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

60 3. Sound Wave Propagation and CharacteristicsFigure 3.15. Complex exponentials portrayed as rotating vectors.Let us now determine the effect of adding two 60-dB signals, which have frequenciesof 1000 and 1100 Hz, respectively. Because the frequencies are not equal,the root-mean-square sound pressure is double that of one wave. The sound pressurelevel increases by( p 2)10 log rmsprms12 = 10 log 2or approximately 3 dB. The combined SPL is 63 dB.We gain a further insight into the phenomenon of beat frequency if we visualizethe complex exponential functions (3.33c) and (3.33d) as rotating vectors inFigure 3.15. Without loss of generality we can define time t = 0 as the instantwhen both vectors lie along the positive real axis, resulting in the maximum soundpressure. The two vectors will be opposed along the real axis when (ω 2 − ω 1 )t =2π. The envelope of the pressure–time curve yields a period τ = 2π/(ω 2 − ω 1 ) =1/( f 2 – f 1 ). The term ( f 2 − f 1 ) is, of course, the beat frequency, which has beenpreviously discussed in Section 3.3.3.17 Sound IntensityAn acoustic signal emanates from a point source in a spherical pattern over anincreasingly larger area. When a closed surface completely surrounding the sourceis defined, the sound power W radiated by the source can be established from∫W = I · dS (3.37)whereI = sound intensity, W/m 2dS = element of surface area, m 2SS = surface area surrounding source