Vibrations and resonance

Free (natural) vibrations. Disturb a body once — pluck it, strike it, pull it aside and let go — and if nothing keeps pushing it, it vibrates on its own at a fixed rate called its natural frequency. This frequency is a property of the body itself: its mass, length, tension, stiffness, or shape decide it, not how hard or gently it was disturbed. A tuning fork struck once and left to ring, or a pendulum given a single push and left to swing, both settle into their own natural frequency regardless of the strength of the initial disturbance.

Damped vibrations. No real vibration continues forever. Air resistance and internal friction quietly remove energy every cycle, so the amplitude of a free vibration steadily shrinks with time until the body comes to rest — this dying-away is damping. A plucked guitar string rings out but grows quieter and quieter; an unpushed swing gradually slows and stops. At this level the important point is that damping mainly reduces amplitude, not the frequency of vibration.

Forced vibrations. If an external periodic force keeps pushing a body again and again at some frequency of its own choosing — a frequency generally different from the body's natural frequency — the body is compelled to vibrate at the driver's frequency, not its own, and the resulting amplitude stays modest. A loudspeaker diaphragm driven by an electrical signal, or a violin's wooden body driven by its vibrating string, are both examples of forced vibration.

Resonance. Resonance is the special case of forced vibration in which the driving frequency exactly equals the body's natural frequency. Every push now arrives in step with the body's own rhythm, so energy is added efficiently on every single cycle instead of being partly wasted or opposed. The result is an amplitude that grows to be much larger than in ordinary, mismatched forced vibration — sometimes large enough to be dangerous, as with bridges, or large enough to be strikingly audible, as with two tuning forks.

At this level, vibrations and resonance are explained purely through energy transfer and timing, with no standing-wave or SHM formulas required. A vibrating body has a fixed natural frequency because its inertia (mass) and its restoring influence (tension, stiffness, elasticity) together decide how quickly it can complete one cycle — heavier or looser systems swing more slowly, stiffer or tighter ones swing faster, and this ratio is fixed however hard the body was first disturbed. When an external force pushes the body only once, or intermittently at random moments, the pushes sometimes add to the body's motion and sometimes oppose it, so energy is delivered inconsistently and the amplitude built up is modest — this is ordinary forced vibration. But when the driving force repeats at exactly the body's own natural frequency, every single push arrives at the same phase of the body's motion — always helping, never opposing. Energy is therefore added on every cycle without being cancelled out on the next, so the amplitude keeps climbing until damping (friction, air resistance) finally balances the energy being poured in. That balance point, at resonance, is far higher than the amplitude reached under any mismatched driving frequency, which is precisely why resonance stands out as dramatically stronger than everyday forced vibration. This timing argument also explains the classic demonstrations. Two tuning forks of equal natural frequency: the sound waves from the struck fork arrive at the second fork as a periodic force at exactly its own natural frequency, so energy builds and it starts ringing audibly on its own. Soldiers marching in step near a bridge's natural frequency: each footfall is a small periodic push, and if the marching frequency matches the bridge's natural frequency, those small pushes accumulate into oscillations large enough to be structurally dangerous — hence the order to break step. A radio tuner: turning the dial adjusts an internal circuit's own natural frequency until it matches the broadcast frequency, so that one station's signal is picked out strongly while all the others, driving at mismatched frequencies, stay weak.

A tuning fork of natural frequency 256 Hz is struck and set vibrating close to a second, identical tuning fork of natural frequency 256 Hz standing nearby. Explain what is observed, then predict what changes if the second fork is replaced by one of natural frequency 288 Hz, and finally state what would happen if the second fork's frequency were adjustable and slowly tuned from 256 Hz up to 288 Hz.

Step 1 — Identify the mechanism. The struck fork radiates sound waves that act as a periodic driving force on the second fork, arriving at the struck fork's own frequency of 256 Hz.

Step 2 — Apply the resonance condition for the matched case. The second fork's natural frequency (256 Hz) is exactly equal to the driving frequency (256 Hz), so the resonance condition is satisfied.

Step 3 — State the matched outcome. The second fork absorbs energy efficiently on every cycle, so its amplitude grows and it starts vibrating audibly on its own, even though nothing touched it directly.

Step 4 — Consider the mismatched 288 Hz case. Here the driving frequency (256 Hz) no longer equals the second fork's natural frequency (288 Hz); the pushes fall out of step with the fork's own motion, so only a weak, barely noticeable forced vibration occurs and resonance does not happen.

Step 5 — Consider the swept case. As the second fork's natural frequency is slowly tuned from 256 Hz toward 288 Hz, it passes through exact equality with the 256 Hz driver only right at the start; as the mismatch grows, the response weakens steadily, so the loudest ringing of the second fork occurs precisely when the two frequencies coincide.

Answer: The 256 Hz fork resonates strongly and rings audibly; the 288 Hz fork shows only weak forced vibration with no audible ringing, and any tuning that moves the second fork's frequency away from 256 Hz weakens its response — resonance is sharply strongest exactly at matched frequency.

Competitive stretch

A swing's parent can push either at random moments or exactly once per full swing (the swing's own natural frequency). Using the same timing argument, rhythmic pushing at the natural frequency is the resonance case: each push consistently adds energy in phase with the motion, so the swing climbs to the greatest height, while random pushes sometimes help and sometimes oppose the swing, keeping the amplitude modest.

• Treating "forced vibration" and "resonance" as the same thing: all resonance is forced vibration, but forced vibration counts as resonance only when the driving frequency exactly equals the natural frequency; any mismatch gives ordinary, smaller-amplitude forced vibration.
• Assuming natural frequency depends on how hard a body is struck or disturbed — it does not; it is fixed entirely by the body's own physical properties (mass, length, tension, stiffness), regardless of the strength of the initial push.
• Believing damping mainly changes the frequency of vibration — at this level, damping is understood chiefly as a steady decay of amplitude with time, not a change in how fast the body vibrates.
• Thinking resonance needs two identical objects like tuning forks — it is a general frequency-matching condition seen just as clearly in a swing being pushed rhythmically or a radio circuit tuned to a broadcast frequency.

Ask the tutor: Not sure whether a described situation counts as free, damped, forced, or resonant vibration — ask the AI tutor to work through a few example scenarios with you.