ) = r )L z), <r<, <z<l, ) by using the results of Exercise 9, Exmple 9., nd Exercise ) in Section 6.. b) by seprtion of vribles. Prt B Vibrtions. ) A vibrting circulr membrne of rdius is given n initil displcement tht is function only of r, nmely, fr), r, nd zero initil velocity. Show tht subsequent displcements of the membrne, if its edge r = is fixed on the xy-plne, re of the form zr, t) = n= A n cos cλ n t J λ n r) J λ n ). Wht is A n? b) The first term in the series in prt ), clled the fundmentl mode of vibrtion for the membrne, is H r, t) = A cos cλ t J λ r) J λ ). Simplify nd describe this mode when =. Does H r, t) hve nodl curves? c) Repet prt b) for the second mode of vibrtion. d) Are frequencies of higher modes of vibrtion integer multiples of the frequency of the fundmentl mode? Were they for vibrting string with fixed ends?. A circulr membrne of rdius hs its edge fixed on the xy-plne. In ddition, clmp holds the membrne on the xy-plne long rdil line from the centre to the circumference. If the membrne is relesed from rest t displcement fr, θ), find subsequent displcements. For consistency, we require fr, θ) to vnish long the clmped rdil line.) 3. Simplify the solution in prt ) of Exercise when fr) = r. See Exmple 9..) 4. A circulr membrne of rdius is prllel to the xy-plne nd is flling with constnt speed v. At time t =, it strikes the xy-plne. For t>, the edge of the membrne is fixed on the

366 SECTION 9. xy-plne, but the reminder of the membrne is free to vibrte verticlly. Find displcements of the membrne. 5. Eqution 9.9 with coefficients defined in 9.3 describes displcements of circulr membrne with fixed edge when oscilltions re initited from rest t some prescribed displcement. In this exercise we exmine nodl curves for vrious modes of vibrtion. ) The first mode of vibrtion is the term d / )R r) cos cλ t. Show tht this mode hs no nodl curves. b) Show tht the mode d / )R r) cos cλ t hs one nodl curve, circle. c) Show tht the mode d 3 / )R 3 r) cos cλ 3 t hs two circulr nodl curves. d) On the bsis of prts ), b), nd c), wht re the nodl curves for the mode d n / )R n r) cos cλ n t? e) Corresponding to n = m =, there re two modes, d / )R r) cos cλ t cos θ nd f / )R r) cos cλ t sin θ. Show tht ech of these modes hs only one nodl curve, stright line. f) Find nodl curves for the modes d / )R r) cos cλ t cos θ nd f / )R r) cos cλ t sin θ. g) Find nodl curves for the modes d / )R r) cos cλ t cos θ nd f / )R r) cos cλ t sin θ. h) On the bsis of prts e), f), nd g), wht re the nodl curves for the modes d mn / )R mn r) cos cλ mn t cos mθ nd f mn / )R mn r) cos cλ mn t sin mθ? 6. The initil boundry vlue problem for smll horizontl displcements of suspended cble when grvity is the only force cting on the cble is y t = g x y ), <x<l, t>, x x yl, t) =, t >, yx, ) = fx), <x<l, y t x, ) = hx), <x<l. See Exercise 6 in Section.3.) ) Show tht when new independent vrible z = 4x/g is introduced, yz,t) must stisfy y t = z ym,t) =, t >, yz,) = fgz /4), y t z,) = hgz /4), z y ), <z<m, t>, z z <z<m, <z<m, where M = 4L/g. b) Solve this problem by seprtion of vribles, nd hence find yx, t). 7. Multidimensionl eigenfunctions for problem 9. re solutions of the two-dimensionl eigenvlue problem W r + r W r + W r θ + λ W =, <r<, <θ, W, θ) =, <θ.

SECTION 9. 367 ) Find eigenfunctions normlized with respect to the unit weight function over the circle r ). b) Use the eigenfunctions in prt ) to solve problem 9.. Potentil, Stedy-stte Het Conduction, Sttic Deflections of Mem- Prt C brnes 8. ) Solve the following boundry vlue problem ssocited with the Helmholtz eqution on circle V + k V =, <r<, <θ k > constnt) V, θ) =fθ), <θ. b) Is V,θ) the verge vlue of fθ) onr =? c) Wht is the solution when fθ) =? 9. Solve the following problem for potentil in cylinder V r + V r r + V =, <r<, <z<l, z V, z) =, <z<l, V r, ) =, <r<, V r, L) =fr), <r<. 3. Find the potentil inside cylinder of length L nd rdius when potentil on the curved surfce is zero nd potentils on the flt ends re nonzero. 3. ) Find the stedy-stte temperture in cylinder of rdius nd length L if the end z = is mintined t temperture fr), the end z = L is kept t temperture zero, nd het is trnsferred on r = to medium t temperture zero ccording to Newton s lw of cooling. b) Simplify the solution when fr) =U, constnt. 3. The temperture in semi-infinite cylinder <r<, z> is in stedy-stte sitution. Find the temperture if the cylindricl wll is t temperture zero nd the temperture of the bse z =isfr). 33. Repet Exercise 3 if the cylindricl wll is insulted. 34. Use seprtion of vribles to find the potentil inside sphere of rdius when the potentil on the sphere is function fφ) ofφ only. Does the solution for Exmple 9.4 specilize to this result? Wht is the solution when fφ) is constnt function? 35. Show tht if the potentil on the surfce of sphere is function fθ) ofθ only, the potentil interior to the sphere is still function of r, φ, nd θ. 36. Find the potentil interior to sphere of rdius when the potentil must stisfy Neumnn condition on the sphere, V, φ, θ) = fφ, θ), φ, <θ. r 37. Find the potentil interior to sphere of rdius when the potentil must stisfy Robin condition on the sphere,

368 SECTION 9. V, φ, θ) l + hv, φ, θ) =fφ, θ), φ, <θ. r 38. Find the stedy-stte temperture inside hemisphere r, z when temperture on z = is zero nd tht on r = is function of φ only. Hint: See Exercise 5 in Section 8.6.) Simplify the solution when fφ) is constnt function. 39. Repet Exercise 38 if the bse of the hemisphere is insulted. Hint: See Exercise 6 in Section 8.6.) 4. Find the bounded potentil outside the hemisphere r, z when potentil on z =is zero nd tht on r = is function of φ only. Hint: See the results of Exercise 5 in Section 8.6.) 4. Find the potentil interior to sphere of rdius when the potentil on the upper hlf is constnt V nd the potentil on the lower hlf is zero. 4. Use the result of Exercise 4 to find the potentil inside sphere of rdius when potentils on the top nd bottom hlves re constnt vlues V nd V, respectively. 43. Find the potentil in the region between two concentric spheres when the potentil on ech sphere is ) constnt; b) function of φ only nd show tht the solution reduces to tht in prt ) when the functions re constnt; c) function of φ nd θ nd show tht the solution reduces to tht in prt b) when the functions depend only on φ. 44. ) Show tht the negtive of Poisson s integrl formul 9.4 is the solution to Lplce s eqution exterior to the sphere r = if V r,φ,θ) is required to vnish t infinity. b) Show tht if V r,φ,θ) is the solution to the interior problem, then /r)v /r, φ, θ) is the solution to the exterior problem. Do this using the result in prt ), nd lso by checking tht the function stisfies the boundry vlue problem. 45. ) Wht is the potentil interior to sphere of rdius when its vlue on the sphere is constnt V? b) Determine the potentil exterior to sphere of rdius when its vlue on the sphere is constnt V, nd the potentil must vnish t infinity. Do this in two wys, using seprtion of vribles, nd the result of Exercise 44. 46. Wht is the potentil exterior to sphere of rdius when the potentil must vnish t infinity nd stisfy Neumnn condition on the sphere, V, φ, θ) = fφ, θ), φ, <θ. r 47. Wht is the potentil exterior to sphere of rdius when the potentil must vnish t infinity nd stisfy Robin condition on the sphere, V, φ, θ) l + hv, φ, θ) =fφ, θ), φ, <θ. r 48. Consider the following boundry vlue problem for stedy-stte temperture inside cylinder of length L nd rdius when the temperture of ech end is zero:

SECTION 9. 369 U r + r r + U =, <r<, <z<l, z Ur, ) =, <r<, Ur, L) =, <r<, U, z) =fz), <z<l. ) Verify tht seprtion of vribles Ur, z) = Rr)Zz) leds to Sturm-Liouville system in Zz) nd the following differentil eqution in Rr): r d R dr + dr dr λ rr =, <r<. b) Show tht the chnge of vrible x = λr leds to Bessel s modified differentil eqution of order zero, x d R dx + dr xr =. dx See Exercise in Section 8.3.) c) Find functions R n r) corresponding to eigenvlues λ n, nd use superposition to solve the boundry vlue problem. d) Simplify the solution in prt c) in the cse tht fz) is constnt vlue U. 49. Solve the boundry vlue problem in Exercise 48 if the ends of the cylinder re insulted. 5. ) A chrge Q is distributed uniformly round thin ring of rdius in the z xy-plne with centre t the origin figure to the right). Show tht potentil t every point on the z-xis due to this chrge is Q V = 4ɛ + r. b) The potentil t other points in x y spce must be independent of the sphericl coordinte θ. Show tht V r, φ) must be of the form V r, φ) = n= A n r n + B ) n n + P r n+ n cos φ). Wht does this result predict for potentil t points on the positive z-xis? c) Equte expressions from prts ) nd b) for V on the positive z-xis nd expnd / + r in powers of r/ nd /r to find V r, φ). 5. Repet Exercise 5 in the cse tht chrge Q is distributed uniformly over disc of rdius in the xy-plne with centre t the origin.