THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

48 3. Sound Wave Propagation and Characteristicsalways zero, since the mean rarefaction equals the mean compression. A simpleway to measure the degree of disturbance is to square the values of the soundpressure disturbance over a period of time, thereby eliminating the counter-effectsof negative and positive disturbances by rendering them always positive. The rootmean-squaresound pressure p rms can be defined byp rms = √ √ ∫ τ(p) 2 0=p2 dt∫ τ0 dt (3.20)where τ is the time interval of measurement and p the instantaneous pressure. Fora simple cosine wave over an interval of period T = 2π/ω, there results√ ∫ Tp0rms =p2 m cos2 k(x − ct) dtT= p m√2. (3.21)The sound pressure as portrayed by the oscillation of the pressure above andbelow the atmospheric pressure is detected by normal human ear at levels as lowas approximately 20 μPa (the SI unit of pressure is the pascal, abbreviated Pa,equivalent to 1.0 N/m 2 ). 1 Because p rms could vary over a wide range of ordersof magnitude, it would be cumbersome to use it as the measure of loudness. Atthe threshold of pain, p rms would reach approximately 40,000,000 μPa! The blastoffpressure in the vicinity of the launching pad of a Titan rocket can exceedup to a thousandfold the threshold of pain (i.e., 40 kPa). It is therefore moreconvenient to use the decibel as the folding-scale measure of loudness. This unit isdefined by( ) 2 ( )prmsprmsL p = 10 log = 20 log(3.22)p 0 p 0whereL p = sound pressure level (dB)log = common (base-ten) logarithmp 0 = 20 × 10 −6 Pa = the reference pressure.From the context of Equation (3.22) it can be established that the doubling ofa root-mean-square pressure corresponds to approximately 6 dB increase in thesound pressure level. In order to determine the sound pressure level from a givenvalue of L p , Equation (3.22) can be rewritten asorp rmsp 0= 10 L p20p rms = 20 × 10 L p20 −61 One standard atmosphere equals 101.325 kPa.