# Sound Intensity and Frequency Relation

The intensity of a wave is proportional to the square of the amplitude of the and the frequency of the wave given by the equation:

$$I = \frac{\overline{P}}{S} = 2\pi^2\rho vf^2A^2$$

Now when talking about sound waves the sound level of a noise (loudness), it is proportional to the intensity of the source. If intensity of a wave is proportional to frequency, why doesn't sound level, in general, depend on frequency. i.e. Higher frequency, higher intensity, higher sound level.

Why is this not the case?

For example if you change the frequency of a sound wave from 500Hz to 15000Hz, this won't necessarily be perceived as a change in loudness. But hasn't the intensity of the sound increased, because the frequency has increased? The loudness of a sound does not seem to depend on frequency. But according to the equation above, intensity DOES depend on frequency. So why does it not change perceived loudness?

• Loudness is not proportional to intensity. hyperphysics.phy-astr.gsu.edu/hbase/sound/loud.html – Farcher Feb 27 '16 at 15:32
• Not sure what you are asking. If you have the same displacement, you will have greater pressure at higher frequency. Sound level depends on the energy in the wave, which depends on the pressure; and pressure is a function of (displacement) amplitude and frequency. – Floris Feb 27 '16 at 16:33
• @Farcher yes it does... but loudness is not SIMPLY intensity as your link points out. – Jane Jacobson Feb 27 '16 at 17:41
• @JaneJacobson Sorry, I should have written "not just proportional". – Farcher Feb 27 '16 at 17:54

What you may seeing in the physics relationships is $$\beta=\log_{10}\left(\frac{I}{I_0}\right)$$ where $\beta$ is the intensity level in units called bels (decibels would be $10\times$ this), $I$ is the intensity of a wave, and $I_0$ is some reference intensity to define a level of $0$ bels.
While you don't explicitly see the frequency in this formula, it's there. It is true that if you have two sources of equal intensity and different frequency, they will have the same level, $\beta$.
Loudness is affected not only by amplitude and frequency, but also by the ear mechanism and the brain. Equal level sounds of differing frequencies have differing loudnesses. You can research Fletcher-Munson curves.