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.1Then a general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as

 (C.11)

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point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables has been replaced by two functions of a single variable in time units.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

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

obeys the ideal wave equation for all twice-differentiable functions and .

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.,

the derivative of the composition with respect to can be expressed according to the chain rule as

where denotes the derivative of with respect to .

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 .

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

By linearity of differentiation, the string slope is given by

Consider only the right-going component, and define

with . By the chain rule,

The left-going component is similar, but with . Thus, the string slope in terms of traveling velocity-wave components can be written as

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

Defining the wave velocity (or phase velocityC.2) as , the wave equation becomes
 (C.13)

Thus

is a solution for all . By superposition,

is also a solution, where and are arbitrary complex-valued functions of arbitrary points in the complex plane.

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 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, or

 (C.15)

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

for all [317]. Equation (C.15) gives us as a simple sum of the traveling-wave components, and now we need a formula for in terms of them as well. It will be derived in §C.7.3 (see Equations (C.44-C.46)) that we can write

where denotes the partial derivative with respect to as usual. We have

Inverting the two-by-two differential operator matrix yields left- and right-going slope waves as a function of an arbitrary initial slope and velocity:

Integrating both sides with respect to , and choosing the constant of integration to give the correct constant component of , we obtain the displacement-wave components

where

Notice that if the initial velocity is zero, each of the initial traveling displacement waves is simply half the initial displacement, as expected. On the other hand, if the initial displacement is zero and there is a uniform initial velocity (the whole string is moving), the initial displacement-wave components are 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).

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Sampled Traveling Waves
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The Finite Difference Approximation