PHYS 221
General Physics II  Calculus Supplement
Grist
Homework Solution “Waves”
A)
In order to show that
x
ct
yS
i
n C
o
s
L
L
π
=
is a solution of the wave equation
22
1
yy
2
x
ct
∂∂
=
we first take the second partial derivative of
y
with respect to
x:
2
22 2
xc
t
Sin
Cos
yx
LL
Sin
Cos
c
t
x
xL
L
ππ
L
⎛⎞
∂
⎜⎟
∂
⎝⎠
==
−
We next take the second partial derivative of
y
with respect to
t:
2
2
2
t
Sin
Cos
yc
Sin
Cos
tt
L
L
x
c
t
L
∂
∂
−
Now we substitute these solutions into the wave equation and see if it is satisfied:
2
2
1
x
ct
c
x
ct
Sin
Cos
Sin
Cos
L
LLc
LLL
−=
−
The
2
1
c
and
term cancel, so yes,
2
c
x
ct
i
o
s
L
L
=
is a solution of the wave
equation
1
2
x
=
!
B)
At x=0 we get
2
2
01
0
0
ct
c
ct
Sin
Cos
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This note was uploaded on 01/09/2010 for the course PHYS 221 taught by Professor G.r.grist during the Spring '09 term at Skyline College.
 Spring '09
 G.R.GRIST
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