Temporal interference results from the superposition of twowaves having slightly

Question

Temporal interference results from the superposition of twowaves having slightly different frequencies. When the twowaves are observed at a point in space, there is a temporal (time)alternation between constructive and destructiveinterference. If, for example, two tuning forks of slightlydifferent frequencies are struck, one hears a sound of periodicallyvarying amplitude. This phenomenon is called beating, and thenumber of amplitude maxima one hears per second gives the beatfrequency. (a) Let the time-dependent variations of thedisplacement due to two sound waves of slightly differentfrequencies, at a point where kx = /2, be

                  S1 = Sm Sin(/2 - 1t)    and     S2 = Sm Sin(/2 -2t)

Use the principle of superposition to show that the resultingdisplacement is

S(t) = S1+ S2 = [2Smcos't] cost  

where ' = 1/2 (1- 2) and = 1/2 (1 + 2). (b)The derivation in (a) shows that the resultant displacement is acosine function whose angular frequency is andwhose time varying amplitude is the absolute value of the quantityin the brackets. Show that the number of beats per second(periodically varying amplitude) is fbeat =f1- f2. (c) A siren emitting a sound of1000 Hz moves away from you towards the face of a cliff at a speedof 10 m / s. Take the speed of sound in air as 330 m /s. You hear the direct sound of the siren and the soundreflected from the cliff. What is the beat frequency of thesetwo sounds? Is it perceptible (it must be less than 20 Hz tobe perceived)?

Explanation / Answer

the resultant waves should be: S(t)=S1+S2=Sm*[sin(/2-1t)+sin(/2-2t)] =Sm*[cos1t + cos2t] =Sm*2cos[(1-2)/2]*cos[(1+2)/2] let (1-2)/2='     (1+2)/2= =>   S(t)=[2Sm*cos't] cost   in one period, there will be 2 possitions havingamplitude =>the period of beat f'='/=(1-2)/2 =f1-f2 (c)Doppler's effect f1 received by cliff f1=f*c/(c-10) f2 received by the person from the siren f2=f*c/(c+10) => f{beat}=f1-f2= 60.7 Hz >20 Hz => it isperceptible (c)Doppler's effect f1 received by cliff f1=f*c/(c-10) f2 received by the person from the siren f2=f*c/(c+10) => f{beat}=f1-f2= 60.7 Hz >20 Hz => it isperceptible

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