Intensity

Intensity is a concept that you will draw upon in wave, energy and astrophysics calculations. As with Density, this 'interdisciplinary' concept is worth absorbing (!) in its own right.

Key Concepts

Waves transfer energy.

The intensity of a wave is the power incident on a surface per unit area:

$$I={P\over A}$$

• $$I$$ is intensity in $$\text{W m}^{-2}$$
• $$P$$ is power in $$\text{W}$$
• $$A$$ is surface area in $$\text{m}^2$$

Since power is be defined as energy produced per unit time, an equivalent equation for intensity can be derived:

$$I={E\over At}$$

• $$I$$ is intensity in $$\text{J m}^{-2}\text{ s}^{-1}$$
• $$E$$ is energy in $$\text{J}$$
• $$t$$ is surface area in $$\text{s}$$

Note that intensity is proportional to amplitude2 When two waves interfere the amplitudes add together so if each wave had amplitude A the resultant would have amplitude 2A. If the intensity of each was I then the resultant intensity would be 4I.

Essentials

For energy radiating outwards into 3-dimensional space from a point or sphere, the 'surface area' of the wave surface is that of a sphere:

$$A=4\pi r^2$$

• $$r$$ is radius from the source in $$\text{m}$$

The intensity of a wave spreading outward from a point source therefore follows an inverse square law:

$$I\propto {1\over r^2}$$

Or, rewriting this as an equation for locations '1' and '2':

$$I_\text{1}{r_\text{1}}^2=I_\text{2}{r_\text{2}}^2$$

In addition, since the energy of a wave is proportional to the square of the amplitude:

$$I\propto {x_0}^2$$

And, once more, rephrasing as an equation for the intensities and amplitudes at '1' and '2':

$${I_\text{1}\over {x_\text{0,1}}^2}={I_\text{2}\over {x_\text{0,2}}^2}$$