# 9.1.4 Wave Intensity

Which causes more damage to your eyes? Staring into the light emitted by a 1 mW laser pointer or staring into the light emitted by a 10 W filament light bulb? Well, it depends…

The intensity of a wave (at a location) refers to the power per unit area (at that location). Take for example a power source of 10 W. Now think of a point at a distance of 0.5 maway from the source. Do we get all 10 W of power arriving at this point? If the wave propagates uniformly in all directions, the total power of 10 W would have to be transmitted to each and every point that is 0.5 m away from the source. So we are talking about 10 W being spread evenly onto a spherical surface (not volume) of area $\displaystyle 4\pi {{(0.5)}^{2}}=3.14\text{ }{{\text{m}}^{2}}$. The intensity of the wave at any point on this spherical surface is thus 10 W ÷ 3.14 m2 = 3.18 W m-2.

Since the power P is propagated onto $\displaystyle 4\pi {{r}^{2}}$ areas, the intensity of a point source at distance r away is given by

$\displaystyle \displaystyle \mathrm{I}=\frac{P}{{4\pi {{r}^{2}}}}$.

In general, waves can propagate in one, two or three dimensions. Sound wave traveling down a narrow tube is an example of a 1D wave. The circular ripples on a water surface is a 2D wave. The light leaving the Sun is a 3D wave.

3D wave propagation

A 3D wave propagates in 3 dimensions onto spherical surfaces whose area varies with the square of the distance (since surface area of sphere is $\displaystyle 4\pi {{r}^{2}}$). As such we can deduce that for a 3D wave, $\displaystyle \displaystyle \mathrm{I}\propto \frac{1}{{{{r}^{2}}}}$ and $\displaystyle A\propto \frac{1}{r}$ (because $\displaystyle \displaystyle \mathrm{I}\propto {{A}^{2}}$).

2D wave propagation

A 2-D wave propagates onto cylindrical surfaces that expand with r (since circumference of circle is $\displaystyle 2\pi r$). So for a 2-D wave, $\displaystyle \displaystyle \mathrm{I}\propto \frac{1}{r}$ and $\displaystyle A\propto \frac{1}{{\sqrt{r}}}$.

1D wave propagation

A 1-D wave propagates only in one direction along a line, so the power P is propagated within a constant area. For a 1-D wave, both I and A are constant and do not diminish with distance.

Now back to the question of light bulb vs laser. A light bulb is 3-D source. Staring into the light bulb 10 cm away and 100 m away is two completely different experiences because the intensity of the light at those two distances are different by a factor of 1,000,000! A laser beam on the other hand is a 1-D source. The intensity of the laser beam is the same whether 1 m or 100 m away. Ouch.

Video Explanation    Wave Intensity

Concept Test             1821