The superposition principle

The superposition principle states that, if y₁(x, t) and y₂(x, t) are any two solutions of the wave equation (1),

..........(1)

then so is any linear combination

y(x, t) = A₁y₁(x, t) + A₂y₂(x, t) .......(2)

where A₁ and A₂ are arbitrary constants. This result follows at once from the linearity of the wave equation (1), i.e. each term in the wave equation is proportional to y or one of its derivatives: it does not contain quadratic or higher-power terms or product terms such as y(∂y/∂x). (Equations of this type are known as linear equations.) We can see this as follows. Multiplying the first of the following equations

.......(3)

by A1 and the second by A2, and adding the resulting equations gives

..........(4)

Since

it follows that the linear superposition y(x, t), Equation (2), is also a solution of the wave equation (1). This result clearly generalises to the superposition of any number of solutions of the wave equation. These can be any solutions: they do not have to be normal modes. However, for reasons that will become clearer in the course of the following discussions we now choose a general superposition of normal modes.