Wave Intensity

Power per unit area in three-dimensional waves. Theory, formulas, worked examples and interactive graphs.

Conceptual context

When a wave propagates in three dimensions, its energy spreads over increasingly larger surfaces. This geometric dispersion explains why sound weakens as we move away from a source and why the same light power looks dimmer when it illuminates a larger area.

Wave intensity measures that energy concentration: power transported per unit area perpendicular to propagation. Energy flux describes local transport, while intensity $I$ summarizes how much power crosses a measuring surface.

The concept is central in acoustics, optics, and radiometric safety. It lets us compare whispers, loudspeakers, lamps, lasers, and exposure limits through one physical idea: energy delivered per second and per square meter.

🟢 Essential level

Imagine a small lamp at the center of a dark room. Light leaves in every direction. Near the lamp it is concentrated on a small surface and looks bright; farther away, the same emission is spread over more wall and looks weaker.

Intensity $I$ measures how much wave power crosses each square meter. If you move twice as far from an ideal sound source, intensity does not fall to one half, but to one quarter. The reason is not that the source emits less, but that the available spherical surface grows much faster.

This explains why shouting at someone far away feels so ineffective compared with speaking up close: the same voice, at four times the distance, arrives with only one-sixteenth of the original intensity. Doubling the distance twice multiplies the weakening, it does not add it up.

[!TIP] Think of intensity as the physical brightness of the wave. Near the source, energy is concentrated; far away, it is dispersed. Total energy may be conserved while each square meter receives less.

🔵 Formal level

Basic formulas

For a point source emitting power $P$ isotropically in three dimensions, the intensity at distance $r$ is obtained by dividing the total power by the area of the sphere centered on the source:

Intensity from source power

The spreading area $A$ follows spherical geometry:

Sphere area at distance r

This relation expresses the inverse square law. When distance doubles, the receiving surface becomes four times larger and the same power is diluted over four times more area.

Real student questions

Why does intensity decrease with the square of distance and not linearly?

Because total power is distributed over a spherical surface. When distance doubles, that surface quadruples. The same energy per second is spread over more area, so each square meter receives less.

Why do we use decibels instead of always measuring W/m²?

Because the audible range is enormous. The logarithmic scale turns very small and very large intensities into manageable numbers, and it also matches human loudness perception more closely.

How do intensities from two simultaneous sources add?

First add physical intensities, not decibels. Then, if necessary, convert the total result to sound level. Two equal sources do not double the decibel number; they produce a moderate increase.

Why is intensity always positive if acoustic pressure oscillates between positive and negative values?

Because average intensity represents energy transport. Compressions and rarefactions both contribute to wave energy even though they have opposite pressure signs.

Final synthesis

Wave intensity quantifies the spatial concentration of energy transport. It measures power per unit area and lets us compare sound, light, and electromagnetic waves with the same physical unit.

Its basic law for an isotropic point source comes from geometry: the same power is distributed over larger spherical surfaces. In real situations, directivity, absorption, reflections, and near-field conditions must also be considered.

Understanding this quantity requires separating physical intensity, decibel level, and perception. Intensity says how much energy arrives; sound level translates that intensity into a practical scale; human perception adds a physiological response that is not linear.