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# D'Alembert's solution of the Wave Equation

A particularly neat solution to the wave equation, that is valid when the string is so long that it may be approximated by one of infinite length, was obtained by d'Alembert. The idea is to change coordinates from and to and in order to simplify the equation. Anticipating the final result, we choose the following linear transformation

Thus, and we must use the chain rule to express derivatives in terms of and as derivatives in terms of and . Hence,

The second derivatives require a bit of care.

Thus,

and in a similar manner

Thus, (2.6) becomes

This equation is much simpler and can be solved by direct integration. First of all integrate with respect to to give

where is an arbitrary function of . Then integrate with respect to to obtain

where is an arbitrary function of and .

Finally, we replace and by their expressions in terms of and to obtain

 (2.7)

Example 2. .14Verify that is a solution to the wave equation. Here and .

Thus

Therefore,

Note that we may use the trigonometric identities for to obtain

and we remark that this is the product of a function of with a function of . This result will be used later.

D'Alembert's solution involves two arbitrary functions that are determined (normally) by two initial conditions. If the initial conditions are

 (2.8)

Then, from (2.7), we have
 (2.9)

Note that is a function of but at . The tricky part is calculating

but at and . Thus, the functional form of and are obtained by replacing by and .

Example 2. .15If, for example, , then

The initial speed, that is evaluated at , is then

Integrating with respect to , as both and are functions of when , we get
 (2.10)

Subtracting (2.10) from (2.9) gives

Hence, d'Alembert's solution that satisfies the initial conditions (2.8) is

Example 2. .16 can be thought of as a shape", defined by the function , moving to the right. We can see this if we consider the value . Say


Then  at 		 when
when
when


Thus, we are moving to the right to larger values of . Similarily corresponds to a wave propagating to the left.

Example 2. .17 Thus, the solution to the wave equation is

corresponding to a wave travelling to the right (increasing ).


Here 		  is the amplitude,
is the wavenumber - ,
is the wavelength - distance from one crest to the next,
is the angular frequency ,
is the period of the wave - time from one crest to the next to pass afixed point.


Example 2. .18 This gives the solution

The solution is a standing wave as shown in Figure 2.2. The maximum and minimum displacements are at and the nodes (where ) are at

Example 2. .19

Consider the case of the string initially at rest with the above initial displacement. Thus,

where is a constant. In Figure 2.3, .

From d'Alembert's solution we have

where is a constant. Adding the two solutions we have

Hence, the solution is

and the arbitrary constant, , does not appear in the solution. Thus, we may, in fact, take without any loss of generality. So we have 2 equal triangles of height going to the right and the left. This is shown in Figure 2.4.

Next: Separation of Variables Up: Partial Differential Equations of Previous: Modelling: Derivation of the
Prof. Alan Hood
2000-10-30