Bessel Functions. y 2

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1 Bessel Functions Vibrtions of Membrne The governing eqution for the isplcement w, y, t)from the equilibrium position the plne z ) is w c w t ) [ The ssumptions re:- + y i) Tht the ction cross the element DeltS is force T S perpeniculr to S, where T T s S ii) the isplcement w of ny point of the membrne is purely trnsverse. ) iii) tht w + w y ) is smll. ] It cn be shown tht the tension is isotropic t point t point i.e. inepenent of the orienttion of S), inepenently of ii) n iii), n for equilibrium or motion from ii) it cn be shown tht T is uniform over the membrne n c T / mss/unit re. We ssume c constnt. The usul bounry conition for finite membrne) is tht w on the bounry. Simple Hrmonic Vibrtions Then w, y, t) W, y) coswt + ɛ) ) W + k W k w 3) c For circulr membrne complete or nnulr) we use plne polr coorintes r, θ Therefore r r r r r + r θ r r W r + W r θ + k W

2 This is seprble. i.e. we cn fin solutions of the form W rθ) F r)gθ) by substitution we hve r F ) + k + G F r r r r G θ Therefore Therefore G G θ constnt n 4) Gθ) A cos nθ + B sin nθ r r F ) + k r n )F 5) r r [Note tht F r ) + r r k r n )F is the self joint for?????????] r In 5) write kr not the co-orinte) F ) + n )F 6) This is Bessel s Eqution of orer n. The solution of the originl eqution W + k W must be perioic in θ of perio π, for otherwise W woul not be one vlue function of position. Therefore n is n integer. Series solution for F We ssume F m m+c where c is to be foun. ) n m+c) [m + c) n ] m+c ) n F m [m + c) n ] m+c F m n+c+ m m+c m Hence we require m [m + c) n ] m+c + m m+c m this is true if c n ) Inicil eqution c + n )

3 m m + c n ) + m m, 3,... Since, c ±n In the secon, putting c n n + ) In the thir, putting c n, m 3, 5, n + 3) + We suppose n, 3,... Then 3... For m, 4, 6,... in the thir reltion + n) n) + 4 n + ) 4 n + )n + ) ) m m m m!n + )... n + m) Hence we hve one solution tking c n) n ) m ) m m!n + )... n + m) The Bessel function J n ) is efine by tking n Γn + ) J n ) m ) n Γn + ) ) m m!γm + n + ) ) m m!n + )... n + m) ) m ) m The series converges for ll, by the rtio test. So J n ) is n integrl function of. If L n is the opertion in Bessels eqution L n y) L n y) since n ppers s n L n [J n )] for ll n Therefore L n [j n )] i.e. L n J n )] Hence J n ) is solution. J n ) ) m ) n+m m!γm n + When n is non-negtive integer Γm + n + ) is infinite from 3

4 m,,,..., n since Γz) hs poles t z,,,... Therefore Write m n + k J n ) mn ) m ) n+m m!γm n + J n ) ko ) k ) k+n i) n J n ) n + k)!k! Wronskin of J n n J n W J n J n we hve ) ) j ±n + n Therefore { [J n J n J n J n] } i.e. [J nj n J n J n] C constnt). C lim [J nj n J n J n] J n J n ) n rn + ) ) n r n + ) J n n ) n rn + ) J n n r n + ) Therefore J n J n J nj n Therefore C ) n J n J n J ±n [ + ] )!n + ) + [ + ] n Γ + n)γ m) )! n + ) + n Γ + n)γ m) [ + ] Γ n)γn) sin nπ π Therefore J n J n J nj n sin nπ π Thus J n n J n re linerly inepenent when n is not n integer. 4

5 n J n n J n re linerly epenent when n is n integer. Definition of the secon solution [Weber] Y n ) cos nπj n J n sin nπ when n tens to n integer the numertor tens to n the enomintor tens to. Hence when m is n integer we efine Y m ) lim n m Y n ) Therefore Y m ) [cos nπj } n n J n ] sin nπ n nm π sin nπj n + cos nπ J } n n n π cos πn [ π n J n) ) m ] n J n) In prticulr Y ) [ ) π n J n ) ] n J n [ ] π n J n n n n The functions of the secon kin of orer n. They re unboune t, for fro the reltion between J n n n other solution of Bessel s equtions, sy Y n ), we hve [Y n J n Y nj n ] C Y n J N Y nj n c If c then Y n n Y n cn not eist t. Hence the generl solution of Bessels equtions is either A J n ) + B J n ) n not n integer) or A J n ) + B Y n ) ll cses) A solution boune t is necessrily AJ n ) Returning to the membrne problem nm nm. Complete membrne r ). Since W must be boune t r, we hve F r) AJ n kr) n W rθ) AJ n kr) cosnθ + ε) n integer ) Since W on r. AJ n k) A is trivil therefore J n k). This is n eqution for the eigenvlues k, k,.... Hence for given n the vlues of k re given by k j n, j n,... where j n, j n,... re the 5

6 positive zeros of J n ). The llowe frequencies frequencies of norml w moes) re, w kc Therefore pi w c j n, j n,...) n,,.... Annulr Membrne b r. In this cse r is not in the physicl spce. Therefore we must tke F r) AJ n kr) + BY n kr) F ) F b) give:- AJ n k) + BY n k) AJ n kb) + BY n kb) Hence for non-trivil A, B AJ n k) AJ n kb) BY n k) BY n kb) Sketch of J J J ) ) 4 J!) + 4!) + ) ) 3 ) 5 J!! +!3! Note lso J ) J ) PICTURE The symptotic formul for J n ) is J n ) ) cos π ) π 4 + nπ Orthogonl n norml Properties of J j m ) where j, j... re the zeros of j ). We show tht J j m )J j p ) p m 6 J j m ) p m

7 the functions re orthogonl to weight function over [, b]. J α) + α J α) J β) + β J β) [ J β) J α) J α) ] J β) α β )J α)j β) [ Therefore J β) J α) J α) ]} J β) Iα β ) J α)j β) i.e. α β ) J α) J β) J α)βj β) J β)αj α) ) J β)j α) J α)βj β) If α j m β j p m p J j m )J j p ) Therefore J αj β)j α) lim α) + βj α)j β β α α β Numertor β Denomintor β βα j β) αj α) βj β β) β βα αj α) αj [ ] α) α β βj β)) + βj β) J o α) J α) + J α) [J J ] ) Therefore then α j m J j m )J j p ) J j m )δ mp It lso follows tht if f, f,... re the zeros of J ) then J j m)j j p) J j m )δ mp 7

8 Specil Cses of J α n )f) In ) tke α αn zero) β α m m,,... Then J α n )J β) α n In this put β n J ) n so 4) cn lso be obtine s follows: J α n ) αnj α n ) Therefore α n α n β J α n )J β) 3) J α n ) [ ] J α n ) n J α n ) J α n ) J α n ) α n 4) α n J α n ) Net we consier We hve Therefore { I k J α) k J α) α J α) k k k k J α) J α) } k J α){α k + k k } i) ii) Therefore by integrtion over [,] α I k k I k αj α) kj α) Therefore I I,... cn be foun in terms of J α) n J α). I n I cn be foun in terms of the Sturve function J α)qj α)j α) 8

9 Forml Fourier - Bessel Eplntions Assume tht f), efine in [,], possesses n epnsion Then f) A n J α n ) n J α m )f) A m J α n )J α m ) n A m J α m ) J α m ) A m A m J J α m )f) α m ) Initil n Bounry Problem for the Vibrting Membrne We hve, for the isplcement wr, t) in rilly symmetric vibrtions r r r w r w, t) eists n w, t) Also wr, ) fr) ) r c w t r w r, ) t r Choose - unit of length n choose unit on time so tht c. Also replce r by, giving w w t w, t) eists n w, t) w Also w, ) f), ) t Assume w, t) J α n )φ n t) n This stisfies the bounry conitions J α n )φ n t) w, t)j α n ) 9

10 J α n ) φ n t) Therefore n [ J α n ) w t) t J α n ) w ) ] J α n ) w w [ + w ] J α n ) + + w, ) f) t w.) Therefore i w[ α nj α n )] α J α n )φ n t) φt) + α nφt) J α n ) φ n t) A n cosα n t) + B n sinα n t)) w J α n ) f) φ n, φ n ) J α n ) i.e. B n A n J α n )φ n ) J α n ) φ n ) 9 J α n )f) J J α n )f) α n ) Hence we hve the series solution for w, t) w, t) J α n ) cos α n t yj J α n y)fy) y α n )

11 Alterntive proceure Solutions of the ifferentil eqution boune t re J k)[a cos kt + sin kt] This stisfies the bounry conitions w, t) if J k) thus k α, α,... The solution lso stisfies w t ) if B n. Formlly the series A n J α n ) cos α n t stisfies both bounry conitions n n the initil conitions on w t. Therefore s w ) f) thus f) A n J α n ) Therefore Emple A n J α n ) J α n )f). f) J k) J k) k rel n J k) J α n )J k) J k)α n J α n ) J α n )kj k) αn k J k) α nj α n ) αn k k gives J α n ) J α n ) α n J α n )f) J α n ) { α n k J α n ) α n k α n) α } n αn k Hence in this cse: k w, t) J α n ) cos α n t n α n k αn) J α n ) Since α n On) for lrge N, n J α n ) O ) O α n n ) Then the coefficient of J α n ) cosα n t) is O n 5

12 Solution of liner ifferentil eqution by efinite integrl or contour integrl) Preliminry Remrks Consier the ifferentil eqution where Seek solution y φd)y + ψd)y φp) p n + p n n ψp) b p m + b p m b m b e p Kp)ph.inot y C e p Kp)p where Kp) is to be foun n, b or C) re lso to be foun n re inepenent of.) Then Hence φd)y ψd)y then φd)y b b b b b φd)y + ψd)y [e p φp)kp)] b + b φd)e p Kp)p φp)e p Kp)p ψp)e p Kp)p e p φp)kp)p p ep )φp)kp)p b [e p φp)kp)] b e p p φp)kp))p i) Choose Kp) so tht the integrn is zero. i.e. e p { ψp)kp) p φp)kp)p } p ψp) {φp)kp)} ψp)kp) p φp) φp)kp)

13 therefore φp)kp) C ep { } p ψq) φq) q Note: if ll the zeros of φ re simple ψ n φ p r ψ q q r φ logp p r) r Therefore K r n p p r ) r ii) when Kp) is known we choose n b or C) so tht [e p Kp)φp)] b Consier Bessel s eqution on orer n Substitute y n z ) + n y { } y + y + y n y y { n } + n ) { y n ] + n z n ) + n + n z { } n + n + ) + n ) { } + n y n + n + ) + z ) { n+ + n + ) } z 3

14 Hence y stisfies Bessel s eqution if z stisfies { + Consier solution for z of the form b e it Kt)t Then { ) + ) + n + ) } b e it Kt)t + n + ) } z b { t ) + n + )it}kt)e it t [ t e it Kt) i ] b + b [ e it { } ] t Kt) + itn + )Kt) t t i Hence choose Kt) so tht t t )Kt) n )tkt) Kt) c t ) n hence solution of Bessel s eqution is if n b re chosen so tht b y n e it t ) n t [ t ) n+ e it ] b Suppose n > n lso is rel n positive [There is no ifficulty if z is comple] Amissible Pirs of limits i), +) ii), + i ) iii) +, + + i ) PICTURE 4

15 These integrls must be linerly epenent since tht re solutions of secon orer eqution. With proper specifiction of t ) n the reltion is ii) - iii) Consier e it t ) n t roun the contour shown. PICTURE We choose tht brnch of t) n which is rel n positive on AA. i.e. rg t), rg + t) on AA. AS t psses from to b roun AB rg t) ecreses by π ; s t psses from A to B roun A B rg + t) increses by π. Since the integrn eit t ) n is now one-vlue n regulr on n insie the countour, by Cuchy s Theorem we hve We show ) lim h b) lim ɛ {A A} {C C} {A B } We shll then hve i.e. {A A} {B C } {CB} {C C} {A B } {BA} lim ɛ {BA} e it t ) n i +i t e it t ) n e it t ) n i) ii) iii) since the limits if the three integrls eist s ɛ if n > n s h. on CC, t P X P X C X CX h + 4 e it e iu+ih) e +h Therefore CC e h + 4) n s h on AB we hve t ɛe tθ Therefore t n ɛ n e it t ) n is boune in the neighbourhoo of t with boun M sy. Therefore e it t n Mɛ n Thus Mɛ n π AB ɛmπ en s ɛ Similrly s ɛ n > ) B A 5

16 Hence we hve the following solutions of Bessel s eqution i) n e it t ) n t +i ii) n e it t ) n t ++i iii) n e it t ) n t + Series Epnsions of i) i) n i t ) n the series for ll, t bsolutely n uniformly. i) n i) m m! t m t ) n t if m is o it) m t m! t m t ) n t t m t ) n t t m t ) n t u m u) n u u u m u) n u Γn + )Γm + ) Γm + n + ) i) n i) m γn + )Γm + ) m)! Γm + n + ) Γm + ) m)! Γ ) 3 m 3 m )m Γ ) m m! i) Γn + )Γ ) n ) ) m m m!γm + n + ) 6

17 n Γn + ) )Γ J n ) J n ) n Γn + )Γ ) i) Hnbel Functions of orer n Bessel functions of the thir kin) Definition Then we lso efine H ) n ) H ) n ) ) n Γ )Γn + ) iii) ) n Γ )Γn + ) ii) J n ) H) n ) + H n ) )) Y n ) i H) n ) H n ) )) Alterntive integrl representtion of H ) n ) n H ) n ) In the integrl representtion for H n ) ), rg t) π Therefore we write t) ηe πi iη. Where η goes from to through rel vlues s t goes from to + i. Thus Also t) n e πi n ) + t) t) ηe πi + iη ) + t) n n + in )n t) n n η n e πi n ) + iη ) n e it e i+iη) e i e η t e πi η e it t ) n t n e i nπ + pi 4 ) e η η n H n ) ) n nπ i e π 4 ) Γ )Γn + ) 7 e η η n + iη ) n η + iη ) n η I)

18 In the integrl for H n ) ) rgt + ) π πi Therefore t + ) ηe iη where η goes from to s t goes from - to + + i we get similrly n ) n nπ i e π 4 ) Γ )Γn + ) H ) e η η n iη ) n η I) [Note: when is rel n positive, H n ) ), H n ) ) re comple conjugtes. Therefore [H) n ) + H n ) )] is rel, so is i [H) n ) H n ) )].] Finlly substitute η u in both integrls. For rel n positive u goes from to s η goes from to H n ) ) ) H n ) ) ) Asymptotic Epnsions of e i nπ π 4 ) Γ ) Γn + ) e u u n nπ π 4 ) e i Γ ) Γn + ) We consier the cse n i.e. e u u We pply Tylor s formul ft) n r to the function t) n t) r the lst term is writing t 3 3 f r) )t r r! + this gives r t r + r! e u u n e u u n ± iu ) u + iu ) n u iu ) n u ± iu ) n u t t s) n f n) s)s n!) 3 n t t s) n s) n s n )! n t n v) n tv) n v n )! u i u ) i n r 3 n r! 8 u n ) u i) + r n n

19 ) u 3 r n If is rel n positive n n )! u r v) n u ) n i) r i v v u i v + u v 4 ) Therefore Hence u r n ) 3 n n! 3 n n! u n v) n v ) n u n ) n Also e u u Therefore u ) n u i r R n ) 3 r r! e u u rn u e u y r i) r u + Rn ) ) u ) e u u r u Γr + Γ 3 r R n ) e u u n u r n u n! ) n e u u [ u ) n u Γ i ) r e u u n u [ 3 r ] r! i) r + R n Where R n [ 3 n ] n! ) lim n n There the series is the symptotic epnsion of the left hn sie. In fct Rn n )) 9

20 Divergence of the Infinite series The D Alernbert rtio is n + ) n i n + ) n which tens to infity for n. Asymptotic Epnsion of H ) ) H ) ) [ ) H ) 3 ) e i π 4 ) r π r r! [ ) H ) 3 ) e i π 4 ) r π r r! where the reminer fter the term in Write A) [ Γ ) e u u u i B) [ Γ ) e u u u i ) H ) ) e i π 4 ) [A) + ib)] π H ) ) π J ) ) e i π 4 ) [A) ib)] ] ] i) r i) r hs mouls term in n n ) u + ) u H ) ) + H ) ) ) e u u e u u + u i + u i ) [ A) cos π ) B) sin π )] π 4 4 ) u ] ) ] u Y ) i H ) ) H ) ) ] ) [ A) sin π ) B) cos π )] π 4 4 The generl Bessel function of zero orer is A J ) + B Y ) Ccos J ) + sin Y ))

21 From the efinitions of A) n B) [ 3 A) r ] ) r r)! ) r B) r r [ 3 r + ] ) r+ r + )! ) r+ Zeros of Bessel Function of zero orer The zeros re given by cot π 4 ) A) + O Therefore α) B) A) B) 8 + O 3 A) B) ) 8 + O 3 Therefore for lrge, the zeros re pproimtely given by cot π α) 4 i.e. π 4 α k + ) π k lrge integer. α + k + 3 ) π 4 ) Asymptotic Epnsions of H ) ) & H ) ) ) H ) ) e i nπ π 4 ) π H ) ) π where, n). ) e i nπ π 4 ) ) m m, n) i) m ) m m, n) i) m m, n) 4n )4n 3 ) 4n m ) ) m m! There epnsion re only useful when >> n Bessel Functions of orer k + ) k,, We hve ) ) n+m m J n ) m!γn + m + ) ) ) J ) sin J ) cos π π

22 J ) ) ) m m m m!γm + 3 ) m m!γm + 3 ) m m!γ 3 )3 5 m + Γ 3 ) 4 m3 5 m + ) m + )!Γ 3 ) m + )! Γ ) Similrly J ) ) π ) π Γ ) sin sin ) J ) cos π ) m m m + )! H ) H ) ie i π ) k+ ) H ) ) ie i π +i ) k+ Γ Γk + ) ) k+ Γ )k! + ) k+ Γ )k! + e {Polynimil in, egree k} e it t ) k t ) +i e it t ) e it The functions H ) ), H ) ) re clle sphericl Bessel functions. They k+ k+ rise in solution of the wve eqution in sphericl Polr coorintes.

23 Rilly Progressive Wves in two imensions We h for the membrne w c w t + y n foun solutions in the cse of ril symmetry) w [AJ kr) + BY kr)] coswt + ɛ) k w c ssuming the form w fr)e iwt rel prts to be tken eventully) we fin similr form w [A H ) kr) + A H ) kr)]e iwt since ) H ) kr) e ikr π 4 ) s r πkr ) H ) kr) e ikr π 4 ) s r πkr we get ) H ) kr)e iwt e iwt+ r c ) π 4 ) πkr ) H ) kr)e iwt e iwt r c ) π 4 ) πkr The first repents wve converging to the origin with velocity c, the secon wne iverging from the origin with velocity c. 3