A particularly neat solution to the wave equation, that is valid when
the string is so long that it may be approximated by one of infinite length,
was obtained by d'Alembert. The idea is to change coordinates from and
to
and
in order to simplify the equation. Anticipating the final
result, we choose the following linear transformation
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The second derivatives require a bit of care.
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This equation is much simpler and can be solved by direct integration. First
of all integrate with respect to to give
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Finally, we replace and
by their expressions in terms of
and
to obtain
Example 2. .14Verify that
is a solution to the wave
equation. Here
and
.
Thus
Note that we may use the trigonometric identities for
to obtain
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D'Alembert's solution involves two arbitrary functions that are determined
(normally) by two initial conditions. If the initial conditions are
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Example 2. .15If, for example,
, then
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The initial speed, that is
evaluated at
, is then
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Hence, d'Alembert's solution that satisfies the initial conditions (2.8) is
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Example 2. .16 can be thought of as a
``shape", defined by the function
, moving to the right. We can see
this if we consider the value
. Say
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Thenat
when
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when
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when
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Thus, we are moving to the right to larger values of . Similarily
corresponds to a wave propagating to the left.
Example 2. .17
Thus, the solution to the wave equation is
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Hereis the amplitude,
is the wavenumber -
,
is the wavelength - distance from one crest to the next,
is the angular frequency
,
is the period of the wave - time from one crest to the next to pass afixed point.
Example 2. .18
This gives the solution
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Example 2. .19
Consider the case of the string initially at rest with the above initial displacement. Thus,![]() |
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