

Class 15, Friday, February
12,
2010
Reading:
J~, ~
/~ ~ i~. ~
2)
Standing Waves
in
a medium open
at
both
ends (openopen). A tube does not have
to be closed in order to give rise to standing waves. A flute, or
an organ pipe, are
examples
of tabes that are
ope1!'Otl
both ends, and yet can give rise to a standing wave
pattern. How does this compare
to the example described previously?
In this case we need to enforce antiBades at both boundaries, because
the
ends are
aIrowed to vibrate freely. This is in contrast to the closedclosed case, where the ends had
to be held fixed, thus enforcing nodes. Writing down the equation that describes a
standing wave:
y(x,t) =2Asin(kx) cos(wt),
we notice that at x=O, y(O,t)=O for all times, because the sine
of zero is zero. This means
that with sin(kx) we will never manage to adequately describe our new situation.
If we,
however, add an appropriate phase to the sine, we can reproduce an adequate description
of the situation:
y(x,t)= 2Asin(kx+n:/2) COS(Olt)
Since sin(kx+n:l2)=cos(kx), our new expression for a standing wave in this case is
y(x,t)= 2Acos(kx) cos(wt)
Note that both sine and cosine are valid descriptions
of a wave! You are free to choose
whichever is best suited to reproduce the physical situation you are studying. We derive
the normal modes below.
~UrdJ.
.
,.t
x=L
fivdi
t
.(=)
/tn(~L)/
it
MI~)(·
(q)
(k.L)
=
:r /
Ie
L:!
t?;
li;
2,/11""
,/)/I
*.L:
nii
ffsL=
nil
[In''
if
e.i>
...
J
co>
,11=
I.
q
Notice that this condition is identical to that derived for the closedclosed situation!
The first three normal modes look as sketched below. Here you count the nodes in order
to find out what order the normal mode you observe has.
IS:
/