Why is wave intensity defined as displacement^2 rather than just its magnitude?

In summary, intensity is a classical concept defined as power delivered per unit area, which can be illustrated by mechanical waves where every point is in SHM. This is why intensity is calculated as the square of the amplitude, as it is proportional to the energy carried by the wave. This is not just an arbitrary convention in physics, but rather a fundamental principle based on the relationship between power and energy.
  • #1
jeebs
325
4
pretty straightforward question. for a long time I've been blindly calculating that intensity is (displacement from zero)^2, but never questioned why.

So why is this? I get that by squaring you get a positive value, which helps in, say, quantum probability calculations, but what's wrong with just taking a magnitude of a wave at a given point and calling that the intensity?
is it just one of our arbitrary conventions in physics or is there some real reason?
 
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  • #2
Intensity is a classical concept, so let's leave the quantum realm alone for now.
To answer your question, we have to look back at what intensity is defined as: crudely, it is power delivered per unit area (albeit perpendicular to the propagation), which works out to energy per unit time per unit area.

Now, the energy carried by a wave is proportional to the square of its amplitude. This can best be illustrated with mechanical waves, such as a wave on a rope, in which every point is essentially in SHM. Consequently, intensity of a wave is also proportional to (amplitude^2) of the wave.
 

Related to Why is wave intensity defined as displacement^2 rather than just its magnitude?

1. Why is wave intensity defined as displacement^2 instead of just its magnitude?

The square of the displacement of a wave is used to calculate its intensity because it takes into account the direction of the wave. Since waves can move in both positive and negative directions, their magnitudes can cancel each other out when added together. Squaring the displacement ensures that all values are positive and accurately represent the overall intensity of the wave.

2. Can't the magnitude of a wave also represent its intensity?

The magnitude of a wave only represents its amplitude, which is the maximum displacement from its equilibrium position. However, the intensity of a wave also takes into account its direction and the rate at which energy is being transferred. Therefore, using just the magnitude would not accurately reflect the true intensity of a wave.

3. Is there a mathematical reason for using displacement^2 in the calculation of wave intensity?

Yes, the intensity of a wave is directly proportional to the square of its amplitude. This relationship is derived from the wave equation, which describes the relationship between the displacement, velocity, and acceleration of a wave.

4. Are there any other reasons for using displacement^2 instead of just the magnitude of a wave?

Using displacement^2 is also convenient mathematically because it simplifies the equations used to calculate wave intensity. It also allows for a more direct comparison between different waves, as the units for intensity (W/m^2) are the same as the units for displacement^2 (m^2).

5. Does this mean that waves with larger displacements will always have higher intensities?

Not necessarily. While the intensity of a wave is directly proportional to the square of its displacement, there are other factors that can affect the intensity, such as the frequency and wavelength of the wave. So, a wave with a larger displacement may not always have a higher intensity than a wave with a smaller displacement.

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