Class 18, Monday,
February
22, 2010
Reading: page 231
l
t
1.
,
Waves
in two and three dimensions
So far we've only considered linear waves, i.e. waves that only change along one
direction in space. In general, however, waves are three-dimensional phenomena. The
most common example
of a two-dimensional wave are ripples on a pond, excited by a
stone dropped in the water. Each circle corresponding to a maximum displacement
(height
of the ripple above the average level
of the water) constitutes a
wave
front. Each
two successive wave fronts are separated by one wavelength,
A..
In nature you will notice
that the peaks are not evenly distributed. This is because the ripples are actually a pulse,
generated by a one-time event (analogously to once shaking one end
of a rope): the stone
impacting the water surface. A pulse can be described by adding up an infinite number
of
waves
of different frequencies. In some media, the speeds at which the individual
components that make up the pulse travel are independent
of their wavelengths, and so
the pulse can retain its shape as
it
travels through the medium. This is not necessarily
always the case. In other media, the wave speeds depend on the wavelengths, a property
called
displlrsion.
Thus, the components that make up the pulse travel at different speeds.
This changes the shape
of the pulse, and alters the distance between consecutive
maximum displacements. For example, for surface water waves in deep water:
1h
~
fEI
w-
vZ7T
As a result, in dispersive media, one needs to understand how far a pulse can travel
before the information it carries is so altered, that it can no longer be "read".
For the waves we study presently, we will assume that they are non-dispersive. For a
spherical wave, the wave fronts are surfaces
of concentric nested spheres, separated by
one wavelength. For a spherical or circular wave, one can sometimes make the
simplifying approximation that the shape
of the wave front is a line rather than a circle, or
a plane rather than the curved surface
of a sphere. For waves in 3D, this approximation is
called the planar wave approximation. An ant sitting on the surface
of an inflating
balloon, will perceive the balloon as clearly spherical, when it is small in size. As you
keep inflating it, however, the ant will loose track
of the curvature and will think of its
immediate surroundings as a plane. We can make this approximation whenever we are far
enough away from the source of the wave.
11.1