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It can be readily checked (see §G.3 for details)
that the lossless 1D wave equation
(where all terms are defined in Eq. (4.1)) is solved by
any string shape which travels to the left or right with speed
If we denote right-going traveling
waves in general by
and left-going traveling waves by
, where and are arbitrary twice-differentiable
functions, then the general class of solutions to the lossless,
one-dimensional, second-order wave equation can be expressed as
|
(5.2) |
Note that we have
and
(derived in §G.3.1)
showing
that the wave equation is satisfied for all traveling wave shapes
and . However, the derivation of the wave equation itself
assumes the string slope is much less than at all times and
positions (see Appendix F). An important point to note is
that a function of two variables is replaced by two functions
of a single (time) variable. This leads to great reductions in
computational complexity, as we will see. The traveling-wave solution
of the wave equation was first published by d'Alembert in 1747
[93].
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