*Oscillations in which the amplitude decreases with time. Consider the equation of motion mẍ=−kx−cẋ, where the first term on the right-hand side arises from an elastic restoring force satisfying Hooke's law, and the second term arises from a resistive force. The constants k and c are positive. The form of the general solution of this linear differential equation depends on the auxiliary equation mα2+cα+k=0. When c2 < 4mk, the auxiliary equation has non-real roots and damped oscillations occur. This is a case of weak damping. When c2=4mk, the auxiliary equation has equal roots and critical damping occurs: oscillation just fails to take place. When c2>4mk there is strong damping: the resistive force is so strong that no oscillations take place.
http://www.lon-capa.org/~mmp/applist/damped/d.htm An applet exploring the effects of changing parameters on damped oscillations