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4.3: 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 F.2)

    \[ \label{eq:3.6} \Big ( \frac{d^2}{dt^2} + \Gamma \frac{d}{dt} + \omega_0^2 \Big ) x (t) = A \cos \omega t \]

    The quantity in the brackets on the left hand side is a linear operator that can be designated by L where

    \[ \label{eq:3.7} \mathbb{L} x (t) = F (t) \]

    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

    \[ \label{eq:3.8} \mathbb{L} ( x_1 + x_2 ) = \mathbb{L} ( x_1 ) + \mathbb{L}(x_2) \]

    Therefore if there are two solutions \( x_1 (t) \) and \( x_2 (t) \) for two different forcing functions \( F_1 (t) \) and \( F_2 ( t) \)

    \[ \label{eq:3.9} \begin{align*} \mathbb{L}x_1(t) & = & F_1(t) \\ \mathbb{L}x_1(t) & = & F_2(t) \end{align*} \]

    then the addition of these two solutions, with arbitrary constants, also is a solution for linear operators.

    \[ \label{eq:3.10} \mathbb{L} ( \alpha_1 x_! + \alpha _2 x_2 ) = \alpha_1 F_1 (t) + \alpha_2 F_2 (t) \]

    In general then

    \[ \label{eq:3.11} \mathbb{L} \Bigg ( \sum_{n=1}^N \alpha x_n (t) \Bigg ) = \Bigg ( \sum_{n=1}^N \alpha_n F_n (t) \Bigg ) \]

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

    \[ \label{eq:3.12} x(t) = \sum_{n=1}^N \alpha x_n (t) \]

    while the driving force is a linear superposition of harmonic forces

    \[ \label{eq:3.13} F(t) = \sum_{n=1}^N \alpha_n F_n (t) \]

    Thus these linear combinations also satisfy the general linear equation

    \[ \label{eq:3.14} \mathbb{L} x(t) = F(t) \]

    Applicability of the Principle of Superposition to a system provides a tremendous advantage for handling and solving the equations of motion of oscillatory systems.