String Wave Solutions

A solution to the wave equation for an ideal string can take the form of a traveling wave

Show

For a string of length L which is fixed at both ends, the solution can take the form of standing waves:

Show

For different initial conditions on such a string, the standing wave solution can be expressed to an arbitrary degree of precision by a Fourier series

Traveling Wave Solution for String

A useful solution to the wave equation for an ideal string is

Show wave parameters

It can be shown to be a solution to the one-dimensional wave equation by direct substitution:

Setting the final two expressions equal to each other and factoring out the common terms gives

These two expressions are equal for all values of x and t and therefore represent a valid solution if the wave velocity is

Wave velocity for a stretched string

String Traveling Wave Velocity

For a point of constant height moving to the right:

For a point of constant height moving to the left:

From the traveling wave solution, the phase velocity for a string wave is given by:

Traveling Wave Parameters

A traveling wave solution to the wave equation may be written in several different ways with different choices of related parameters. These include the basic periodic motion parameters amplitude, period and frequency.

Equivalent forms of wave solution:

The wave velocity is determined by the properties of the medium and is independent of the other parameters, but it can be determined from measurements of the frequency and wavelength. The following calculation allows you to specify any two of the quantities in the wave relationship v = fl and then initiate the calculation by clicking on the active text for the quantity you wish to calculate.

Note: The position, velocity and acceleration are recalulated only when you click on either velocity, frequency or wavelength in the active formula above.

Plane Wave Expressions

A traveling wave which is confined to one plane in space and varies sinusoidally in both space and time can be expressed as combinations of

It is sometimes convenient to use the complex form

which may be shown to be a combination of the above forms by the use of the Euler identity

In the case of classical waves, either the real or the imaginary part is chosen since the wave must be real, but for application to quantum mechanical wavefunctions such as that for a free particle, the complex form may be retained.