2214
2010 Physics Robert E. Thorne
Fall 2011
Lecture 7
Recap
Standing Waves:
- simple harmonic motion at each point
- amplitude of SHM varies with position x, i.e. Am =
- phase of SHM is the same for all x
- due to interference of waves traveling in opposite directions
- boundary conditions are essential in producing the reflected waves
Standing waves on a string
String of length L, fixed at x=0, L so that the y displacement y(0,t) = y(L,t)=0 for all t. The
standing wave wavelengths and frequencies are:
The frequency f1 corresponding to the longest wavelength standing
move is called the fundamental mode of oscillation. The
frequency fn is called the nth harmonic. (Note that this is a second
and different use of the word harmonic. Harmonic motion means
sinusoidal motion. But a harmonic is one of a series of sinusoidal
oscillations that have different frequencies, usually related by
integers.)
The nth standing wave mode has n+1 nodes and n loops (or antinodes). As we go from mode n
to mode n+1, we add one more node and one more loop. Loops are also called antinodes.
We indicated that these standing waves result from an interference of waves traveling in opposite
directions, produced by reflection at boundaries. What is the round-trip time for a wave to
propagate from one end of the string to the other, be reflected, and return to its starting point?
The round trip time is the period of the lowest frequency standing wave mode. For standing
wave modes of higher n and higher frequencies,
Physics 2214
2010 Robert E. Thorne
.
Fall 2011
so that
period of nth harmonic Tn
Trt
n
A wave launched from one end returns in phase after n oscillations of the standing wave.
Standing Sound Waves in Pipes
As for harmonic standing waves on a string, we can write
where
is the position-dependent amplitude of the sinusoidal
oscillations of the pressure. We can write a similar expression for the displacement
the relation
using
What are the boundary conditions in this case?
At an open end of the pipe, the air molecules are free to expand in all directions, and the pressure
there is just atmospheric pressure
Thus,
at an open end. The excess pressure has a
node, and from
we then expect the displacement to have an antinode.
At a closed end, s=0 because the air molecules cannot displace into the wall. The excess
pressure
has an antinode.
Pipe with two open ends:
The two lowest frequency standing wave modes are
sketched at left. Their frequencies are:
fundamental
and
2nd harmonic
Physics 2214
2010 Robert E. Thorne
So, just as for standing waves on a string fixed at each end, we have:
Fall 2011
Pipe with one open end, one closed end:
At the closed end, s=0 and
is an antinode, and at the open end, s is an antinode and
The standing wave modes with the longest wavelengths / lowest frequencies are:
=0.
fundamental
and
third harmonic
Because of the non-symmetric boundary conditions,
we lose the even harmonic standing wave modes.
So
Our demo of a loudspeaker sending sound into a glass graduated cylinder shows that this
prediction agrees with experiment.
Physics 2214
2010 Robert E. Thorne
Fall 2011
Standing waves and resonance
Recall the response of an underdamped oscillator to a sinusoidal drive at frequency
:
For fixed drive amplitude, there is a range of drive frequencies near
where we get a much
bigger response than at other frequencies. If we push our system at the frequency that it naturally
likes to oscillate at, the energy from successive pushes will add up and we get a big amplitude of
motion. We call this a resonance.
This oscillator is a one-degree-of freedom system - we need only consider one point in the
oscillator (a single coordinate y(t)) to describe its motion.
Strings, drumheads, pipes, beams, buildings, etc. are extended objects with many degrees of
freedom. We can think of them as a bunch of tiny masses connected together by springs. Each
point in the object has its own motion, and to describe the motion at each point we in general
need continuous functions of position, e.g., y(x,t) or s(x,t).
At the standing wave frequencies for strings and pipes, we observed a much larger amplitude of
motion for a given drive amplitude than at other frequencies, suggesting that standing waves are
resonances an of extended system. In fact, thats just what they are. If we launch a wave down
a spring, it will reflect at the boundary and return to the launch point. If we launch another wave
just as the reflected wave returns, the two wave amplitudes will add up, and the total amplitude
will grow. If we keep launching waves in synchrony in this way, the amplitude will eventually
build up to a value determined by the amount of damping. If we launch waves a little before or a
little after the return of the reflected wave, the waves will interfere destructively and the total
amplitude will be small.
Consequently, if we drive a string at different frequencies d and measure the maximum
amplitude of local motion that results, we expect to see something like what is shown below:
Physics 2214
2010 Robert E. Thorne
Fall 2011
Consequently, unlike for a one degree of freedom (e.g., mass on a spring) system which will only
show a single resonance, an extended system will show standing wave resonances at many
frequencies.
The "amplitude" of the motion is a more complicated notion for an extended system, because the
amplitude of the motion will in general be different at each position x, and the pattern of motion
will be different at each resonance. For example, in the fundamental standing wave mode of a
string, the midpoint of the string has the largest amplitude of motion; in the second harmonic
mode, the midpoint has no motion at all, i.e., it is a node. We might be better off plotting "total
energy of motion" or some other more global measure of motion, but for now we'll stick with
"amplitude."
Similarly, if we launch a sound wave down a pipe, we'll get a big amplitude if the waves
reflected from its ends interfere constructively.
For more general kinds of objects, we'll get standing waves and resonances at frequencies that
lead to constructive interference of waves that are reflected by the object's boundaries or by
changes in its composition.
The damping of each mode and thus the height and width of its resonance curve will depend on
the kinds of motions involved in the mode and how they interact with loss-generating processes
in the system.
Unlike for a string, for most extended objects,
there will be no simple relationship between the
frequencies of different standing wave modes.
There will be lots of different resonant modes, and
there may be overlap in the resonance curves, so
that a sharp resonance pops out of a broader one.
Physics 2214
2010 Robert E. Thorne
Fall 2011
The different resonant modes may involve qualitatively different kinds of motions. For example,
consider a hollow cylinder. There can be standing waves that run along the length of the
cylinder, and standing waves that run around its circumference, and torsional waves that involve
a twisting along the axis of the cylinder.
Nonsinusoidal Drives and Responses
So far we've learned how to calculate the response of simple systems - a mass on a spring, a
string - to sinusoidal drives. We've learned how to describe and analyze sinusoidal traveling
waves and standing waves. But in the real world, the systems we are dealing with are usually
quite complicated. We drive them with more complex waveforms than sinusoids, and their
response is usually even more complex. How do we analyze oscillations and waves in this case?
First, let's learn how to describe more complex periodic waveforms.
Fourier Analysis
Any periodic function
of x of period L can be expressed as a sum of sinusoids
where
and
We can rewrite this in the following alternate forms:
Similarly, we can express any periodic function of t of period T as a sum of sinusoids with
frequencies
For even functions of x,
the coefficients of the sine terms must be zero by
symmetry, and
is described by a sum of cosines. For odd functions
, the
cosine terms are zero and and
is described by a sum of sines.
Physics 2214
2010 Robert E. Thorne
Fall 2011
Here are some Fourier series of common waveforms: (pictures from Wolfram Math)
Square wave:
Triangle wave:
Sawtooth wave:
where
and L is the period.
The simulation at http://phet.colorado.edu/simulations/sims.php?sim=Fourier_Making_waves is
quick and easy way to develop your inituition about Fourier series.