We will first define resonance in the case where b = 0, meaning there is no damping. In this case, resonance occurs when the frequency of the external force is the same as the natural frequency of the system. When such a situation occurs, the external force always acts in the same direction as the motion of the oscillating object, with the result that the amplitude of the oscillation increases indefinitely. When there is a damping force present, resonance occurs at a slightly different frequency and, though the amplitude does increase rapidly, the damping force prevents the increase from being infinite.

Any structure--a building, a bridge, a wine glass--has what is called a resonant frequency. If an external force is applied to such a structure at its resonant frequency, its amplitude of oscillation will increase greatly. A popular phenomenon is the case of a woman breaking glass by screaming. What breaks the glass is not the force of the scream, but the frequency at which the woman screams. If the frequency happens to be the resonant frequency of the glass, the particles in the glass will vibrate at increasing frequency, until the glass shatters. Engineers and builders must take into account the resonant frequency of the structures they design and construct, so as to prevent the destruction of a given structure by a natural oscillating force (such as wind or sound or tides).

Without going into complex mathematics, this is the most we can do with the topic of resonance. A qualitative understanding of resonance, however, gives us a good understanding of this complex motion.