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Harmonic Oscillator Phase

In classical mechanics, a harmonic oscillator is a system which, whendisplaced from its equilibrium position, experiences a restoring force,F, proportional to the displacement, x according to Hooke's law:
where k is a positive constant.


IfF is the only force acting on the system, the system is called a simpleharmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).


If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. 


Depending on the friction coefficient, the system can:
Oscillate with a frequency smaller than in the non-damped case, and an amplitude decreasing with time (underdamped oscillator). Decay exponentially to the equilibrium position, without oscillations (overdamped oscillator). 


If an external time dependent force is present, the harmonic oscillator is described as a driven oscillator.Mechanical examples include pendula (with small angles of displacement), masses connected to springs, and acoustical systems. Other analogous systems include electrical harmonic oscillators such as RLC circuits. 


The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.


Simple harmonic oscillator
A simple harmonic oscillator is an oscillator that is neither driven nor damped. Its motion is periodic— repeating itself in a sinusoidal fashionwith constant amplitude, A. Simple harmonic motion SHM can serve as a mathematical model of a variety of motions, such as a pendulum with small amplitudes and a mass on a spring. It also provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.


In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its period T, the time for a single oscillation, its frequency, f, the reciprocal of the periodf = 1⁄T (i.e. the number of cycles per unit time), and its phase, φ, which determines the starting point on the sine wave. The period and frequency are constants determined by the overall system, while the amplitude and phase are determined by the initial conditions (position and velocity) of that system. Overall then, the equation describing simple harmonic motion.