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The motion generated by the simple harmonic oscillator istermed a "free oscillation;" once set into oscillation , tha motion would never case. This is, of course , an oversimplication of the actual case in which dissipative or frictional forces would eventually demp the motion to the point that oscillations would no oonger occur. It is possible to analyze the motion ub such a case by incorporating into the differential equation a term which represents the dampening force. Now, it does not seem reasonatle that the damping force should in general depend on the displacement, but it could be a function of the velocity or perhaps of some higher time derivative of the displacement. It is frequently assumed that the damping force is a linear function of the Velocity, Fc=(alpha)V. We will consider here only one-dimensial damped oscillations so that we can represent the damping term by -bx. The paramater b must be positive in order that the force