For problems with so-called harmonic oscillations, given by for example the following differential equation
$$ m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), $$where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.
If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find the particular solution \( x_{pn}(t) \) to the above differential equation for each \( F_n \). The particular solution for the entire driving force is then given by a series like
$$ \begin{equation} x_p(t)=\sum_nx_{pn}(t). \tag{1} \end{equation} $$This is known as the principle of superposition. It only applies when the homogenous equation is linear. Superposition is especially useful when \( F(t) \) can be written as a sum of sinusoidal terms, because the solutions for each sinusoidal (sine or cosine) term is analytic.
Driving forces are often periodic, even when they are not sinusoidal. Periodicity implies that for some time \( t \) our function repeats itself periodically after a period \( \tau \), that is
$$ \begin{eqnarray} F(t+\tau)=F(t). \end{eqnarray} $$One example of a non-sinusoidal periodic force is a square wave. Many components in electric circuits are non-linear, for example diodes. This makes many wave forms non-sinusoidal even when the circuits are being driven by purely sinusoidal sources.