# Confusion between two equations for sound intensity

The definition of sound intensity is $$I=P/S$$, where $$P$$ is the power generated by the wave source and $$S$$ is the area over which the energy is transferred. Assuming conservation of mechanical energy of a sound wave, intensity for a point source is: $$I=P/4\pi r^2$$ ($$P$$ = source power, $$r$$ = radius). The intensity gets smaller when the radius, so the distance from the source gets longer. That's clear to me.

But then there is an equation that links intensity and amplitude. It looks like this: $$I=(1/2)\rho v \omega^2 A^2$$, where $$I$$ = intensity, $$ρ$$ = air volume density, $$v$$ = sound velocity, $$ω$$= angular frequency and $$A$$ = amplitude.

The whole problem is that there is no $$S$$ in the second equation so there is no area. This is not consistent with the first equation because we know that sound intensity depends on the area.

I am going to give an example. Let's say that we have a speaker and the intensity one meter from it is $$5 \,\text{W}/\text{m}^2$$. Then we measure it further away and it is $$2 \,\text{W}/\text{m}^2$$. Which number would we calculate with the second equation $$I=(1/2)\rho v \omega^2 A^2$$?

As far as I'm concerned the second equation assumes that the intensity is constant whatever distance from the speaker. Where am I wrong?

• Remember that the amplitude will decrease the further from the source.
– Rich
Commented Apr 12 at 2:12

Let's say that we have a speaker and the intensity one meter from it is $$5 \,\text{W}/\text{m}^2$$. Then we measure it further away and it is $$2 \,\text{W}/\text{m}^2$$. Which number would we calculate with the second equation $$I=(1/2)\rho v \omega^2 A^2$$?

You would get $$5 \,\text{W}/\text{m}^2$$ at 1 m distance and $$2 \,\text{W}/\text{m}^2$$ a the other distance. Because the amplitude $$A$$ will decrease with distance from the source in exactly the right way to be consistent with the decreasing intensity.

If you know the other terms ($$\rho$$, $$v$$, and $$\omega$$) for the medium, you could use the second equation to determine the amplitude at each distance from your intensity measurements

As far as I'm concerned the second equation assumes that the intensity is constant whatever distance from the speaker. Where am I wrong?

Amplitude is not constant, it decreases as you get further from the source.

Using the first equation is fine if you know the power of the emitter and how far away you are, and there are no reflections, etc. What do you do if you do not know these? You have to measure the kinetic energy of the air molecules due to the sound. To do this you must consider a small volume of air, the speed of the air molecules due to the sound and so on. This depends on the amplitude of the sound and its frequency. As I mentioned in my comment, the amplitude decreases with distance from the source. The small volume is determined by considering a small area that the sound passes through in one period of oscillation. (That's where the area appears in the derivation.) Usually the formula is presented as $$I = \frac{(\Delta p_{max})^{2}}{2 \rho v}$$ derivation here
If you do a dimensional analysis the two formula are identical - $$\frac{W}{m^2}$$