It can be readily checked (see §C.3 for details)
that the lossless 1D wave equation
(where all terms are defined in Eq.
![$ \,$](http://www.dsprelated.com/josimages_new/pasp/img196.png)
(
6.1)) is solved by
any string shape which travels to the left or right with speed
If we denote right-going
traveling waves in general by
![$ y_r(t-x/c)$](http://www.dsprelated.com/josimages_new/pasp/img377.png)
and left-going
traveling waves by
![$ y_l(t+x/c)$](http://www.dsprelated.com/josimages_new/pasp/img378.png)
, where
![$ y_r$](http://www.dsprelated.com/josimages_new/pasp/img1314.png)
and
![$ y_l$](http://www.dsprelated.com/josimages_new/pasp/img1315.png)
are arbitrary twice-differentiable functions, then the general class
of solutions to the lossless, one-dimensional, second-order
wave
equation can be expressed as
![$\displaystyle y(t,x) = y_r(t-x/c) + y_l(t+x/c). \protect$](http://www.dsprelated.com/josimages_new/pasp/img1316.png) |
(7.2) |
Note that we have
![$ {\ddot y}_r= c^2y''_r$](http://www.dsprelated.com/josimages_new/pasp/img1317.png)
and
![$ {\ddot y}_l= c^2y''_l$](http://www.dsprelated.com/josimages_new/pasp/img1318.png)
(derived in §
C.3.1) showing that the
wave
equation is satisfied for all traveling wave shapes
![$ y_r$](http://www.dsprelated.com/josimages_new/pasp/img1314.png)
and
![$ y_l$](http://www.dsprelated.com/josimages_new/pasp/img1315.png)
.
However, the derivation of the
wave equation itself assumes the string
slope
![$ \vert y^\prime\vert$](http://www.dsprelated.com/josimages_new/pasp/img1319.png)
is much less than
![$ 1$](http://www.dsprelated.com/josimages_new/pasp/img138.png)
at all times and positions
(see §
B.6). An important point to note is that a
function of two variables
![$ y(t,x)$](http://www.dsprelated.com/josimages_new/pasp/img373.png)
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
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7.1
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