6soln6 - Physics 315: Oscillations and Waves Homework 6:...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Physics 315: Oscillations and Waves Homework 6: Solutions 1. Making use of the identity cos( ) cos cos + sin sin , we can write the traveling wave solution ( x, t ) = A cos( k x t ) (1) in the form ( x, t ) = A cos( k x ) cos( t ) + A sin( k x ) sin( t ) . (2) But, this is just the sum of two standing wave solutions. Likewise, making use of the identity cos cos (1 / 2) [cos( )+cos( + )], we can write the standing wave solution ( x, t ) = A cos( k x ) cos( t ) (3) in the form ( x, t ) = ( A/ 2) cos( k x t ) + ( A/ 2) cos( k x + t ) . (4) But, this is just the sum of two counter-propagating traveling wave solu- tions. Finally, making use of the identities cos( ) cos cos + sin sin and cos( + ) cos cos sin sin , we can write the following super- position of traveling waves, ( x, t ) = A cos( k x t ) + A R cos( k x + t ) , (5) in the form ( x, t ) = A cos( k x ) cos( t ) + A sin( k x ) sin( t ) + A R cos( k x ) cos( t ) A R sin( k x ) sin( t ) (6) = A (1 + R ) cos( k x ) cos( t ) + A (1 R ) sin( k x ) sin( t ) . Of course, this is just a superposition of two standing waves. 2. If a transmission line of characteristic impedance Z carries a current I ( x, t ) = I i cos( k x t ) + I r cos( k x + t ) , (7) then the corresponding voltage is [see Eq. (7.104) in lecture notes] V ( x, t ) = I i Z cos( k x t ) Z I r cos( k x + t ) . (8) If the line is open circuited at x = 0 then this implies that I (0 , t ) = 0 (since current cannot flow through an infinite resistance). Hence, it follows from (7) that I (0 , t ) = I i cos( t ) + I r cos( t ) = 0 , (9) which implies that I r = I i . (10) Hence, the current takes the form I ( x, t ) = I i [cos( k x t ) cos( k x + t )] = 2 I i sin( k x ) sin( t ) , (11) where use has been made of a trigonometric identity. Likewise, the voltage is written V ( x, t ) = I i Z [cos( k x t ) + cos( k x + t )] = 2 I i Z cos( k x ) cos( t ) ....
View Full Document

This note was uploaded on 11/02/2010 for the course PHY 59372 taught by Professor Fitzpatrick during the Spring '10 term at University of Texas at Austin.

Page1 / 7

6soln6 - Physics 315: Oscillations and Waves Homework 6:...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online