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Waves in Two and Three Dimensions
6/2/08
Michael Fowler
Introduction
So far, we’ve looked at waves in one dimension, traveling along a string or sound waves going
down a narrow tube.
But waves in higher dimensions than one are very familiar—water waves
on the surface of a pond, or sound waves moving out from a source in three dimensions.
It is pleasant to find that these waves in higher dimensions satisfy wave equations which are a
very natural extension of the one we found for a string, and—very important—they also satisfy
the
Principle of Superposition
, in other words, if waves meet, you just add the contribution from
each wave.
In the next two paragraphs, we go into more detail, but this Principle of
Superposition is the crucial lesson.
The Wave Equation and Superposition in One Dimension
For waves on a string, we found Newton’s laws applied to one bit of string gave a differential
wave equation,
22
1
2
y
y
x
vt
∂
∂
=
∂
∂
and it turned out that
sound waves in a tube satisfied the same equation
.
Before going to higher
dimensions, I just want to focus on one crucial feature of this wave equation: it’s
linear
, which
just means that if you find two different solutions
( )
1
,
yx
t
and
( )
2
,
t
then
()
( )
12
,,
t yx
t
+
is
also a solution, as we proved earlier.
This important property is easy to interpret
visually
: if you can draw two wave solutions, then at
each point on the string simply add the displacement
( )
1
,
t
of one wave to the other
( )
2
,
t
—
you just add the waves together—this also is a solution.
So, for example, as two traveling waves
moving along the string in opposite directions meet each other, the displacement of the string at
any point at any instant is just the sum of the displacements it would have had from the two
waves singly.
This simple addition of the displacements is termed “interference”, doubtless
because if the waves meeting have displacement in opposite directions, the string will be
displaced less than by a single wave.
It’s also called the
Principle of Superposition
.
The Wave Equation and Superposition in More Dimensions
What happens in higher dimensions?
Let’s consider two dimensions, for example waves in an
elastic sheet like a drumhead.
If the rest position for the elastic sheet is the (
x
,
y
) plane, so when
it’s vibrating it’s moving up and down in the
z
direction, its configuration at any instant of time
is a function
( )
zxyt
.
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In fact, we could do the same thing we did for the string, looking at the total forces on a little bit
and applying Newton’s Second Law.
In this case that would mean taking one little bit of the
drumhead, and instead of a small stretch of string with tension pulling the two ends, we would
have a small
square
of the elastic sheet, with tension pulling all around the edge.
Remember that
the net force on the bit of string came about because the string was curving around, so the
tensions at the opposite ends tugged in slightly different directions, and didn’t cancel.
The
2
/
2
y
x
∂∂
term measured that curvature, the rate of change of slope. In two dimensions, thinking
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 Fall '07
 MichaelFowler

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