Theory & learning
Every idea as a card: definition, key points, formulas, examples and the mistakes to avoid.
Sound is energy from vibration
Sound is a form of energy that produces the sensation of hearing. It is always produced by a vibrating object — a rapid to-and-fro motion.
- Sound needs a material medium (solid, liquid or gas), so sound waves are mechanical waves.
- A vibrating object moving forward compresses the air in front of it, creating a compression.
- Moving backwards it leaves a region of low pressure, a rarefaction.
- A repeating series of compressions and rarefactions travels outward as a longitudinal wave.
No medium, no sound. Light can cross vacuum; sound cannot.
Did you know? A bell ringing inside a vacuum jar can be seen vibrating but not heard at all.
Wavelength, frequency, period, amplitude, speed
Five quantities describe every sound wave. Learn them together — they are linked by one equation.
- Wavelength (λ): distance between two consecutive compressions or two consecutive rarefactions. Unit: metre.
- Frequency (f): number of complete vibrations per second. Unit: hertz. It is fixed by the source and does not change when the wave enters another medium.
- Time period (T): time for one complete wave to pass a point. Unit: second.
- Amplitude (A): maximum displacement of a medium particle from its mean position. Unit: metre.
- Speed (v): distance travelled by the wave per second. Unit: m/s. In the same conditions it is the same for all frequencies.
Wave equation linking speed, frequency and wavelength.
Time period is the reciprocal of frequency.
Frequency does not change when sound moves from air into water — the speed and wavelength change instead.
Free vibration with constant amplitude
The periodic vibrations of a body in the absence of any external force are called natural or free vibrations. The frequency with which it vibrates is its natural frequency and the corresponding period is its natural period — both are decided only by the body itself.
- Amplitude and frequency both stay constant, so the total energy of the body is conserved and no energy is lost.
- These vibrations are simple harmonic: the only force acting is the restoring force, which is proportional to the displacement and directed towards the mean position.
- Truly constant-amplitude vibration is only possible in vacuum; any medium offers resistance, so in practice free vibrations always die out.
- Natural frequency depends on the size, shape, mass, tension and material (elasticity) of the body — not on how hard you disturb it.
- Striking the body harder increases the amplitude and the loudness, but the frequency (pitch) stays exactly the same.
- For a simple pendulum the period depends only on length and gravity, and is independent of the mass of the bob and of the amplitude (for small swings).
- A stretched string can vibrate in several modes (fundamental and overtones); the lowest frequency mode is the fundamental note.
Angular natural frequency of a loaded spring.
Natural period of a loaded spring.
Natural period of a simple pendulum of length l.
Natural frequency in hertz — vibrations completed each second.
Fundamental frequency of a stretched string of length l, tension T and mass per unit length m.
Natural frequency does not depend on the amplitude. Hitting a tuning fork harder makes a louder note, never a higher-pitched one.
In a stretched string the modes have frequencies in the ratio 1 : 2 : 3 and wavelengths in the ratio 6 : 3 : 2.
Did you know? A pendulum clock runs slow when taken up a mountain: g is smaller there, so T = 2π√(l/g) becomes longer.
Amplitude that dies away
The periodic vibrations of a body of decreasing amplitude in the presence of a resistive force are called damped vibrations.
- Two forces act: the restoring force and the frictional or resistive force of the medium.
- Each vibration loses some energy as heat, so the amplitude falls steadily.
- How fast it dies depends on the viscosity and density of the medium and on the shape and size of the body.
- The frequency of damped vibration is slightly less than the natural frequency.
Cosine gives the rhythm; the exponential envelope shrinks the amplitude.
Every real natural vibration in a medium is in fact a damped vibration.
Vibration under an external periodic force
Vibrations of a body under the influence of an external periodic force are called forced vibrations. Three forces act: restoring, resistive and the external driving force.
- The body gradually gives up its own natural frequency and vibrates at the frequency of the applied force.
- The amplitude depends on how close the driving frequency is to the natural frequency, and it does not change with time.
- If the two frequencies are far apart the amplitude is very small.
- The energy lost to damping is continuously replaced by the external force.
Forced vibration is not the same as resonance. Resonance is the special case when the two frequencies are exactly equal.
A special case of forced vibration
When the frequency of the external periodic force equals the natural frequency of the body, the body vibrates with a greatly increased amplitude. This is resonance, and the large vibrations are resonant vibrations.
- Conditions: the applied frequency must exactly equal the natural frequency, and the force must actually produce forced vibration in the body.
- The amplitude at resonance is limited by the frictional forces present — less damping gives a taller, sharper peak.
- At resonance the body vibrates in phase with the driver and radiates a large amount of energy, so a loud sound is heard.
- Resonant vibrations continue for a long time after the external force stops; ordinary forced vibrations die at once.
Amplitude of a driven oscillator; maximum when ω = ω₀.
Soldiers break step on a suspension bridge to avoid driving it at its natural frequency.
Did you know? The Tacoma Narrows Bridge collapse of 1940 is the most famous resonance disaster.
Natural vs damped vs forced
The examiner's favourite table. Compare frequency, amplitude and energy across the three.
| Feature | Natural | Damped | Forced |
|---|---|---|---|
| Definition | Periodic vibrations with no external force acting | Periodic vibrations of decreasing amplitude in a resistive medium | Vibrations under an external periodic force |
| Frequency | Depends on the size and shape of the body; stays constant | Slightly less than the natural frequency; decrease depends on damping | Equal to the frequency of the applied force |
| Amplitude | Constant with time | Falls steadily and the vibrations finally stop | Depends on the applied frequency; constant with time |
| Energy | No loss of energy | Some energy lost as heat in each vibration | Loss made up by the external force |
| Example | Pendulum swinging in vacuum | Pendulum swinging in air or water | Tuning fork stem pressed on a table top |
Sound that comes back
The repetition of sound caused by the reflection of sound waves is called an echo. Because a sound sensation persists in the brain for about 0.1 s, the reflected sound must arrive at least 0.1 s later to be heard separately.
- In air at 344 m/s, sound covers 34.4 m in 0.1 s, so the reflector must be at least 17.2 m away.
- Closer than about 17 m, the reflected sound merges with the original.
- Repeated reflections that prolong sound in a hall are called reverberation; rolling thunder is a natural example.
- In sea water (v ≈ 1400 m/s) the minimum distance becomes about 70 m.
- The reflector must be large compared with the wavelength, and the original sound loud and short.
Distance to the reflector, since the sound travels there and back.
Did you know? Bats fly far slower than sound, which is why their echoes reach them in time to steer.
Loudness, pitch and quality
Two sounds are distinguished by loudness, pitch (shrillness) and quality (timbre). Each has a subjective sensation and an objective, measurable partner.
- Loudness is the sound energy reaching the ear per second; it depends on amplitude, and is a sensation. Its objective partner is intensity, measured in W/m². Loudness is measured in phons, sound level in decibel.
- Loudness ∝ (amplitude)², falls as 1/(distance)², and grows with the vibrating surface area, the density of the medium and the presence of a resonating body.
- Pitch depends on frequency: higher frequency gives a shriller note. Pitch is subjective; frequency is objective.
- Quality depends on the wave form. Two sounds of the same amplitude and frequency from different instruments differ in wave shape.
- A musical sound has a regular, periodic wave form; noise has an irregular wave form with sudden changes in amplitude.
Loudness and intensity are not the same. Intensity is measurable; loudness also depends on the listener's ear.