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

58 3. Sound Wave Propagation and Characteristics3.15 Particle Displacement and VelocityInvoking Equation (2.21) and inserting the wave equation solution (3.1), we obtainthe expression for the particle velocity u:u =− 1 ∫ ∂ρρ ∂x dt =−p mcos k(x − ct)ρcu =p(3.32)ρ cwhere ρ = quiescent density of air = 1.18 kg/m 3 at a normal room temperature of22 ◦ C and atmospheric pressure of 101.3 kPa and c = speed of sound = 344 m/s.The term ρc is the characteristic or acoustic impedance for a wave propagating inair in a free-field condition. The value of ρc at standard conditions of temperatureand pressure is 40.7 rayls or 407 MKS rayls. The dimensional unit rayl is definedas follows:1 rayl = 1.0 dyne s/cm 3The particle displacement x for a cosine wave function can be found by simplyintegrating Equation (3.32) with respect to time:x = p msin k(x − ct)ρc2 It is interesting to note that at 0 dB, the threshold of human hearing, the oscillationof an air molecule covers an rms amplitude that is approximately only one-tenththe diameter of a hydrogen atom.The particle acceleration is obtained from the differentiation of Equation (3.32)with respect to time, and for a cosine wave function the acceleration isdudt= p sin k(x − ct)ρ3.16 Correlated and Uncorrelated SoundCorrelated sound waves occur when they have a precise time and frequency relationshipbetween them. An example of correlated sound waves is the output oftwo identical loudspeakers located in the same plane, consisting of a pure tonesupplied by a single amplifier connected to both loudspeakers. Most of the soundwaves that we hear are generally uncorrelated.Consider two sound waves which are detected at a point in space:p 1 = P 1 cos (ω 1 t + φ 1 )p 2 = P 2 cos (ω 2 t + φ 2 )(3.33a)(3.33b)