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

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Physics 315: Oscillations and Waves Homework 6: Solutions 1. Making use of the identity cos( α β ) cos α cos β + sin α sin β , we can write the traveling wave solution ψ ( x,t ) = A cos( kx ωt ) (1) in the form ψ ( x,t ) = A cos( kx ) cos( ωt ) + A sin( kx ) sin( ωt ) . (2) But, this is just the sum of two standing wave solutions. Likewise, making use of the identity cos α cos β (1 / 2) [cos( α β )+cos( α + β )], we can write the standing wave solution ψ ( x,t ) = A cos( kx ) cos( ωt ) (3) in the form ψ ( x,t ) = ( A/ 2) cos( kx ωt ) + ( A/ 2) cos( kx + ωt ) . (4) But, this is just the sum of two counter-propagating traveling wave solu- tions. Finally, making use of the identities cos( α β ) cos α cos β + sin α sin β and cos( α + β ) cos α cos β sin α sin β , we can write the following super- position of traveling waves, ψ ( x,t ) = A cos( kx ωt ) + AR cos( kx + ωt ) , (5) in the form ψ ( x,t ) = A cos( kx ) cos( ωt ) + A sin( kx ) sin( ωt ) + AR cos( kx ) cos( ωt ) AR sin( kx ) sin( ωt ) (6) = A (1 + R ) cos( kx ) cos( ωt ) + A (1 R ) sin( kx ) sin( ωt ) . Of course, this is just a superposition of two standing waves.

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2. If a transmission line of characteristic impedance Z carries a current I ( x,t ) = I i cos( kx ωt ) + I r cos( kx + ωt ) , (7) then the corresponding voltage is [see Eq. (7.104) in lecture notes] V ( x,t ) = I i Z cos( kx ωt ) ZI r cos( kx + ωt ) . (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( ωt ) + I r cos( ωt ) = 0 , (9) which implies that I r = I i . (10) Hence, the current takes the form I ( x,t ) = I i [cos( kx ωt ) cos( kx + ωt )] = 2 I i sin( kx ) sin( ωt ) , (11) where use has been made of a trigonometric identity. Likewise, the voltage is written V ( x,t ) = I i Z [cos( kx ωt ) + cos( kx + ωt )] = 2 I i Z cos( kx ) cos( ωt ) . (12) Note that the current at a given point on the line oscillates as sin( ωt ) whereas the corresponding voltage oscillates as cos( ωt ) = sin( ωt + π/ 2). In other words, the current and the voltage oscillations are everywhere π/ 2 radians out of phase. The energy flux down the line is I = IV . Hence, the mean energy flux is (I) = 4 I 2 i Z sin( kx ) cos( kx ) ( sin( ωt ) cos( ωt
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6soln6 - Physics 315 Oscillations and Waves Homework 6...

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