ECE 6341 Sprig 16 Prof. David R. Jackso ECE Dept. Notes 8 1
Cylidrical Wave Fuctios Helmholtz equatio: ψ + k ψ = ψ ρφ,, z = A or F ( ) z z ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z + + + + k ψ = Separatio of variables: ψ ρφ,, z = R ρ Φ( φ) Z( z) let ( ) ( ) Substitute ito previous equatio ad divide by ψ.
Cylidrical Wave Fuctios (cot.) ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z + + + + k ψ = ( ) ( ) ψ ρφ,, z = R ρ Φ( φ) Z( z) R 1 R 1 Φ Z Φ Z + Φ Z + RZ + RΦ k R Z + Φ = ρ ρ ρ ρ φ z let R 1 R 1 Φ Z R ρ R ρ Φ Z Divide by ψ + + + + = k 3
Cylidrical Wave Fuctios (cot.) R 1 R 1 Φ Z R ρ R ρ Φ Z + + + + = k (1) or Z R 1 R 1 Φ = k Z R ρ R ρ Φ f( z ) g( ρφ, ) Hece, f (z) = costat = - k z 4
Cylidrical Wave Fuctios (cot.) Hece Z = k Z z { ± jk } z z Z( z) = hkz ( ) = e,si( kz),cos( kz) z z z Next, to isolate the φ -depedet term, multiply Eq. (1) by ρ : ρ R 1 R 1 Φ + + + k + k ρ = R ρ R ρ Φ z 5
Cylidrical Wave Fuctios (cot.) Hece Φ 1 R R = ρ k + kz Φ ρ R R () f ( φ ) g( ρ) so Hece, Φ = costat = Φ { e ± jφ } Φ= h( φ) =,si( φ),cos( φ) 6
Cylidrical Wave Fuctios (cot.) From Eq. () we ow have 1 R R = ρ k + kz ρ R R The ext goal is to solve this equatio for R(ρ). First, multiply by R ad collect terms: ( ) z ρ R + ρr + ρ k k R R = 7
Cylidrical Wave Fuctios (cot.) Defie k k ρ k z The, ( ) ρ R + ρr + kρρ R = Next, defie x = kρρ yx ( ) = R( ρ) Note that dr dy dx R ( ρ) = = = y( xk ) dρ dx dρ ρ ad R ( p) = y ( x) k ρ 8
Cylidrical Wave Fuctios (cot.) The we have + + = x y xy x y Bessel equatio of order Two idepedet solutios: J ( x), Y ( x) Hece Therefore y( x) = AJ ( x) + BY ( x) { } ρ ρ R( ρ) = J ( k ρ), Y ( k ρ) 9
Cylidrical Wave Fuctios (cot.) Summary ψ ρφ,, z = R ρ Φ( φ) Z( z) ( ) ( ) { ± jk } z z z z Z( z) = e,si( kz),cos( kz) Φ= { ± j e φ,si( φ),cos( φ) } { } ρ ρ R( ρ) = J ( k ρ), Y ( k ρ) k = k ρ k z 1
Refereces for Bessel Fuctios M. R. Spiegel, Schaum s Outlie Mathematical Hadbook, McGraw-Hill, 1968. M. Abramowitz ad I. E. Stegu, Hadbook of Mathematical Fuctios with Formulas, Graphs, ad Mathematical Tables, Natioal Bureau of Stadards, Govermet Pritig Office, Teth Pritig, 197. N. N. Lebedev, Special Fuctios & Their Applicatios, Dover Publicatios, New York, 197. 11
Properties of Bessel Fuctios 1 1.8 = () J is fiite.6.4 = 1 = J (x) J( x) J1( x). J(, x)..4.43.6 1 3 4 5 6 7 8 9 1 x 1 1
Bessel Fuctios (cot.).51 1 1 = = 1 = Y Y( (x) Y1( x) Y(, x) 3 () Y is ifiite 4 5 6 6.6 7 1 3 4 5 6 7 8 9 1 x 1 13
Bessel Fuctios (cot.) Small-Argumet Properties (x ): J ( x) Ax, 1,,... J ( x) Ax, = 1,,... Y ( x) Bx, Y x C x ( ) l ( ), = For order zero, the Bessel fuctio of the secod kid Y behaves as l(x) rather tha algebraically. 14
Bessel Fuctios (cot.) No-Iteger Order: yx ( ) = J( x), J ( x) { } Two liearly idepedet solutios Note: Bessel equatio is uchaged by J ( x) is a always a valid solutio These are liearly idepedet whe is ot a iteger. J ( x) Ax, J ( x) A x as x 1 15
Bessel Fuctios (cot.) Symmetry property = J ( x) = ( 1) J ( x) Y ( x) = ( 1) Y ( x) The fuctios J ad J - are o loger liearly idepedet. 16
Bessel Fuctios (cot.) Frobeius solutio : J ( x) ( 1) x = ( ) k = k! + k! k z! =Γ z+ 1 ( ) + k This is valid for ay (icludig = ). Ferdiad Georg Frobeius (October 6, 1849 August 3, 1917) was a Germa mathematicia, best kow for his cotributios to the theory of differetial equatios ad to group theory (Wikipedia). 17
Bessel Fuctios (cot.) Defiitio of Y J( x)cos( π ) J ( x) Y ( x) si( π ) -, -1,, 1, (This defiitio gives a ice asymptotic behavior as x.) For iteger order: Y ( ) lim ( ) x = Y x 18
Bessel Fuctios (cot.) From the limitig defiitio, we have, as : ( k ) 1 k x 1! x 1 Y( x) = J( x) l γ π + π k! k = 1 k 1 x ( 1) ( k) ( k) π Φ +Φ + k = k! + k! ( ) k+ (Schaum s Outlie Mathematical Hadbook, Eq. (4.9)) where 1 1 1 Φ ( p) = 1 + + + + ( p> ) 3 p ( ) Φ = 19
Example Bessel Fuctios (cot.) Prove: J ( x) = ( 1) J ( x) J J ( x) ( x) ( 1) ( ) ( 1) ( ) k + k x = k= k! + k! k + k x = k= k! + k! Deote: k = + k J ( x) ( 1) ( k ) ( k ) + k + k x = k = +!!
Bessel Fuctios (cot.) Example (cot.) J ( x) ( 1) ( k) ( k) + k + k x = k=! +! Plot of Γ fuctio (from Wikipedia) Note that! =Γ + 1 =, = 1,, 3... ( ) 1
Bessel Fuctios (cot.) Example (cot.) Hece J ( x) ( 1) ( k) ( k) + k + k x = k=! +! J ( x) ( 1) ( k) ( k) + k + k x = k=! +!
Bessel Fuctios (cot.) Example (cot.) Hece, we have J ( x) ( 1) ( ) k + k x = k= k! + k! J ( 1) ( k) ( k) k + k x ( x) = ( 1) k=! +! so J ( x) = ( 1) J ( x) 3
Bessel Fuctios (cot.) J as x From the Frobeius solutio ad the symmetry property, we have that 1 J ( x) ~ x 1,, 3,...! 1 J( x) ~ x,1,,... =! J ( x) = ( 1) J ( x) 1 J ( x) ~ ( 1) x,1,,... =! 4
Bessel Fuctios (cot.) Y as x x Y ( x) ~ l γ, γ.577156 π + = 1 Y ( x) ~ ( 1)!, > π x Y cosπ 1 x ( x)~, < si π! ( + 1) / 1 Y ( x) ~ ( 1)! π x = 1,,3,... Y ( x) = ( 1) Y ( x) 1 Y ( x) ~ ( 1 ) ( 1)! π x = 1,,3,... 5
Bessel Fuctios (cot.) Asymptotic Formulas x J ( x) ~ cos x π x Y ( x) ~ si x π x π π 4 π π 4 6
Hakel Fuctios ( 1) H ( x) J ( x) + jy ( x) ( ) H ( x) J ( x) jy ( x) As x H x e π x π π ( ) ( 1 + j x ) 4 ( )~ H x e π x π π ( ) ( j x ) 4 ( )~ Icomig wave Outgoig wave These are valid for arbitrary order. 7
Fields I Cylidrical Coordiates 1 1 E = ( A) F jωµε ε 1 1 H = A+ F µ jωµε ( ) A= za ˆ or F= zf ˆ z z We expad the curls i cylidrical coordiates to get the followig results. 8
TM z Fields TM z : ψ = A z Ez 1 = + jωµε z k ψ H z = E ρ = 1 ψ jωµε ρ z H ρ = 1 ψ µρ φ E φ = 1 ψ jωµερ φ z H φ 1 ψ = µ ρ 9
TE z Fields TE z : ψ = F z E ρ 1 ψ = ερ φ H ρ = 1 ψ jωµε ρ z E φ = 1 ψ ε ρ H φ = 1 ψ jωµερ φ z E z = Hz 1 = + k jωµε z ψ 3