It is easily shown that the lossless 1D wave equation is solved by any string shape which travels to the left or right with speed . Denote right-going traveling waves in general by and left-going traveling waves by , where and are assumed twice-differentiable.C.1Then a general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as
The next section derives the result that and , establishing that the wave equation is satisfied for all traveling wave shapes and . However, remember that the derivation of the wave equation in §B.6 assumes the string slope is much less than at all times and positions. Finally, we show in §C.3.6 that the traveling-wave picture is general; that is, any physical state of the string can be converted to a pair of equivalent traveling force- or velocity-wave components.
An important point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables has been replaced by two functions of a single variable in time units. This leads to great reductions in computational complexity.
Traveling-Wave Partial Derivatives
Because we have defined our traveling-wave components and as having arguments in units of time, the partial derivatives with respect to time are identical to simple derivatives of these functions. Let and denote the (partial) derivatives with respect to time of and , respectively. In contrast, the partial derivatives with respect to are
Denoting the spatial partial derivatives by and , respectively, we can write more succinctly
where this argument-free notation assumes the same and for all terms in each equation, and the subscript or determines whether the omitted argument is or .
Now we can see that the second partial derivatives in are
These relations, together with the fact that partial differention is a linear operator, establish that
Use of the Chain Rule
To apply the chain rule to the spatial differentiation of traveling waves, define
Then the traveling-wave components can be written as and , and their partial derivatives with respect to become
and similarly for .
Because is an eigenfunction under differentiation (i.e., the exponential function is its own derivative), it is often profitable to replace it with a generalized exponential function, with maximum degrees of freedom in its parametrization, to see if parameters can be found to fulfill the constraints imposed by differential equations.
Substituting into the wave equation yields
Setting , and extending the summation to an integral, we have, by Fourier's theorem,
for arbitrary continuous functions and . This is again the traveling-wave solution of the wave equation attributed to d'Alembert, but now derived from the eigen-property of sinusoids and Fourier theory rather than ``guessed''.
We verified in §C.3.1 above that traveling-wave components and in Eq.(C.14) satisfy the ideal string wave equation . By definition, the physical string displacement is given by the sum of the traveling-wave components, or
Thus, given any pair of traveling waves and , we can compute a corresponding string displacement . This leads to the question whether any initial string state can be converted to a pair of equivalent traveling-wave components. If so, then d'Alembert's traveling-wave solution is complete, and all solutions to the ideal string wave equation can be expressed in terms of traveling waves.
The state of an ideal string at time is classically specified by its displacement and velocity
It will be seen in §C.7.4 that state conversion between physical variables and traveling-wave components is simpler when force and velocity are chosen as the physical state variables (as opposed to displacement and velocity used here).
Sampled Traveling Waves
The Finite Difference Approximation