3.3: Linearity and Superposition
( \newcommand{\kernel}{\mathrm{null}\,}\)
An important aspect of linear systems is that the solutions obey the Principle of Superposition, that is, for the superposition of different oscillatory modes, the amplitudes add linearly. The linearly-damped linear oscillator is an example of a linear system in that it involves only linear operators, that is, it can be written in the operator form (appendix 19.6.2)
(d2dt2+Γddt+ω20)x(t)=Acosωt
The quantity in the brackets on the left hand side is a linear operator that can be designated by L where
Lx(t)=F(t)
An important feature of linear operators is that they obey the principle of superposition. This property results from the fact that linear operators are distributive, that is
L(x1+x2)=L(x1)+L(x2)
Therefore if there are two solutions x1(t) and x2(t) for two different forcing functions F1(t) and F2(t)
Lx1(t)=F1(t)Lx1(t)=F2(t)
then the addition of these two solutions, with arbitrary constants, also is a solution for linear operators.
L(α1x!+α2x2)=α1F1(t)+α2F2(t)
In general then
L(N∑n=1αnxn(t))=(N∑n=1αnFn(t))
The left hand bracket can be identified as the linear combination of solutions
x(t)=N∑n=1αnxn(t)
while the driving force is a linear superposition of harmonic forces
F(t)=N∑n=1αnFn(t)
Thus these linear combinations also satisfy the general linear equation
Lx(t)=F(t)
Applicability of the Principle of Superposition to a system provides a tremendous advantage for handling and solving the equations of motion of oscillatory systems.