1. Consider the following standing wave, with period
T
= 2
π/ω
and wave
length
λ
= 2
π/k
,
y
1
=
A
cos
kx
sin
ωt
a) Suppose this solution describes a standing wave on a string that
has one end at
x
= 0 and the other at
x
=
L
. The end at
x
=
L
is tied down so that
y
= 0 at that point for all times. Find the
relation between
λ
and
L
for the three lowestfrequency standing
waves (i.e. the ﬁrst three normal modes of vibration).
b) Plot the 1st and 2nd normal modes from a) at time
t
=
T/
4 on
the axis provided below, labeling each clearly.
c) Now consider a second standing wave solution,
y
2
=
A
sin
kx
cos
ωt
.
Is the superposition,
y
=
y
1
+
y
2
a standing wave or a traveling
wave? You will ﬁnd it useful to rewrite
y
using one of the trig.
identities below and argue by inspection.
sin(
x
±
y
) = sin
x
cos
y
±
cos
x
sin
y
cos(
x
±
y
) = cos
x
cos
y
∓
sin
x
sin
y
Solution:
a) We must have:
y
(
L,t
) =
A
cos
kL
sin
ωt
= 0.
A
cannot be zero because
there would then be no standing wave. sin
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 COHN
 Physics, Standing wave, sin kx cos

Click to edit the document details