Intensity and the decibel scale

Sound intensity

Sound intensity is power transmitted per unit area:

I=rac{P}{A}

It is measured in watts per square meter. Intensity describes the physical rate at which sound energy flows through a surface.

For a point-like source radiating uniformly in all directions, intensity decreases with distance according to

I=rac{P}{4pi r^2}

This inverse-square relationship occurs because sound power spreads over the surface of a sphere.

Why decibels are used

Human hearing covers an enormous range of intensities. The quietest sound a typical young human can hear is around

$I_0=10^{-12} W/m^2$

Painfully loud sounds can be near or above

$1 W/m^2$

This is a factor of a trillion. A linear scale would be awkward, so sound level is measured using a logarithmic scale.

Sound intensity level

The sound intensity level in decibels is

ight)$$ where $I_0=10^{-12} W/m^2$ is the reference intensity. If $I=I_0$, then $$eta=0 dB$$ This does not mean no sound energy; it means the sound is at the reference threshold. ## Interpreting decibel changes Because the decibel scale is logarithmic, adding decibels corresponds to multiplying intensity. If intensity increases by a factor of 10, sound level increases by 10 dB: $$10log_{10}(10)=10$$ If intensity increases by a factor of 100, sound level increases by 20 dB. A 3 dB increase corresponds approximately to doubling intensity because $$10log_{10}(2)approx 3$$ Perceived loudness is biological and more complicated, but a 10 dB increase is often perceived roughly as about twice as loud. ## Distance and sound level If distance from a point source doubles, intensity decreases by a factor of 4: $$Ipropto rac{1}{r^2}$$ The decibel change is $$10log_{10}left(rac{1}{4} ight)approx -6 dB$$ So doubling distance from an ideal point source reduces sound level by about 6 dB. ## Pressure amplitude and sound level Sound can also be described by pressure amplitude. Since intensity is proportional to pressure amplitude squared, $$Ipropto (Delta P)^2$$ sound pressure level uses $$eta=20log_{10}left(rac{Delta P}{Delta P_0} ight)$$ The factor 20 appears because pressure is squared in intensity. ## Hearing safety High sound levels can damage hearing, especially with long exposure. Damage depends on intensity, duration, frequency, and individual sensitivity. Ear protection reduces intensity reaching the ear. Physics helps explain why both loudness and exposure time matter. A moderately loud sound for many hours can be harmful, while a very loud impulse can cause immediate damage. ## Multiple sources Two identical independent sound sources do not usually double the decibel level. They double intensity, which increases level by about 3 dB. This surprises many students because decibels are logarithmic. If one source produces intensity $I$, two incoherent identical sources produce approximately $2I$: $$Deltaeta=10log_{10}(2)approx 3 dB$$ ## The big idea Sound intensity is physical energy flow per area. The decibel scale compresses the huge range of human hearing into a practical logarithmic measure. Understanding decibels requires thinking multiplicatively: equal decibel steps correspond to equal intensity ratios, not equal intensity differences.