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Physics LibreTexts

3.3: Linearity and Superposition

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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(Nn=1αnxn(t))=(Nn=1αnFn(t))

The left hand bracket can be identified as the linear combination of solutions

x(t)=Nn=1αnxn(t)

while the driving force is a linear superposition of harmonic forces

F(t)=Nn=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.


This page titled 3.3: Linearity and Superposition is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

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