A common approach to solving differential equations is to assume sinusoidal solutions in the form of complex exponentials.
It is easy to verify that
is a general solution to the wave equation, where is the (complex) amplitude,
is known as the wave number, and is the wavelength.
This solution can be represented in a more general form, attributed to d'Alembert in 1747, of
.
represents a wave traveling in the positive direction with a velocity . Similarly, represents a wave traveling in the negative direction with the same velocity. Each component is generally referred to as a traveling wave.
The functions and are arbitrary and of fixed shape (given our assumed lossless medium) ... see waves on string simulation.
This implies that waves can propagate in two opposite directions in a one-dimensional medium.
Figure 2:
Traveling waves on a string.
When two or more waves pass through the same region of space at the same time, the actual displacement is the vector (or algebraic) sum of the individual displacements.