Wave Packet Solution

Since the traveling wave solution to the wave equation

is valid for any values of the wave parameters, and since any superposition of solutions is also a solution, then one can construct a wave packet solution as a sum of traveling waves:

For a continuous range of wave vectors k, the sum is replaced by an integral:

Wave Packet Details

A wave packet solution to the wave equation, like a pulse on a string, must contain a range of frequencies. The shorter the pulse in time, the greater the range of frequency components required for the fast transient behavior. This requirement can be stated as a kind of uncertainty principle for classical waves:

The actual numbers involved depend upon the definition of the pulse width, but creating short pulses inherently requires a large frequency bandwidth. This is similar in nature to the uncertainty principle in quantum mechanics.

Wave Packets from Discrete Waves

If discrete traveling wave solutions to the wave equation are combined, they can be used to create a wave packet which begins to localize the wave. This property of classical waves in mirrored in the quantum mechanical uncertainty principle.