Wave Intensity



The intensity of a wave is proportional to the square of its amplitude. I shall look at the meaning of this sentence here.

First, what is the intensity of a wave? If you shake one end of a rope, you make a wave motion. This wave travels along the rope, and carries kinetic and potential energy with it. Use a marker to put a mark on one point on the rope. Every second, a certain amount of energy travels pass this mark. This is the intensity for the wave on the rope. It is the energy per unit time that passes a point.

The energy at each small segment of the rope is proportional to its amplitude. This is because the segment moves up and down, as if it is attached to a spring. We know from spring oscillation that the total energy kx²/2 is proportional to the square of the amplitude. (k is the spring constant and x the displacement.) This means that the total energy of the segment is also related to its amplitude in the same way. This is explained in more details in the section on Wave Energy.

Therefore the intensity, which is just the energy flow per second, is also proportional to the square of the amplitude.

What about the intensities of sound wave and light wave? These are a bit more complicated because they are in three dimensions.

When a sound wave propagates through the air, the intensity is related to the energy flow per second. However, we cannot just think of energy flow through a point. We need to consider the energy flow through an area. To make things simple, we take this as the area perpendicular to the wave velocity, or sound direction. The full definition for intensity for sound wave is the energy flow through 1 m² of the area in 1 second.

But where is this energy hidden? Imagine a tiny volume of air that lies in the path of the sound wave. This volume oscillates along the direction of the sound. It has kinetic energy. It is forced to move back and forth by the changing pressure of the air on either side, which has potential energy. The surrounding air acts like a spring, and we can think of this tiny volume of air like a piece of wood attached to the spring. Like the wood, the total energy of this volume of air is proportional to the square of its amplitude. In air, the bigger the displacement, the bigger the pressure change. So the amplitude can refer to either maximum displacement or maximum change in pressure.

What about light wave? Light is an electromagnetic wave. This means that it is made up of an electric field and a magnetic field that oscillates together. An oscillating electric field just means that the field increases and decreases in strength at any point in air or vacuum.

Where is the energy hidden? There is energy in the electric field and magnetic field. If you put an electron in an electric field, the electron will move. The energy of the field is proportional to the square of the field strength. In the light wave, the amplitude is the maximum strength of the oscillation electric field. So the intensity must be proportional to the square of the amplitude of the electric field.

The magnetic field strength in light wave is proportional to the electric field strength. So we can also say that the light intensity is proportional to the square of the amplitude of the magnetic field.




Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 25 May 2011.