Loudness and intensity

Loudness is a sensation, not a measurement you can point an instrument at directly. It lives in the listener — shaped by ear sensitivity, attention, and distance from the source — which is why the "same" sound can seem louder to one person than another.

Intensity is the physical quantity behind that sensation: the amount of sound energy passing through a unit area, perpendicular to the direction of travel, every second. Intensity is what a sound-level meter actually measures, and it is the true, objective cause of loudness. Intensity depends on how vigorously the source is vibrating. A source with a larger amplitude of vibration pushes more energy into the surrounding air each cycle, so it radiates greater intensity. Since intensity grows with the square of the amplitude, doubling the amplitude of vibration multiplies the intensity by four, not two:

where: $I$ is intensity (sound energy per unit area per unit time) and $A$ is the amplitude of vibration of the source. This relation is not required for ICSE numerical work, but it explains why even a modest increase in how hard you strike, pluck, or blow something produces a noticeably louder sound — the effect on intensity is squared, not linear. Intensity also falls off as you move away from the source, because the same total sound energy has to spread over an ever-larger area as it travels outward:

where: $r$ is the distance from the source. This inverse-square fading is why a shout that is deafening up close becomes a faint murmur across a field, and why standing closer to any source — a speaker, a drum, a person talking — makes it sound louder without the source itself changing at all.

The decibel (dB) is the practical unit in which loudness or sound level is expressed. At ICSE level it is introduced only by name and by rough scale — a whisper sits around 20 dB, ordinary conversation around 60 dB, and heavy traffic around 80–90 dB — with no logarithmic formula or calculation expected.

Noise pollution is loud, unwanted sound in the environment — from traffic, construction, machinery, or loudspeakers — intense or persistent enough to damage hearing, disturb sleep, raise stress, and cut into concentration.

Step 1. Start from the chain already built: amplitude drives intensity, and intensity drives the sensation of loudness. Anything that increases the vibration amplitude of a source — hitting harder, blowing stronger, plucking further — pushes more energy per second into the surrounding air, so the intensity rises, and the listener's ear registers a louder sound.

Step 2. The inverse-square fading with distance is not a special property of sound; it is simple geometry. Picture the sound energy leaving the source spread evenly over an expanding sphere. As the sphere's radius doubles, its surface area quadruples, so the same energy is now spread four times thinner — intensity falls to a quarter. This is exactly why moving twice as far from a loudspeaker does not merely halve the loudness; the physical intensity drops much faster than distance alone suggests, even though the ear's perceived loudness change feels more gradual.

Step 3. The decibel exists precisely because the ear can handle an enormous range of intensities — from the faintest audible whisper to the threshold of pain — a range too vast to describe conveniently on a simple linear scale. Rather than write out intensities that differ by factors of a trillion, the decibel scale compresses that huge range into small, manageable numbers, which is why a "10 dB louder" jump can represent a large real increase in sound energy even though the number itself only ticked up by 10. ICSE Class 10 does not require the underlying logarithmic formula — only the names and rough dB values of everyday sounds.

Step 4. This same reasoning explains why noise pollution control works at three different points along the chain, not just one: reduce the amplitude at the source (quieter machines, silencers, regular maintenance), block or absorb the intensity in transit (green belts, noise barriers, sound-absorbing insulation), or protect the listener directly (earplugs, sound-proofed windows). Because loudness ultimately traces back through intensity to amplitude and distance, any measure that lowers one of those two quantities reduces loudness — which is why effective noise control never relies on a single trick.

A drum is struck gently, producing a certain loudness. It is then struck twice as hard with the same stick, doubling the amplitude of the membrane's vibration. A listener who was originally 2 m from the drum also moves back to 4 m. Explain what happens to the intensity of the sound reaching the listener, and hence to the loudness heard, giving reasons at each stage.

Step 1 — Effect of doubling the amplitude. Since intensity varies with the square of amplitude, doubling the amplitude alone would multiply the intensity by $2^2 = 4$ — the harder strike, on its own, makes the sound four times as intense at any fixed distance.

Step 2 — Effect of doubling the distance. Since intensity varies inversely with the square of distance, moving from 2 m to 4 m alone would divide the intensity by $2^2 = 4$ — the extra distance, on its own, cuts the intensity to a quarter.

Step 3 — Combine both changes. The two effects are equal and opposite: intensity is multiplied by 4 from the harder strike and divided by 4 from the greater distance, so the two changes cancel and the intensity reaching the listener's ear is essentially unchanged.

Step 4 — Loudness. Because intensity at the listener's ear stays the same, the loudness heard is also essentially the same as before — even though both the strike and the distance changed. This shows loudness depends on the combination of amplitude and distance, not on either one alone.

Answer: The harder strike (amplitude doubled) would quadruple the intensity, but moving to double the distance would cut it to a quarter, so the two effects cancel and the listener hears essentially the same loudness as with the original gentle strike at the closer distance.

Competitive stretch

If the same listener had instead stayed at the original 2 m distance when the drum was struck harder, the intensity reaching them would rise by the full factor of 4 (from amplitude alone), so the sound would be noticeably louder — the cancellation in the worked example only happens because the distance change was chosen to exactly offset the amplitude change.

• Treating loudness and intensity as the same thing — intensity is the objective sound energy per unit area per second; loudness is the ear's subjective perception of it, and the two can diverge between listeners.
• Confusing loudness (linked to amplitude) with pitch (linked to frequency) — a loud sound need not be high-pitched, and a soft sound need not be low-pitched; they are independent characteristics of sound.
• Assuming intensity falls off in direct proportion to distance — it actually falls off with the square of distance (inverse-square law), so doubling distance cuts intensity to a quarter, not a half.
• Trying to recall or apply a decibel formula such as $\beta = 10\log(I/I_0)$ — this is beyond the ICSE Class 10 syllabus; only the name "decibel" and rough everyday dB values are required.

Ask the tutor: Want a few more amplitude-distance combination problems, or a quick rundown of real-world noise-control case studies to test yourself on?