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Unformatted text preview: PHYS 302: Unit 11 Viewing Notes – Athabasca University PHYS 302: Vibrations and Waves – Unit 11 Viewing Notes
In all examples of explicit propagating waves thus far, the medium has been non-dispersive, which means
characterized by a single speed of wave propagation. A number of important media are non-dispersive. For example,
sound in air is characterized by a single speed. However, in some cases, waves are subject to dispersion, and the
speed of propagation may depend on the wavelength or frequency of the wave. In fact, one case already discussed
—beads on a string—does show dispersion (particularly that of five beads in Unit 7). At the time we discussed it, we
had not progressed to travelling waves, so we solved it purely as a case of a normal mode or standing wave. Recall
that for n beads there exist n modes, and their frequencies are n 20 sin 2( n1) . These correspond to standing
waves with displacement y p , n Ap ,n cos n t at the position of particle p. The amplitude for that particle is Ap ,n Cn sin , where C is the amplitude associated with mode n. Although each particle has its own
N 1 n amplitude, Cn is the same for all particles in that mode. It determines the overall amplitude of the sine curve on
which the particles lie as they oscillate in unison at the frequency ωn. In Unit 8 we saw that standing waves
correspond to travelling waves coming from opposite directions. In the lecture, Dr. Lewin makes the connection that
allows us to re-analyze this problem to find that those travelling waves are actually dispersive.
The case of N=5 with beads of mass M separated by l to give total length L is examined, and we are reminded that n 20 sin 2( n1) (2:10), with 0 N T
Ml . If the sin is made equal to its maximum possible value of 1, it is clear that the maximum possible frequency is ωmax=2ω0. (Note, however, that this does not correspond to a
physically possible motion since at this frequency all the amplitudes would be 0, as described on p. 142 of French.
You may also see this by putting n=N+1 in the amplitude equation above). At time 3:00, Dr. Lewin makes an important conceptual leap that likely needs elaboration. He states that the boundary conditions require kn nL . We
will justify this to a greater degree than is done in the lecture. Recall that the preferred, symmetric form of the wave
equation solution is y ( x, t ) A sin(kx t ) for the wave propagating toward -x. (Recall that travelling waves
moving in opposite directions are needed to make a standing wave or normal mode, but here we will only consider
one of them). To make the normal mode more similar to a wave, we combine the results for normal modes repeated
above to get y p , n Cn sin cos t . We note that the time dependence is the same as in the wave solution,
N 1 n and we can connect the position via x if we express the bead position as a fraction of L. Since the string has length L,
and this corresponds to N+1 lengths of string, the position of bead p is x y p ,n Cn sin cos t C
N 1 n n p
N 1 L , that is, x
N 1 . Thus, sin n x cos n t . Although it is not worth doing the algebra involving both
L travelling waves here, the x dependence in the wave and the normal mode clearly requires that kn nL . This, of course, does go back to boundary conditions. Having established this, we can conclude that the speed v n
kn cannot be the same for all frequencies, since the frequency goes up with the sin of n while the wavenumber goes up directly
with n. In fact, we can write v nn k 2 L0 sin 2 ( n1)
n . The speed of propagation is lower for higher frequencies, since the sin curve flattens out. This is clear in a plot of ω versus k (called a dispersion relation). On such a plot, the slope
of the line from the origin to the point (k,ω) is the speed (phase velocity). Adding two waves y1 A sin(k1 x 1t ) and y2 A sin(k2 x 2t ) , with phase velocities v1 11 and v2 22 respectively
(8:20), the sum y y1 y2 2 A sin k1 k2
2 x 1 2 t cos
2 k1 k2
2 x 1 2 t (9:50), using the now-familiar
2 “sin of the sum times cos of the difference” identity. Please recall that in the text, the similar development (p. 232)
uses a different definition of k, which results in less symmetric equations unless you use the form with ν shown (f in
the video lecture; frequency in Hz). What comes next is algebraically true for any frequencies, but if the frequencies
are close together, physical meaning is attached later. We can write that 1 y 2 A sin kx t cos 2k x 2 t , PHYS 302: Unit 11 Viewing Notes – Athabasca University where k is the average wavenumber k k1 k2 , ω is the average frequency 1 2 , and the differences are
2 k k1 k2 and 1 2 (11:10). This is clearly similar to the beat equation, with a fast oscillating
travelling wave at the phase velocity characterized by the sin part, v p , and a new velocity called the group
velocity characterizing the cos part: vg k (12:15). The name group velocity comes from the fact that a packet of waves would travel at this speed. In a nondispersive medium, it does not matter if one takes the ratio of differences
of frequency over difference of wavenumber or just the ratio of frequency over wavenumber, so both the group and
phase speeds (respectively) are the same. This is not true in a dispersive medium. Spatially, the sin pattern repeats
after 2k wavelength, while the cos curve has wavelength 4 . The shorter sin waves move with the phase
velocity while the envelope moves with the group velocity. The overall effect is like moving beats, and if the
medium is dispersive, the envelope will move at a different speed than the individual waves (15:30), either faster or
slower as determined by the dispersion relation (ω-k diagram). On the ω-k diagram, a straight line indicates a
nondispersive medium, with the phase and group velocities equal (16:30). For strings, such a nondispersive ω-k
diagram applies. In the case of a downward curving dispersion relation, the slope at larger k is smaller than at
small k, so the group velocity is smaller than the phase velocity. For an upward curving dispersion relation, the slope
at larger k is larger than at small k, so the group velocity is larger (the opposite). It is even possible to have changing
slope on the ω-k diagram, with a phase velocity moving in the opposite direction from the group velocity in some
regimes of k (20:20).
In the demo of bars overlapping with slightly different (5%) spacing of bars on transparencies, the overall pattern
moves 20 times faster than does an individual sheet in such a case (24:00).
We now return to strings, where the solution of the wave equation 1 2 y 2 y requires v v 2 t 2 x 2 T so that the velocity of propagation does not depend on frequency, but only on the tension and mass loading of the wire (27:30).
Thus, the string is a nondispersive medium with ω=vk or ω2 =v2k2. However, some approximations were made in
this derivation, mainly to neglect stiffness. In reality, the equation should be ω2=v2k2+αk4, which gives a curve that
bends upward. Dr. Lewin gives the example of a piano string, for which
L=1 m, and μ=10 g/m=10-2 kg/m. The square of the speed is T 2 T k 2 k 4 , α=10-2, T=250N, 2.5 104 m/s, and the tenth harmonic is considered. The fundamental would consist of one-half wave in the string length L, so here
10 2 L 10 m instead.
k10 10 10 m-1. With these numbers, we can solve that ω2=2.5×107+104, which is slightly higher than the value would be for a non-stiff wire. This gives ω≈5000 rad/s, or f≈800 Hz (35:20). What is important here is that
the extra term results in a 0.2% increase (1/6 Hz) of the tenth harmonic above what it would be without dispersion.
A stiffer wire would have a larger increase. The tenth harmonic is slightly higher than ten times the fundamental,
representing a sharpening of the frequency.
————————For a demo, Dr. Lewin uses a “toy model” dispersion relation ω2=v2k2(1+αk2). Although this superficially resembles
the dispersion relation of a stiff wire, it is actually quite different (36:30). The initial condition is six waves, 2.5%
separated in wavelength. For the waves lined up, you can see (on the screen) a function sharply peaked at the origin.
In the nondispersive case (α=0), the pulse envelope moves along at the same speed as all six waves, and nothing
changes in its shape. All of the waves making up the pulse envelope move at the same speed. If the phase velocities
are different, this “marching in step” does not happen. Instead, the component waves change how they line up, and
the pulse breaks down (in this example the group velocity is larger than the phase velocities) (43:30), but the pulse
breaks down slowly since the dispersive term is small (α=0.001).
In the next demo, α is negative and the group velocity ends up negative. This case is more dispersive (46:30). If we
think in Fourier space, the shape of a nondispersive case comes back to its original form since all the components
(harmonics) repeat in one full period. If there is dispersion, lining up after one period of any of the component 2 PHYS 302: Unit 11 Viewing Notes – Athabasca University waves is no longer possible, so the pulse shape changes. In travelling waves, the sharp features come from high
frequencies, and these will be dispersed quicker if the medium is dispersive, smoothing out a sharp pulse. The effect
of this rapid dispersion is demonstrated by looking (51:00) first without dispersion, which gives back the original
shape, and then with dispersion (α=0.01), in which case the pulse shape degrades very quickly (52:50). (You can
imagine that this feature is undesirable in a digital communication system based on transmitting sharp square
The phase velocity is v p and the group velocity is vg k d
dk , both of which depend on the shape of the dispersion relation. Since the limit of small frequency differences was never taken, the group velocity form shown
here was not really derived. However, if we consider small frequency differences in the analysis above, the
derivative form of group velocity formula follows as a limit. Ironically, though water waves are too complicated to
consider in this course, their well-known dispersion relation is a good one to use for examples of dispersion. This is
S 2 gk k 3 for deep water waves (shallow water effects are different, and the depth criterion is relative to the wavelength), with S as the surface tension and ρ the density. For fresh water, S=0.072 N/m and ρ=1000 km/m3. If
the wavelength is greater than 1 cm, the surface tension is unimportant. For wavelength 1 m, the term gk≈62, while
the last term is an insignificant 0.02 (both terms are in SI units) (56:00). It turns out that the velocity is about 1.25
m/s and the frequency in Hz is also about 1.25—reasonable sounding numbers to someone who has seen such
waves. These results come from the analysis on the board (57:30) that vp k g
k gk so that the phase velocity . This is proportional to the square root of the wavelength, that is, longer waves go faster (recall what you know about tsunami waves). The group velocity, involves a derivative vg d
dk 1 ( gk ) 2 g 1 k
k 1 v p : the group velocity is half the phase velocity. If you carefully
2 observe the ripple that goes out from a rock thrown into a deep pond, you can see this: the individual waves of the
ripple move through the ripple envelope going outward! In the opposite case (short wavelengths), where surface
tension is dominant, the dispersion relation becomes S 2 k 3 and v p k S k , which is proportional to the inverse of the wavelength, so that short wavelength waves move faster (the opposite of the deep water case)
(1:00:30). The nondispersive nature of shallow water waves may be somewhat intuitive, since these are essentially
pressure waves. Fortunately, sound—pressure waves in air—is nondispersive. If it was not, we could not
communicate via sound nor enjoy music.
The discussion of Maxwell Equations (1:02:00 to 1:03:00) does not apply to this course. We will, however, study
some aspects of electromagnetic waves without deriving their origin. You should realize that the speed of
electromagnetic waves in vacuum is c=3×108 m/s and that they are nondispersive. They range from radio waves
(with f=106 Hz and λ=300 m) through radar waves (with f=1010 Hz and λ=3 cm) up to “visible” light. If light
interacts with matter, its speed is slower than in a vacuum. The actual speed depends on two “permeabilities” that
would be studied in a course in electromagnetism such as MIT’s 8.02 (1:05:00). In common materials capable of
transmitting light, κm is almost always 1, but κe varies widely. In water, κe varies from about 78 over the range 0 to
1010 Hz, to about 1.77 for visible light (5×1014 Hz). The speed of light is c , so it varies from about c/9 to c/1.33
me over the large span of frequencies (1:07:40). Even within the range of light at frequencies near 1015 Hz, the speed
changes with frequency, so that light in materials is dispersed.
We will only look at EM radiation in metal pipes (waveguides) in a cursory way. However, we should understand
the final example, which uses 10 GHz (1010 Hz) or 3 cm wavelength radar waves. If the dimension of a gap is less
than half the wavelength, waves cannot pass. The dispersion relation is n c na 2 k z2 (1:11:40), which is reminiscent of equations for cavities for sound, with a as the width of the gap. Using this equation (not derived in d
this course), the velocities along the gap (the z direction) are v pz k (phase) and vgz dk (group) (1:12:00). The
z 3 z PHYS 302: Unit 11 Viewing Notes – Athabasca University lowest frequency possible occurs with n=1. (Note that the graph has a slight error; it should have kz.) The frequency must exceed a certain minimum or cutoff value min ca for propagation to occur. For these waves with fixed
frequency, this places the condition on a that it must be greater than 1.5 cm for these 3 cm waves. The cutoff is
dramatic in the demo. Only a squeeze through 1.5 cm is needed to completely stop the waves.
In the dispersion relation, the phase velocity is greater than the speed of light at all points. It goes to infinity at kz=0.
The group velocity goes to 0 at the cutoff. 4 ...
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This note was uploaded on 02/18/2011 for the course PHYS 320 taught by Professor Martinconner during the Spring '10 term at Open Uni..
- Spring '10