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SECTION 10.7 THE WAVE EQUATION
Suppose we have an elastic string (violin string, guitar string, guy wire, electric power
line, .
. . ) stretched and fastened at its ends
x
= 0 and
x
=
L
. Set this string in motion, by
plucking for example, so that it vibrates in a vertical plane. Let
u
(
x,t
) denote the vertical
displacement of the very very very small piece of string at location
x
at time
t
.
Now we use Newton along with a whole bunch of simplifying assumptions, among which
are
•
damping eﬀects such as air resistance are negligible,
•
each point along the string moves only in a vertical line,
•
the weight of the string is negligible,
•
T
cos
θ
=
T
when
θ
is small
•
a
2
=
tension
linear density
is constant.
We get (pp. 653654) the
onedimensional wave equation
a
2
u
xx
=
u
tt
.
This applies not just to guitar strings, but to nearly any wave problem, including ocean
waves, sound waves, electromagnetic waves, or waves in a solid body, at least if we generalize
slightly. For example, the motion of a drumhead is governed by
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This note was uploaded on 11/28/2010 for the course M 56840 taught by Professor Schurle during the Spring '10 term at University of Texas at Austin.
 Spring '10
 SCHURLE

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