Sound intensity and power emitted by a speaker

Problem statement:

Assuming that the acoustic efficiency of a speaker is 1% and that it is plugged to a 40 W amplifier, find the sound intensity level in decibels at a distance of 1 m from the speaker (assuming the speaker emits spherical waves). What will the sound intensity level be at a distance of 10 meters from the speaker?

Givens: Audibility threshold I0= 10-12 W/m2.

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Solution:

To solve this problem we are going to use the formula for the sound intensity.

For a spherical wave, the power emitted by the source is spread over the surface of the wavefront (a sphere), and therefore its intensity is:

Thus the intensity of the sound wave decreases with the distance from the source.

Now we have to take into account the fact that the sound power emitted by the speaker is not the same as the power of the amplifier. The relationship between the two depends on the quality of the equipment; in this problem we will assume that the second is 1% of the first.

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In the figure below is shown the speaker as well as the surface over which the energy is spread:

At a distance R = 1 m, the sound intensity will be:

The sound intensity level (Li) is defined as:

where I0 is the audibility threshold.

At a distance of one meter from the source (the speaker) the sound intensity level will be:


We now determine the sound intensity at a distance of ten meters from the speaker:

And the corresponding sound intensity level will be:

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