ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1
Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x
Spherical Wave Fuctios (cot.) I spherical coordiates we have ψ = ψ ψ ψ ψ r siθ 1 1 1 = + + r r r r siθ θ θ r si θ φ Hece we have 1 1 1 r ψ ψ ψ siθ k 0 (1) + ψ + + = r r r r siθ θ θ r si θ φ Usig separatio of variables, let ( ) ( ) ψ = Rr ( ) H θ Φ φ () 3
Spherical Wave Fuctios (cot.) After substitutig Eq. () ito Eq. (1), divide by ψ : 1 R 1 H r siθ + rr r r rsiθh θ θ 1 1 Φ + + k = r si θ Φ φ 0 At this poit, we caot yet say that all of the depedece o ay give variable is oly withi oe ter. 4
Spherical Wave Fuctios (cot.) Next, ultiply by r si θ : si θ R siθ H r + siθ R r r H θ θ 1 Φ + + kr θ = Φ φ si 0 (3) Sice the uderlied ter is the oly oe which depeds o, It ust be equal to a costat, φ Hece, set 1 Φ φ Φ = (4) 5
Spherical Wave Fuctios (cot.) Hece, cos φ Φ ( φ ) = (5) si φ I geeral, w (ot a iteger). Now divide Eq. (3) by si θ ad use Eq. (4), to obtai 1 R 1 H r + siθ R r r Hsiθ θ θ si + kr = θ 0 (6) 6
Spherical Wave Fuctios (cot.) The uderlied ters are the oly oes that ivolve θ ow. This tie, the separatio costat is custoarily chose as ( 1) + 1 d siθ dh = ( + 1 ) (7) Hsiθ dθ dθ si θ I geeral, (ot a iteger) To siplify this, let x = cosθ dx = siθ dθ d d = siθ dθ dx ad deote yx ( ) = H( θ ) 7
Spherical Wave Fuctios (cot.) Usig x = cosθ d d siθ dθ = dx ( ) 1/ siθ = 1 x 1 d siθ dh = + 1 Hsiθ dθ dθ si θ ( ) y 1 ( 1 x ) 1/ ( ) 1/ d ( ) 1/ ( )( ) 1/ 1 x 1 x 1 1 x y dx + ( + 1) = 0 ( 1 x ) 8
Spherical Wave Fuctios (cot.) Cacelig ters, y 1 ( 1 x ) 1/ ( ) 1/ d ( ) 1/ ( )( ) 1/ 1 x 1 x 1 1 x y dx + ( + 1) = 0 ( 1 x ) Multiplyig by y, we have d dx ( ) 1 x y + ( + 1) y= 0 (8) ( 1 x ) 9
Spherical Wave Fuctios (cot.) Eq. (8) is the associated Legedre equatio. The solutios are represeted as yx ( ) P = Q ( x) ( x) Associated Legedre fuctio of the first kid. Associated Legedre fuctio of the secod kid. = order, = degree If = 0, Eq. (8) is called the Legedre equatio, i which case yx ( ) 0 P ( x) P( x) = = 0 Q ( ) Q ( x) x Legedre fuctio of the first kid. Legedre fuctio of the secod kid. 10
Spherical Wave Fuctios (cot.) Hece: H ( θ ) P = Q (cos θ ) (cos θ ) To be as geeral as possible: w H ( θ ) P = Q w w (cos θ ) (cos θ ) 11
Spherical Wave Fuctios (cot.) Substitutig Eq. (7) ito Eq. (6) ow yields 1 d R dr r + + kr = dr dr ( ) 1 0 Next, let x = kr dx d dr = = k dr d k dx ad deote yx () = Rr () 1
Spherical Wave Fuctios (cot.) We the have 1 d R dr r + + kr = dr dr ( ) 1 0 d x = kr = k dr d dx k k ( + ) + x = 1 d x dy y dx k dx 1 0 d ( xy ) + x ( + 1) y = 0 dx 13
Spherical Wave Fuctios (cot.) or d ( xy ) + x ( + 1) y = 0 dx + + ( + 1) = 0 x y xy x y spherical Bessel equatio Solutio: b (x) Note the lower case b. 14
Spherical Wave Fuctios (cot.) ad let Hece ( ) Deote = 1/ b x x g x ( ) yx ( ) = b x ( ) 1/ 1 3/ b ( x) = x g x g 1 1 3 b ( x) = x g x g x g + x g 4 1/ 3/ 3/ 5/ 3 x g x g + x g + x g x g 4 ( ) 3/ 1/ 1/ 1/ 1/ ( ) 1/ + x + 1 x g( x) = 0 15
Spherical Wave Fuctios (cot.) Multiply by 1/ x 3 x g xg + g + ( xg g) + x ( + 1 ) g( x) = 0 4 Cobie these ters or 1 xg + xg g + xg ( + 1) g = 0 4 Use 1 1 + ( + 1) = + 4 Cobie these ters Defie γ + 1 16
Spherical Wave Fuctios (cot.) ( ) We the have γ x g + xg + x g = This is Bessel s equatio of order γ. 0 Hece g Jγ ( x) = Yγ ( x) so that b 1 J+ 1/( x) π = x Y + 1/( x) added for coveiece 17
Spherical Wave Fuctios (cot.) Defie π j( x) J+ 1/( x) x π y( x) Y+ 1/( x) x The j ( ) () ( ) kr R r = b kr = y ( ) kr 18
Suary ( ) ( ) ψ + k ψ = 0 ( φ ) ( φ ) j kr P (cos ) cos θ ψ = y kr Q (cos ) si θ π b( x) = B+ 1/( x) x I geeral, w 19
Properties of Spherical Bessel Fuctios = π b( x) = B+ 1/( x) x Bessel fuctios of half-iteger order are give by closed-for expressios. J ( x) k + k ( 1) x! ( 1) k = 0 k! ( + k)! z =Γ z+ = This becoes a closed-for expressio! = + 1/ 0
Properties of Spherical Bessel Fuctios (cot.) Exaples: J1/( x) = si x π x ( ) J 1/( x) = cos x π x ( ) ( x) si J3/ ( x) = cos x π x x ( x) ( ) cos J 3/ ( x) = + si x π x x ( ) J( x)cos( π ) J ( x) Y ( x) si( π ) 1/ 1/ ( ) Y ( x) = J x 3/ 3/ ( ) Y ( x) = J x 1
Properties of Spherical Bessel Fuctios (cot.) Start with: k ( ) ( ) Proof for = 1/ J1/( x) = si x π x ( ) k ( ) ( ) + k 1/+ k 1 x 1 x J ( x) = J1/( x) = k= 0 k! + k! k= 0 k! 1/ + k! Hece k ( 1) ( k) x x J1/( x) = k = 0 k! 1/ +! k + 1
Properties of Spherical Bessel Fuctios (cot.) Exaie the factorial expressio: Note: x = x( x ) ( ) =! 1! 1/! π / 1 + 1/! = + 1/ 1/ 3/... 3/! ( k ) ( k )( k )( k ) ( ) ( k 1/)( k 1/)( k 3/ )...( 3/ )( π /) = + 1 = + = = = ( k 1)( k 1)( k 3 )...( 3 )( π / ) + 1 4...4 ( k 1 )!( π /) ( k)( k )( k ) 1 + k 1... 1 ( k 1 )!( π /) ( k)( k )( k ) 1 + k k! ( k 1 )!( π /) k k k k 3
Properties of Spherical Bessel Fuctios (cot.) Hece ( 1) x x J1/( x) = k k = 0 ( ) ( ) 1 k + 1! π / k! k k! k k + 1 Hece, we have k ( 1) ( k + ) π x x J1/( x) = k = 0 1! k + 1 k + 1 4
Properties of Spherical Bessel Fuctios (cot.) k ( 1) ( k + ) π x x J1/( x) = k = 0 1! k + 1 k + 1 or k = 0 k ( 1) ( k + ) π x J 1/( x ) = x 1! k + 1 We the recogize that π x J x x ( ) si 1/ = ( ) 5
Properties of Legedre Fuctios Relatio to Legedre fuctios (whe w = = iteger): ( ) / d P ( x) = 1 x P( x) dx ( ) / d Q ( x) = 1 x Q( x) dx These also hold for. For w (ot a iteger) the associated Legedre fuctio is defied i ters of the hypergeoetric fuctio. 6
Properties of Legedre Fuctios (cot.) Rodriguez s forula (for = ): 1 d P x x! dx ( ) ( = 1) Legedre polyoial (a polyoial of order ) ( ) ( ) 0 P0 x = x 1 = 1 1 d P1 ( x) = ( x 1) = x dx 1 d 1 P x = x 1 = 3x 1 8 dx ( ) ( ) ( ) 7
Properties of Legedre Fuctios (cot.) P x = > Note: ( ) 0, This follows fro these two relatios: ( ( ) 1 ) / d P x = x P( x) dx 1 d ( ) ( 1) P x = x = polyoial of order! dx 8
Properties of Legedre Fuctios (cot.) w=, = ( ( ) 1 ) / d Q x = x Q( x) dx Q ( ) x = ifiite series, ot a polyoial (ay blow up) ( 1) Q ± = (see ext slide) The Q fuctios all ted to ifiity as x ± 1 θ 0, π Recall : x = cosθ 9
Properties of Legedre Fuctios (cot.) Lowest-order Q fuctios: Q 0 ( x) 1 1+ x = l 1 x Q Q 1 ( x) ( x) x 1+ x = l 1 1 x 3 x 1 1+ 3 l x = x 4 1 x 30
Properties of Legedre Fuctios (cot.) Negative idex idetities: ( ) ( ) P x P x ( + 1) = (This idetity also holds for.) ( ) π ( ) ( ) ( ) Q + x = P x + Q x ( 1) 1 31
Properties of Legedre Fuctios (cot.) P ( x) Q ( x) = (see Harrigto, Appedix E) ifiite series = ifiite series P ( x) ( ) ( ) ( ) ( ) ( ) ( ) ( ) N ( ) N 1 +! 1 x si π 1! +! 1 x = = 0!! π = + 1! N = largest iteger less tha or equal to. P P (1) = 1 ( 1) = Q ( x) ( ) cos( π ) ( ) ( π ) π P x P x = si Q ( ± 1) = Both are valid solutios, which are liearly idepedet for (see ext slide) 3
Properties of Legedre Fuctios (cot.) Proof that a valid solutio is P ( x) d dx The ( ) 1 x P ( x) + ( + 1) P ( x) 0 ( 1 x ) = Let t = x d ( 1 t )( 1 ) P ( t) + ( + 1) P ( t) 0 = dt ( 1 t ) d dx d = dt or (t x) d dx ( ) 1 x P ( x) + ( + 1) P ( x) 0 ( 1 x ) = Hece, a valid solutio is P ( x) 33
Properties of Legedre Fuctios (cot.) P ( x) ad P ( x) are two liearly idepedet solutios. Valid idepedet solutios: P Q (cos θ ) (cos θ ) or P P (cos θ ) ( cos θ ) We have a choice which set we wish to use. 34
Properties of Legedre Fuctios (cot.) = P ( x) = ( 1) P ( x) (They are liearly depedet.) I this case we ust use P Q (cos θ ) (cos θ ) 35
Properties of Legedre Fuctios (cot.) Suary of z-axis properties (x = cos (θ )) = P 1 = 1 P 1 = 1 ( ) ( ) Q ( ± 1) = Q ( ± 1) = P 1 = 1 P 1 = ( ) ( ) ( ) 36
Properties of Legedre Fuctios (cot.) z P ( ) x allowed P ( x) allowed y x x = cosθ Q ( ) ( ) x ad Q x are ot allowed o ± zaxis. 37
Properties of Legedre Fuctios (cot.) Outside or iside sphere z z x y Iside hollow coe y Oly P (x) is allowed x Both P (x) ad P (x) are allowed 38
Properties of Legedre Fuctios (cot.) z Outside coe x y Oly P (x) is allowed 39
Properties of Legedre Fuctios (cot.) z x y Oly P (x) is allowed Iside iverted coe Note: The physics is the sae as for the upright coe, but the atheatical for of the solutio is differet! 40
Properties of Legedre Fuctios (cot.) z y x Both P (x) ad P (x) are allowed Outside iverted coe 41
Properties of Legedre Fuctios (cot.) z Outside bicoe x y P (x) ad P (x) are allowed Q (x) ad Q (x) are allowed 4