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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 counterpropagating 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 ) ....
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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.
 Spring '10
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