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$\bullet$ Twosmall stereo speakers are driven in step by the same variable-frequency oscillator. Theirsound is picked up by a microphone arranged as shown in Figure $12.43 .$ For what frequencies does their sound at the speakers produce (a) constructive interference, ( b) destructive interference?

a) Any frequency that is an integer number of $(819 \mathrm{Hz})$ will produce constructive interference.b) The frequencies that will produce destructive interference are $f=n 409.5 \mathrm{Hz} ;(n=1,3,5, \ldots)$

01:21

Eric M.

Physics 101 Mechanics

Chapter 12

Mechanical Waves and Sound

Periodic Motion

Mechanical Waves

Sound and Hearing

Simon Fraser University

Hope College

University of Sheffield

McMaster University

Lectures

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So here in this problem, we are given this microphone on two speakers Speaker One speaker to the distance off the speaker. One from the microphone were given US 4.5 middle on additions off. The do speaker has given us 2 m. Now we have to calculate, for what frequencies does the sound produced by the speaker have constructive indifference on? In the second case, they have destructive indifference. So we'll be solving the A part first, which is for constructive and difference. So we know that for constructive indifference, bath difference that is. Delta X is given by end times Lambda When Lambda is the Wavelength or Delta X. Basically, it is equals two x two minus 61 This X two is the distance off the second speaker with from the microphone and x one is the distance off the speaker from the microphone. We have X one, but we don't have extra for that. Extra would be this length. This is X two. Now we can use the Pythagoras turn to calculate the value of X two so extra would be calls to under route 4.5 square plus two square. So from here we'll get the value of X to us. Nearly 4.92 m. No deal. Tex will be called to extra minus X one. So 4.9 to minus 4.5. It would be calls to 0.42 m. We have calculated Will takes. Now we know that the wavelength lambda It is a cold too. We over if, where we is. A velocity off sound wave on f is the frequency, Andi. We can plug it the value of Lambda here. So Delta X is recalls toe end times Linda, we have we over If so, from here we'll get the frequency. Yes, If it calls to end times we overdid. It takes where we This V is the velocity off the sound wave Onda we we know its value is 3 44 media per second. So we have all the religions. You can plug it here. So if is equals to end times we we have 3 44 media for second, divided by Delta X is 0.42 So from here we'll get this value us nearly endgames 8. 20 hertz. So this is the frequency for constructive indifference So this is the answer for the first case. Now next it will be calculating the frequency for the destructive indifference. And here in his equals two, one, 23 and soon Now, for destructive and difference Delta X, we have end plus one by two times Linda and Lambda. We know it is equals to be over if seven plugged, hair them down. So Delta X is it calls Toe end plus 1/2. The over. If so, from here, expression off F comes out to be as end plus one by two times we over Del takes. We have all the valleys. Looking back, it here to calculate the frequency it will be called true and plus 1/2. And we we have 3 44 m per second. Divided by Delta X is 0.42 m. So from here, we'll get the answer us to end plus one times 4 10, 410 hertz. So this is the frequency for destructive indifference on. In this case, energy equals two 012 three and so on. So this was the answer for the B. But I hope you have understood the problem. Thank you.

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