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$x''+\gamma x'+w_0^2x=0$

That is the general equation for damped harmonic motion. What is the term or name that describes gamma ?

Is it called the damping constant ? I know its the ration between the resistive coefficient (b) and mass of the system (m) but what do we actually call it ?

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I changed your tag from "harmonic functions" to "differential equations." Harmonic functions are something completely different. –  icurays1 Apr 28 '13 at 19:07

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The second order differential equation arises from the application of Newton's Second Law.

$$\sum F = ma$$ In the case of oscillatory systems, such as a spring, there are two forces exerted onto the spring. The restoring force $-kx$ and the damping force $-bv$ where $v$ is the velocity of the spring. Note that the assumption of linear resistive force is only an approximation, and at higher velocities drag is actually proportional to the square of velocity.

$$\begin{align} -kx-bv &= ma\\ m\ddot{x} + b\dot{x}+kx&=0\\ \ddot{x}+\frac{b}{m}\dot{x}+\frac{k}{m}x &= 0 \end{align}$$ which is the equation you had above.

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It's called damping ratio, damping coefficient of damping constant. it measures how the oscillations of the system decay after an initial force is applied. You can calculate it with the expression: $$\gamma=\frac{c}{\sqrt{km}}$$ where $c$ is the friction coefficient, $m$ the mass of the oscillating object and $k$ the elastic constant corresponding to Hooke's law. If $\gamma>1$ we say that the oscillator is overdamped.

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