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The energy carried by a wave is proportional to the square of its amplitude: E ∝ A². Intensity (power per unit area, I = P/A) is likewise proportional to A². For a point source radiating in three dimensions, intensity decreases as the inverse square of distance: I ∝ 1/r². This inverse-square law is a purely geometric result — the same power spreads over a larger spherical surface as r increases. Amplitude therefore decreases as 1/r.
Measure the loudness (approximately ∝ intensity) of a sound source at distances 1m, 2m, and 4m from the source. Verify that intensity roughly quarters each time distance doubles.
From kinetic energy, you know that energy scales with the square of velocity: KE = ½mv². Waves carry energy by making particles oscillate, and the maximum speed of those oscillating particles is proportional to amplitude A — a larger displacement means faster oscillatory motion. Combining these two ideas gives the foundational result: wave energy is proportional to the square of amplitude, E ∝ A². This quadratic relationship is the reason a wave twice as tall carries four times the energy, not twice. The same logic applies to intensity — the power delivered per unit area of wavefront — so I ∝ A² as well. Doubling the amplitude of a sound wave quadruples its perceived loudness in physical terms, which is why acoustics uses a logarithmic decibel scale to make the range of human hearing manageable.
Intensity is defined as power per unit area: I = P/A (where A here is area, not amplitude). Imagine a point source radiating power uniformly in all directions. At distance r, that fixed total power P is spread over the surface area of a sphere: 4πr². Since I = P/(4πr²), intensity falls off as 1/r². This is the inverse-square law, and it is a purely geometric result — the energy doesn't disappear, it just spreads over an ever-larger surface as the wavefront expands. Double the distance from a campfire and the warmth you feel drops to one-quarter; move three times as far and it drops to one-ninth.
The amplitude consequence follows directly. Since I ∝ A² and I ∝ 1/r², combining them gives A ∝ 1/r. Amplitude decreases linearly with distance for a spherical wave in three dimensions. This is why you can shout across a room but not across a football field — the same vocal power spreads over a sphere that is thousands of times larger in area. Whispers carry even less power to begin with, making the 1/r² fall-off doubly unforgiving.
It is important to note the geometric assumption embedded in the inverse-square law. It holds for waves that radiate isotropically into three dimensions. Surface waves on water, confined to two dimensions, spread over a circular perimeter 2πr rather than a spherical surface 4πr², so intensity falls as 1/r instead of 1/r². Waves on a string are one-dimensional — they carry the same intensity everywhere along the string, ignoring damping. Whenever you apply the inverse-square law, verify that the wave is genuinely radiating in three-dimensional open space; beams, waveguides, and acoustic horns deliberately break the 3D assumption to prevent the energy loss that would otherwise occur.