Loudness and amplitude

Loudness is the sensation of a sound being soft or loud — a measure of how much sound energy the ear perceives arriving from a source. It lets us tell a whisper apart from a shout even when both are spoken in the same voice.

Amplitude is the maximum displacement of a vibrating particle (or of the source itself) from its rest, or mean, position. On a displacement-versus-time graph of a sound wave, amplitude is the height measured from the central rest line up to the top of a crest — not the full crest-to-trough span. The governing rule linking the two is simple and one-directional:

• Larger amplitude → louder sound.
• Smaller amplitude → softer sound.

When a source vibrates with a bigger swing — plucking a string harder, striking a drum with more force, speaking more forcefully — the particles of the medium are pushed further from their rest position with every oscillation. Physically, this larger swing carries more energy outward from the source each second, and the ear registers this extra energy as greater loudness. The energy carried by a vibrating source does not grow in step with amplitude; it grows much faster, roughly with the square of the amplitude:

where: $I$ is the intensity of the sound (the energy reaching a given area each second) and $A$ is the amplitude of vibration. The proportionality tells you that doubling the amplitude does not merely double the sound energy — it quadruples it, because energy depends on amplitude squared. You will not need to calculate $I$ in absolute units at this stage; the useful fact is only the ratio it implies, and it explains why even a modest increase in how hard you strike or pluck something produces a noticeably louder sound. It is essential to keep loudness separate from pitch. These are two independent characteristics of the same wave:

• Amplitude tells you how big each vibration is → controls loudness.
• Frequency tells you how fast the vibrations repeat → controls pitch.

Because the two vary independently, all four combinations are physically possible: loud and low-pitched (a loud drum), loud and high-pitched (a loud whistle), soft and low-pitched, and soft and high-pitched. Changing how hard you strike a source changes only its amplitude, never its natural frequency.

A vibrating source pushes on the medium around it. When the source swings through a larger distance, it displaces more particles of the medium by a larger amount with each cycle, doing more work on the medium every time it moves. That extra work is carried away from the source as sound energy, spreading outward through the air as a wave. The ear behaves like a tiny detector of this arriving energy: a bigger flow of energy per second striking the eardrum produces a stronger sensation, which we call a louder sound. This is also why a source's frequency has nothing to do with loudness. Frequency is fixed by the physical properties of the source itself — the length and tension of a string, the size and tightness of a drumhead, the shape of a tuning fork — and stays the same no matter how gently or forcefully the source is set vibrating. Only the amplitude, the size of each swing, changes with how much effort goes into starting the vibration. That is why striking the same tuning fork gently or hard always produces the same musical note, but at very different loudness. Amplitude is not the only thing that affects the loudness a listener actually hears: sound spreads out and weakens with distance from the source, and different ears have different sensitivities. But the amplitude of vibration at the source remains the fundamental physical cause of loudness, and it is the property tested when you are asked to read or compare amplitudes on a graph.

A guitarist plucks a string gently, producing a wave of amplitude $2\ \text{mm}$. She then plucks the same string, at the same fret, much harder, producing a wave of amplitude $6\ \text{mm}$. In both cases the note produced is the same. Compare the loudness of the two plucks, and state what stays the same and what changes.

Step 1. Identify what stays the same. The string, its length and its tension are unchanged between the two plucks, so it vibrates at the same natural frequency both times — this is why the same musical note is heard in both cases. Pitch is unaffected by how hard the string is plucked.

Step 2. Identify what changes. The harder pluck sets the string swinging through a bigger distance from its rest position before it settles back, so the amplitude of vibration is larger: $6\ \text{mm}$ compared with $2\ \text{mm}$, three times as large.

Step 3. Compare the loudness using the amplitude-energy rule. Since sound energy is proportional to the square of amplitude, $I \propto A^2$, tripling the amplitude increases the energy radiated, and hence the loudness reaching the ear, by a factor of $3^2 = 9$.

Step 4. State the conclusion. The harder pluck produces a note of the same pitch as the gentle pluck, but roughly nine times louder, because its amplitude is three times as large and loudness scales with the square of amplitude.

Competitive stretch

If a third pluck produced a wave carrying only $\dfrac{1}{4}$ of the sound energy of the original $2\ \text{mm}$ pluck, what amplitude would that correspond to? Since $I \propto A^2$, halving the amplitude cuts the energy to one-quarter, $\left(\tfrac{1}{2}\right)^2 = \tfrac{1}{4}$. So the amplitude would be $1\ \text{mm}$, half of the original $2\ \text{mm}$ — a gentler pluck than even the first one.

• Confusing loudness with pitch. Striking or plucking something harder changes loudness (amplitude), never pitch (frequency) — they are independent properties of the same wave.
• Assuming a bigger source is always louder. Loudness depends on the amplitude of vibration, not the physical size of the source — a small source vibrating with a large amplitude can be louder than a large source vibrating gently.
• Misreading amplitude on a graph. Amplitude is the height from the rest (central) line to a crest, not the full crest-to-trough distance — a very common measuring error.
• Treating amplitude and loudness as proportional in the same ratio. Loudness (sound energy) grows with the square of amplitude, so doubling amplitude quadruples the energy, not just doubles it.

Ask the tutor: Want to practise reading amplitude off different sound-wave graphs or work through more loudness comparisons using the amplitude-squared rule? Ask your AI tutor to walk through it with you.