In this worksheet we will study two partial differential equations that are very important in physics.
Many wave motions
can be described by the linear wave equation. We shall do problems
concerning waves on a string, but the equation we study
has many other applications. For example atomic vibrations
in solids, light waves, sound waves and water waves
are all described by similar equations. The linear
wave equation for the waves on a string is the partial differential
A second partial differential equation that is
very important in physics is the diffusion equation.
Atoms in a gas diffuse around in a manner described by this
equation. Similarly pollutants in the ground often diffuse through the
soil. This motion is very different than wave motion. In general
each physical system has ranges of parameters where the
motion is ``diffusive'' or ``wavelike''. In solids for example
motion is wavelike at short times and over long distances
(e.g. sound waves), but diffusive on long times and short distances
(atomic hops). The diffusion equation is given by,
Problem 1 - Wave Phenomena
(i) Standing waves.
Consider waves on a string of length L=1. The
transverse displacement at each end of the string is fixed
at zero. Check that the two solutions:
, satisfy the wave equation. If we
seek the ``fundamental mode'', how
are kand related to v and the length of the string?
Set k=k0 and
(ie. the values for the
y(x,t)=y1(x,t)+y2(x,t) gives rise to standing waves.
Animate the solution y1and the solution y. Show
your animation to a TA (don't try to print it out)
Now consider two solutions of the form
Check that the linear superposition of these two propagating waves
produces a beat pattern.
How does the beat frequency depend on
Almost all functions can
be written as a superposition of sine and cosine waves.
As an example, consider the linear superposition of
sine waves such that;
Problem 2 - Diffusion.
Check that satisfies the diffusion equation. Animate the plots of c(x,t) for different values of t. Notice that the amplitude of c(x,t) decays with time, this is the essence of ``diffusion''. In contrast, in the linear wave equation, the wave amplitude remains constant, it propagates instead of spreading. In reality there is some ``damping'' of waves, and this is modeled by adding a ``diffusion term'' to the wave equation (like the ``damping term'' we sometimes add to Newton's equation). Show a TA your animation, but don't try to print it out