mt1soln_08

# mt1soln_08 - anti-node node node 3 4 anti-node node node 3...

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Unformatted text preview: anti-node node node 3 4 ! anti-node node node 3 4 ! Physics 9B-C Midterm 1 Solutions Cole, UC Davis 20 points per problem/120 points total. [1]   a) both  b)  frequency  c) right  d) greater bulk modulus   e) faster  f) decrease  g) 0   h) false  i) away from  j) antinodes. [2]   (20 pts)  Method 1:   Start by applying the derivatives and using the chain rule.  Let  u  =   x  +   vt:    ∂ y ∂ x = dy du ∂ u ∂ x = dy du      so that   ∂ 2 y ∂ x 2 = d du ∂ y ∂ x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∂ u ∂ x = d 2 y du 2 Similarly for  dt, ∂ y ∂ t = dy du ∂ u ∂ t = + v dy du        so that  ∂ 2 y ∂ t 2 = d du ∂ y ∂ t ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∂ u ∂ t = v 2 d 2 y du 2 Plugging into the wave equation:     ∂ 2 y ∂ t 2 − v 2 ∂ 2 y ∂ x 2 =     õ    v 2 d 2 y du 2 − v 2 d 2 y du 2 =   which is  0. Method 2:   The first order wave equation for waves moving to the left is   ∂ y ∂ t − v ∂ y ∂ x = ,  and is  easier to use.  If  y  is a solution to this equation, it will also be a solution to the wave equation.   is a solution to this equation, it will also be a solution to the wave equation....
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mt1soln_08 - anti-node node node 3 4 anti-node node node 3...

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