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

3.3 Complex Waves 35Figure 3.3. Addition of wavesA and B with equal amplitudes but slightly differing frequencies.The sum of the two sine waves yields an envelope C which has a beat frequencyequal to the difference between the frequencies of the superimposed waves A and B.When the argument of the cosine assumes integer values of π, the amplitude ofthe complex wave is a maximum that is equal to 2A 0 . Continuing the reasoningfurther, it is established that the amplitude of the complex wave vanishes when theargument of the cosine takes on integer odd values of π/2, i.e.,2π ( f 1 − f 2 )t2=(2n − 1)π2(n = 1, 2, 3,...) (3.4)A graph of the envelope of this transient amplitude modulation is given inFigure 3.3. The modulation or beat frequency is simply the frequency difference( f 1 – f 2 ) between the two superposed waves. To demonstrate this, let us solveEquation (3.4) for those times t n when the amplitude of the superimposed soundpressure is zero,t n = 2n − 1 (n = 1, 2, 3,...)2( f 1 − f 2 )Now consider in a general fashion the time difference between two consecutivebeats, namely, the nth and the (n + 1)th:t n+1 − t n ==2(n + 1) − 12( f 1 − f 2 )− 2n − 12( f 1 − f 2 )1f 1 − f 2(3.5)