Problem 2: In the examples of wave motion that we studied in the
lesson, the waves traveled at a constant velocity. Such waves are
called nondispersive due to the linear relationship between the
speed of the wave and its length. This means that they don't ever
slow down, but always maintain the same speed. In the real world, the
velocity of the waves changes as a function of the properties of the
media that it travels through, primarily due to friction. Such waves
are called---you guessed it--- dispersive waves and the form of
the wave changes over time.

(a) Given the original description, the velocity of the wave, v, was set as
a constant value. Now we will modify this to allow the velocity to be a
function of the wavelength, where

v(2&pgr;/&lgr;) = a + b(2&pgr;/&lgr;)

Let a = 1.7 and b = (0.2). Redefine the equation for the
one-dimensional wave u such that the new value of the velocity has
been incorporated. Hint: You can use the definition that we already
gave for the one-dimensional wave and modify it. You can use the same
values for A and &lgr; as in the example.

(b) Now construct a graph which has the original expression for the
one-dimensional wave that was illustrated in the lesson and also
includes the modified wave expression from part (a) on the same plot,
for a value of t = 1.0 Are there any differences? What about at
t=3.0?

(c) In order to investigate any possible difference further,
animate the plot you made in (b) of both waves as a function of time
on the same graph. Use t = 1 up to 20 or 30 and approximately 50
frames. Describe what happens and how.

(d) Now repeat steps (a) and (b) for the two-dimensional wave and
report what you observe. Hint: Again, you can build directly from the
examples given in the lesson.

(Just for fun) You can compute the phase velocity of the wave by
graphically comparing the distance the wave travels from one ``crest'' to
another. You can also measure the ``dispersiveness'' by comparing the same
instance of time between the dispersive wave and nondispersive wave on the
same graph.

Prepare a report giving the answers to these questions.
You don't have to hand in any plots, but be sure to tell us
how the plots and animations that you did helped you to answer the questions.
Submit your report for grading.