Resonance

• Every oscillating system has its own natural frequency (f₀) — the rate it swings/vibrates at when left alone.
• Resonance happens when an external driving frequency (f) matches f₀ exactly — the amplitude shoots up to a maximum.
• In real life, damping (resistive forces like friction/air resistance) always removes energy, so oscillations die down over time — but the frequency stays constant even as amplitude shrinks.
• There are three types of damping: light (slow exponential decay, still oscillates), critical (fastest return to rest, no oscillation), and heavy (slow return to rest, no oscillation).
• Free oscillations = no external force, system just vibrates at f₀. Forced oscillations = a periodic external force keeps replacing lost energy, and the system is forced to move at the driver's frequency.
• A resonance curve (amplitude vs driving frequency) peaks sharply at f = f₀. More damping → the peak gets lower, wider, and shifts slightly left.
• Damping can also occur through plastic deformation of ductile materials — the material absorbs the oscillation's kinetic energy by permanently stretching (e.g. a climbing rope).

Picture pushing a friend on a swing. The swing has its own natural rhythm — say, it swings back and forth once every 2 seconds no matter how hard you push it (amplitude doesn't change that rhythm, only how high it goes). That built-in rhythm is called the natural frequency, symbol f₀.

Now imagine you give it a tiny push every single time it swings back towards you — timed exactly with that 2-second rhythm. Each push adds a bit more energy at exactly the right moment, so the swing goes higher and higher. The frequency at which you are pushing is called the driving frequency, symbol f.

When your pushing frequency (f) becomes exactly equal to the swing's natural frequency (f₀), something dramatic happens: the amplitude increases hugely compared to any other push rate. This maximum-amplitude condition is resonance.

Why does this happen physically? At resonance, the driving force and the oscillator's velocity are always in phase — the push always happens exactly when the object is moving in the same direction as the push. This means energy is transferred from the driver to the oscillating system most efficiently possible. If the driving frequency doesn't quite match, some pushes end up fighting against the motion instead of helping it, so energy transfer is less efficient and the amplitude increase is smaller.

Real examples of resonance

• An organ pipe — air resonates down the column, setting up a stationary (standing) wave.
• radio