A traveling wave is a self-propagating disturbance of a medium that moves through space transporting energy and momentum. Examples include waves on strings, waves in the ocean, and sound waves. Waves also have the property that they are a continuous entity that exists over the an entire region of space; this distinguishes them from particles, which are localized objects. There are two basic types of waves: longitudinal waves, in which the medium is displaced in the direction of propagation (sound waves are of this type), and transverse waves, in which the medium is displaced in a direction perpendicular to the direction of propagation (electromagnetic waves and waves on a string are examples). It is important to remember that the individual 'bits' of the medium do not advance with the wave; they oscillate about an equilibrium position. Consider, for instance, a wave on a string: if the string is given a flick upwards from one end, any particular bit of string will be observed to move upwards and downwards, but not in the direction of the wave (see ).

Figure: % Traveling wave on a string.

Figure %: Transformation between moving and stationary axes.

ψ(x, t) = f (x - vt) |

This is called the

We now want to find a partial differential equation to define all waves. Since *ψ*(*x*, *t*) = *f* (*x'*) we can take
the partial derivative with respect to *x* to find:

= = |

and the partial derivative with respect to

= = ±v |

since

= ±v |

Then taking second derivatives with respect to

= | |||

= ±v |

But = so:

= v^{2} |

So finally we can combine the last equation with our expression for the second derivative with respect to

= |

This is the second order partial differential equations that governs all waves. It is called the

One set of extremely important solutions to the differential wave equation are sinusoidal functions. These are called the harmonic waves. One of the reasons they are so important is that it turns out that any wave can be constructed from a sum of harmonic waves--this is the subject of Fourier analysis. The solution in its most general form is given by:

ψ(x, t) = A sin[k(x - vt)] |

(we could, of course, equally well choose a cosine since the two functions only differ by a phase of

λ = |

T = = |

As usual, the frequency,

One property of the differential wave equation is that it is linear. This means that if you find two solutions
*ψ*_{1} and *ψ*_{2} that both satisfy the equation, then (*ψ*_{1} + *ψ*_{2}) must also be a solution. This
is easily proved. We have:

= | |||

= |

Adding these gives:

+ | = | + | |

(ψ_{1} + ψ_{2}) | = | (ψ_{1} + ψ_{2}) |

This means that when two waves overlap in space, they will simply 'add up;' the resulting disturbance at each point of overlap will be the algebraic sum of the individual waves at that location. Moreover, once the waves pass each other, they will continue on as if neither had ever encountered the other. This is called the principle of superposition. When waves add up to form a greater total amplitude than either of the constituent waves it is called

Figure %: Constructive interference.

Figure %: Destructive interference.