Intensity of a sound wave. Decibels

The expression derived to calculate the intensity of a harmonic wave propagating along a rope can also be used to calculate the intensity of a harmonic sound wave. In this case the amplitude of the wave will be sm, that is, the maximum displacement of the medium particles from their equilibrium positions.

The intensity is defined as power per unit area, and in this case is given by:

where ρ is the density of the medium, ω the angular frequency of the wave and v its velocity (or speed). The SI unit for intensity is watts per square meter (W/m2).

We can now substitute into the expression above the angular frequency ω as a function of the frequency ν:

And therefore the intensity of a sound wave is:

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The wave function of a sound wave can also be written in terms of the pressure variations from the equilibrium pressure Δp within the medium. The relationship between the amplitudes of both wave functions is given by:

And substituting into the expression for the intensity:

And simplifying we get:

For a spherical wave, the energy produced by the source and carried by the wave spreads out over the spherical surface area, and therefore its intensity will be:

From the previous expression it can be seen that for a spherical wave intensity decreases as the inverse square of the distance from the source.

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When a sound wave reaches the human ear, the pressure variations propagating through air make the eardrum vibrate (in red in the figure below), and these vibrations are sent to the brain in the form of electric nerve impulses. The brain then processes and interprets them as sounds.

Human ear can detect sounds of intensities ranging from 10-12 W/m2 (known as audibility threshold) to 10 W/m2 (known as threshold of pain ). Both are represented in the figure below in green and red respectively.

As it can be seen in the graphs, intensity depends both on the frequency and amplitude of the wave, in accordance with the expression derived in this page.

Sound intensity level. Decibel

As we have just seen, the range of sound intensities that humans can hear is very large, so a more convenient way to measure the loudness of sound is in a logarithmic scale. The sound intensity level (Li) is defined as:

where I0 is the audibility threshold (I0 = 10-12 W/m2).

The sound intensity level is measured in decibels (dB). Human ear can hear sounds ranging from 0 to 120 dB.

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