PHY 9B - Physics 9B-C Midterm 1 Solutions Cole UC Davis 20...

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Physics 9B-C Midterm 1 Solutions Cole, UC Davis 20 points per problem/120 points total. [1] a) both b) steel c) left d) | A 1 - A 2 | 2 e) both f) above g) dimmer h) converging i) virtual j) antinodes. [2] a) (8 pts) Plug the solution into the wave equation. First calculate the second derivatives: ! ! " " y t A kx t = # # ( ) cos & ! ! " " 2 2 2 y t A kx t = # # ( ) sin Also, ! ! " 2 2 2 y x k A kx t = # # ( ) sin . Plug into the wave equation: - 2 A sin( kx - t) + u 2 k 2 A sin( kx - t ) + å A sin( kx - t ) = 0 Divide by the A sin and solve for : = u k 2 2 + $ which is the dispersion relation. b) (4 pts) v p = " $ k u k = + 2 2 c) (4 pts) v g = d dk d dk u k u k u k " $ $ = + = ( ) + 2 2 1 2 2 2 2 2 õ v g = u k u k 2 2 2 + $ d) (4 pts) Write the phase velocity in terms of ¬ using k = (2 π )/¬: v p = u 2 2 2 4 + $ % & Thus, longer wavelengths will have a greater phase velocity. [3] a) (2 pts) There are a total of 4 nodes--one at each end and two in the middle, so the mode looks like the third harmonic( n=3 in the figure). b) (4 pts) & = = 2 3 40 L cm k = = = 2 2 0 4 15 7 % & % .
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