### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Class Note for ECE 6341 with Professor Jackson at UH 2

### View Full Document

## 16

## 0

## Popular in Course

## Popular in Department

This 39 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

## Reviews for Class Note for ECE 6341 with Professor Jackson at UH 2

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/06/15

Spherical Wave Funotons Consider solving Vzw kzw O in spherical coordinates Spherical Wave Functions cont In spherical coordinates we have Vzw VVw 2 12 a r2 awj 2 a sin awj 2 12 8 r 8r 8r r s1n 986 86 r sun 9 69 Hence we have 2 g am 21 a smearx 2 12 a r 6r 6r r sm686 86 r sm 6 6 Using separation of variables let w RrH6 2 Spherical Wave Functions cont After substituting Eq2 into Eq 1 divide by 11 1 a 28R 1 a 8H 2 r 2 S1116 r R 8r 8r r stH 819 819 1 1 ach r2 sin2 19 CD 8 k20 At this point we cannot yet say that all of the dependence on a given variable is within only one term Spherical Wave Functions cont Next multiply by r2 sin2 6 sin2 9 8 28R sin6 8 8H r sm R 8r 8r H 86 86 1 821 c1gt M2 18 sinz 6 0 3 Since the underlined term is the only one which depends on It must be equal to a constant Hence set i azq m2 4 CD 5 Spherical Wave Functions cont Hence cos m CD 5 m sin m In general r not an integer Now divide Eq 3 by sin2 9 and use Eq 4 to obtain 1 a 2 6R 1 a 6H r sma R 6r 6r Hsinlt9 66 66 2 m k2 20 6 sin26 Spherical Wave Functions cont The underlined terms are the only ones that involve 6 now This time the separation constant is customarily chosen as n n 1 2 1 isini9d Hj m2 nn1 7 P131116 d6 d6 5111 6 In general not an integer To simplify this let dx sini9di9 i sini91 d6 dx and denote yx 1 109 Spherical Wave Functions cont 1 Since sin 6 1 x2 2 we have 1 d dH m2 31119 2 nn1 P131116 d6 d6 sm 6 nn10 I xz Spherical Wave Functions cont Canceling terms 1 x221 x2i11x2 y nnl0 1362 Multiplying by y we have 8 Spherical Wave Functions cont Eq 8 is the associated Legendre equation The solutions are represented as 131mm y x Q5100 If m 0 Eq 8 is called the Legendre equation in which case P300 an x y Q2ltxgt Qnx Spherical Wave Functions cont Hence Relation to Legendre functions when w m integer For m gt w not an integer the associated Legendre function is defined in terms of the hypergeometric function 11 Spherical Wave Functions cont Rodriguez s formula for z n Legendre polynomial a polynomial of order n Spherical Wave Functions cont Note This follows from these two relations Pro 1 x2 PM 1 d Pquot x 2 dxquot x2 1n polynomial of order n Spherical Wave Functions cont Lowestorder Q functions 1 1 x Q0 x Z 5111 The Q functions all tend to infinity as l x x gti1 x 1x x 2 111 1 gt 9 09 Q1 2 1x 3x2 1 1x 3x 1n Q2 4 l x 2 Spherical Wave Functions cont To be as general as possible I l U m gtw Spherical Wave Functions cont Substituting Eq 7 into Eq 6 yields ii rzd R nn1k2 20 Rdr dr Nextet dxzkdr i 1 dr dx and denote yx Rr Spherical Wave Functions cont We then have 1d 261 Rdr r j nn1k2r2 0 dr Spherical Wave Functions cont or spherical Bessel equation Spherical Wave Functions cont Denote and let 9 X x 28100 1 2 3 b39xx 2gnquot x 2gn39Zx 2g 2 1 3 1 1 1 ngn quot 9628 41x zgn2ngn39 x 28 1 x2 nn1x3gnx 0 Hence Spherical Wave Functions cont 1 M ultiply by X 4 A combine these terms 1 or ngnquotxgn39 Zgn x2gn nn1gn 0 Use lnn1j nl2 4 2 ngn quot xgn 39ign ngn 39 gn x2 nn1gnx O Spherical Wave Functions cont We then have ngn quot xgn 39 x2 j2gn 0 This is Bessel s equation of order y 506 Hence gn Y x 7 so that b iJn12xF n J Yn12x 2 added for convenience Spherical Wave Functions cont De ne Then Summary V2wk2w20 Mk1quot yn 0 H J Pn cos 9 Qnmltcosm I cosm sinm J bn x 2 g Bn12 x BTW Q3106 In general Mg l xz m 1 x23 dm dxm dm dxm PM QnOC m gtw I l gtU Properties of Spherical Bessel Functions bnx iBnHZCX V2x Bessel functions of halfinteger order are given by closedform expressions JUx 2 016 gm z rz1 k0 Uk 2 l becomes a simple function Properties of Spherical Bessel Functions cont Examples Jl2 x i sin x xx J12 x 1 cos x xx W J x 1 cos x sin 32 xx x J U x cosu7z J U x Y x E sinu7z q Y12 x J 12x 13205 J 32 x Properties of Spherical Bessel Functions cont Proof for 1 12 00 ijlZ k J12xkk12k 2 Hence x 00 1k i 2k1 EJ12Xkr12kr 2 Properties of Spherical Bessel Functions cont Examine the factorial expression k12k12k 12k 3232 i k12k 12k 3232 2k12k 12k 33 1T 2 NIH 2k1JZ 2k2k 22k 44 2k1T 2k 1kk 1k 22 2k1ZGT 2k 1k Properties of Spherical Bessel Functions cont x 00 1k i 2k1 Hence JEJl2x k If 2 Simplifying we have k39 22k lk39 Properties of Spherical Bessel Functions cont Properties of Legendre Functions 13rltx1 x2 dm m Pn x m integer dx Rodriguez s formula 1 d 12xn n 2 nldx x2 1n polynomial of order n always finite Properties of Legendre Functions QZ x1x2 Qn x m integer Q x infinite series may blow up For Q x the field blows up on the i2 axis xzcos 9 x1 1 gt 192071quot Spherical Wave Functions cont Lowestorder Q functions Q0 x ln1 x l x Q2 x 3x2 11n1x3x Spherical Wave Functions cont Negative index P n1 X P x This Identity also holds for u n Q n1x 7T1 1100 Q X Properties of Legendre Functions cont see Harrington Appendix E P x infinite series Q0 X infinite series P006 lmumll xjmSinEU7r m l Umul1 xjm 2 1 m2 2 N largest integer less than or equal to o QM5Pultxgtcosltmgt PUlt xgt 2 sin U7Z39 linearly independent for u 72 n Properties of Legendre Funct39ons cont Summary of zaxis properties Properties of Legendre Functions oont Z Pquot x allowed PU x allowed y Qnx and Qux are not allowed on 2 axis Properties of Legendre Functions cont POW and PUx are two linearly independent solutions Note the next slide proves that PU x is always a valid solution Valid independent solutions Properties of Legendre Functions cont To see this use the Legendre equation 1 x2y39uuly20 yxPUx d d Let yxft t x 3 Then 6 1t2 1f39uulf 0 I ft P00 Hence a valid solution is PU x Properties of Legendre Functions cont Pn x 1 13406 In this case we must use ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 20

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "I made $350 in just two days after posting my first study guide."

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.