CHAPTER 3.
Harmonic oscillations

The simplest example of a simple harmonic oscillator is a mass $m$ on a horizontal frictionless plane, coupled to a spring with constant $k$ attached on the other end to a wall. The mass can oscillate with a unique frequency $\omega_0$ determined by the mass itself and the spring constant. The moti0n

The equation of motion

Newton’s second law is
\begin{equation}
m\ddot{x}=-kx-\alpha\dot{x}\ .
\end{equation}
This equation is usually written in the form
\begin{equation}
\ddot{x}+\frac{\alpha}{m}\dot{x}+\frac{k}{m}x=0\ .
\end{equation}
It is also convenient to rewrite the equation in the form
\begin{equation}
\ddot{x}+\frac{2}{\tau}\dot{x}+\omega_0^2x=0\ .
\end{equation}
where
\begin{equation}
\frac{2}{\tau}=\frac{\alpha}{m}\quad ;\quad \omega_0^2=\frac{k}{m}\ .
\end{equation}

Solution of the equation of motion

the characteristic equation is
\begin{equation}
\lambda^2+\frac{2}{\tau}\lambda+\omega_0^2=0\ .
\end{equation}
The roots are
\begin{equation}
\lambda_{\pm}=-\frac{1}{\tau}\pm\sqrt{\frac{1}{\tau^2}-\omega_0^2}\ .
\end{equation}

Supercritical damping

When the expression under the square root is positive
\[\frac{1}{\tau^2}-\omega_0^2>0\ ,\]
the solution is
\begin{equation}
x(t)=C_+e^{\lambda_+t}+C_-e^{\lambda_-t}\ .
\end{equation}

Critical damping

In this case
\[\frac{1}{\tau^2}-\omega_0^2=0\ ,\]
and the solution is
\begin{equation}
x(t)=(C_1t+C_2)e^{-t/\tau}\ .
\end{equation}

Sub-critical damping

In this case we have
\[\frac{1}{\tau^2}-\omega_0^2<0\ ,\] and the solution is \begin{equation} x(t)=Ce^{-t/\tau}\cos(\omega t+\phi)\ . \end{equation} where \begin{equation} \omega=\sqrt{\omega_0^2-\frac{1}{\tau^2}}\ . \end{equation}