. Given that the speed of a sine wave is given by v = w/k,
we expect each component of this series to travel at the same speed
(the n cancels) and so the square wave does not change shape as it
travels. The dependency of w on k is called a dispersion relation, w(k)= vk, and is linear in this case (no dispersion). In real
life, however, it is often the case that the angular frequency is not a linear function of the wave vector, k
in which case the individual components of the Fourier series travel at
different speeds. If different frequencies of a wave travel at
different speeds the effect is called dispersion. For white light (a combination of frequencies) traveling
through a glass prism, different colors travel at different speeds and
are therefore bent by different amounts. This is how a prism separates
white light into colors.|
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19.1. The simulation adds the first four
components of the Fourier series for a traveling square wave with no dispersion. Play the
simulation and describe what happens to the shape as time goes on.
19.2. Given that the speed of a sine wave is v = w/k,
what is the speed of the first four components of the square wave: y(x,t) = sin(1*x-1*t) + sin(3*x-3*t)/3 + sin(5*x-5*t)/5 +
sin(7*x-7*t)/7
19.3. Click 'reset' and then change the
frequency of the second term from 3 to 2.95, click 'set' and then
'play'. This will cause the second term to have a slightly different
speed. What is this new speed for the second term?
19.4. How does the initial shape compare with the shape in question
19.1? What happens to the shape of the square wave in this case as
time goes on?
19.5. Based on the previous simulation,
explain what would happen to a digital signal (which is basically a
series of square waves) traveling down a cable (either wires or optical
fiber) where there is a small amount of dispersion.
19.6. All cables (fiber optical or
metal) have some dispersion. Why is there a limit to how long a cable
can be before a signal traveling on it has to pass through a relay
(where the signal is amplified and 'cleaned up')?