1. The following expression is proposed as a solution of the wave equation:
y
(
x, t
) =
A
sin(
kx
+
φ
) cos(
ωt
)
,
where
A
,
k
,
ω
, and
φ
are constants.
a) Show that it is a solution of the wave equation, or more precisely,
find the condition (involving some of the four constants) that en
sures it is a solution.
b) Suppose that this expression represents the motion of a string that
is tied down at two ends (a standing wave). One end is at
x
= 0
and the coordinate
y
is zero at that point for all times. What does
this condition determine about any of the four constants?
c) The other end of the string at
x
=
L
is also tied down so that
y
is
zero for all times. What does this condition determine about any
of the four constants?
Solution:
a) The wave equations is on the front cover:
∂
2
y
∂x
2
=
1
v
2
∂
2
y
∂t
2
∂y
∂x
=
kA
cos(
kx
+
φ
) cos(
ωt
);
∂
2
y
∂x
2
=

k
2
A
sin(
kx
+
φ
) cos(
ωt
) =

k
2
y
(
x, t
)
∂y
∂t
=

ωA
sin(
kx
+
φ
) sin(
ωt
);
∂
2
y
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 Spring '08
 COHN
 Physics, Vibrating string, wave equation, Electromagnetic wave equation

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