At the other extreme of physical pain, are soft, tactile impressions that are just above the threshold of perception. Imagine a gentle wind finding its way through body hair or the lightest of drizzles falling on one’s bare shoulders. The acoustic equivalent of such pleasures is the distant sound of rustling leaves. At about 10 decibels it has less than 10 times the intensity of the faintest audible sound. And it has only one of ten trillionths of the power per area of a sound wave that could cause pain. It seems that we have evolved not only to react to extreme stimuli that endanger us but also to be rewarded when the environment toys with the lower limits of our senses.
Although the above numbers are mathematically accurate—-they are based on the formula which equates decibels with 10 times the logarithmic ratio of a sound’s intensity to our threshold intensity of 10-12 watts/meter2—they exaggerate what our eardrums actually experience.
Before delving further, just what is sound? When a leaf moves back and forth, or some other body vibrates, the oscillation causes a periodic disturbance of the surrounding air molecules. The movement in one direction causes molecules of the air (or other transmitting medium) to bunch up, increasing pressure in one region. As the object reverts to its original position, a region of less crowding among molecules occurs as well. Meanwhile, the crowded region transmits its kinetic energy to other particles of the medium, repeating the pattern of high and low densities as the pressure wave propagates away from its source.
As a result, what’s more appropriate in comparing sound-strength is the use of pressure amplitude, which is proportional to the square root of a sound’s intensity. For example if our slightly rustling leaves come in at 10 dB and a pneumatic sidewalk-breaker next to us is painfully wreaking havoc at 120 dB, first we figure out the intensity ratio from the exponential version of the decibel-log formula:
I2/I1 = 10 0.1(dB2 – dB1) =10 0.1(120 – 10) = 1011.
Then by taking the square root of 1011, we obtain the pressure ratio— about 316 000. That’s still a huge factor for how much louder the pneumatic drill is in comparison to the distant sound of gently moving leaves.
A little more number-crunching can make us appreciate how wonderfully sensitive our ears are, at least when they are working well. The exact pressure amplitude(ΔPm) of a rustling sound can be calculated from:
ΔPm = √(2Iρv) ,
where I is the intensity, and ρ and v are the density and velocity of the medium. For air at room temperature the latter numbers are 1.2 kg per meter cubed and 343 m/s, and for a 10 dB-sound, I = 10-11 W/m². Thus ΔPm is 9.07 X 10-5 pascals. Comparing it to the atmospheric pressure of 101.3 kPa, we can actually detect a difference of 1 part in a billion, and it’s enough to give us pleasure! When you consider that it takes a few parts per million of psychotropic substances to buzz the brain, something that many marvel at, to me our acute sensitivity to sound seems more remarkable, while no drugs are necessary.