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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