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D'Alembert Derived
Setting
, and extending the summation to an integral,
we have, by Fourier's theorem,

(C.14) 
for arbitrary continuous functions
and
. This is again the travelingwave solution of the wave
equation attributed to d'Alembert, but now derived from the
eigenproperty of sinusoids and Fourier theory rather than
``guessed''.
An example of the appearance of the traveling wave components shortly
after plucking an infinitely long string at three points is shown in
Fig.C.2.
Figure C.2:
An infinitely long string,
``plucked'' simultaneously at three points, labeled ``p'' in the
figure, so as to produce an initial triangular displacement. The
initial displacement is modeled as the sum of two identical triangular
pulses which are exactly on top of each other at time 0
. At time
shortly after time 0
, the traveling waves centers are
separated by
meters, and their sum gives the trapezoidal
physical string displacement at time
which is also shown. Note
that only three short string segments are in motion at that time: the
flat top segment which is heading to zero where it will halt forever,
and two short pieces on the left and right which are the leading edges
of the left and rightgoing traveling waves. The string is not
moving where the traveling waves overlap at the same slope. When the
traveling waves fully separate, the string will be at rest everywhere
but for two halfamplitude triangular pulses heading off to plus and
minus infinity at speed
.

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