Chapter 31
Strings, Air Columns
225
31
Strings, Air Columns
Be careful not to jump to any conclusions about the wavelength of a standing
wave.
Folks will do a nice job drawing a graph of Displacement vs. Position
Along the Medium and then interpret it incorrectly.
For instance, look at the
diagram on this page.
Folks see that a half wavelength fits in the string segment
and quickly write the wavelength as
L
2
1
=
λ
.
But this equation says that a whole
wavelength fits in half the length of the string.
This is not at all the case.
Rather
than recognizing that the fraction
2
1
is relevant and quickly using that fraction in
an expression for the wavelength, one needs to be more systematic.
First write
what you see, in the form of an equation, and then solve that equation for the
wavelength.
For instance, in the diagram below we see that one half a
wavelength
λ
fits in the length L of the string.
Writing this in equation form
yields
L
=
λ
2
1
.
Solving this for
λ
yields
L
2
=
λ
.
One can determine the wavelengths of standing waves in a straightforward manner and obtain
the frequencies from
f
v
λ
=
where the wave speed
v
is determined by the tension and linear mass density of the string.
The
method depends on the boundary conditions—the conditions at the ends of the wave medium.
(The wave medium is the substance [string, air, water, etc.] through which the wave is traveling.
The wave medium is what is “waving.”)
Consider the case of waves in a string.
A fixed end
forces there to be a node at that end because the end of the string cannot move.
(A node is a
point on the string at which the interference is always destructive, resulting in no oscillations.
An antinode is a point at which the interference is always constructive, resulting in maximal
oscillations.)
A free end forces there to be an antinode at that end because at a free end the wave
reflects back on itself without phase reversal (a crest reflects as a crest and a trough reflects as a
trough) so at a free end you have one and the same part of the wave traveling in both directions
along the string.
The wavelength condition for standing waves is that the wave must “fit” in the
string segment in a manner consistent with the boundary conditions.
For a string of length
L
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 Fall '08
 RABE
 Physics, Wavelength, Standing wave, Air Columns

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