## Traveling-Wave Solution

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.1}Then 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point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables has been replaced by two functions of a single variable in time units.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. The traveling-wave solution of the wave equation was first published by d'Alembert in 1747 [100]. See Appendix A for more on the history of the wave equation and related topics.

### 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

*spatial*partial derivatives by and , respectively, we can write more succinctly

*second*partial derivatives in are

*linear operator*, establish that

### Use of the Chain Rule

These traveling-wave partial-derivative relations may be derived a bit more formally by means of the*chain rule*from calculus, which states that, for the composition of functions and ,

*i.e.*,

### String Slope from Velocity Waves

Let's use the above result to derive the*slope*of the ideal vibrating string From Eq.(C.11), we have the string displacement given by

*string slope*is given by

*chain rule*,

### Wave Velocity

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. In the case of the one-dimensional ideal wave equation (Eq.(C.1)), with no boundary conditions, an appropriate choice of eigensolution is

(C.12) |

Substituting into the wave equation yields

*wave velocity*(or

*phase velocity*

^{C.2}) as , the wave equation becomes

(C.13) | |||

Thus

*superposition,*

### D'Alembert Derived

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''. 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.

### Converting Any String State to Traveling Slope-Wave Components

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, orThus, 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

*slope waves*as a function of an arbitrary initial slope and velocity:

*unbounded*as the string length goes to infinity. Related discussion appears in Appendix E. 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).

**Next Section:**

Sampled Traveling Waves

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The Finite Difference Approximation