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point to note about the traveling-wave solution of the 1D wave equation is that a function of two variables has been replaced by two functions of a single variable in time units.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

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

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.*,

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

*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, 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

*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

**Previous Section:**

The Finite Difference Approximation