Question

To practice Problem-Solving Strategy 17.1 for wave interference problems. Two loudspeakers are placed side by side a distance d = 4.00 m apart. A listener observes maximum constructive interference while standing in front of the loudspeakers, equidistant from both of them. The distance from the listener to the point halfway between the speakers is l = 5.00 m . One of the loudspeakers is then moved directly away from the other. Once the speaker is moved a distance r = 60.0 cm from its original position, the listener, who is not moving, observes destructive interference for the first time. Find the speed of sound v in the air if both speakers emit a tone of frequency 700 Hz .

Answer 1

Complete Question

The compete question is shown on the first uploaded question

Answer:

The speed is  $v = 350 \ m/s$  

Explanation:

From the question we are told that

   The  distance of separation is  d =  4.00 m  

  The distance of the listener to the center between the speakers is  I =  5.00 m

  The change in the distance of the speaker is by $k = 60 cm = 0.6 \ m$

    The frequency of both speakers is $f = 700 \ Hz$

Generally the distance of the listener to the first speaker is mathematically represented as

       $L_1 = \sqrt{l^2 + [\frac{d}{2} ]^2}$

       $L_1 = \sqrt{5^2 + [\frac{4}{2} ]^2}$

        $L_1 = 5.39 \ m$

Generally the distance of the listener to second speaker at its new position is  

          $L_2 = \sqrt{l^2 + [\frac{d}{2} ]^2 + k}$

       $L_2 = \sqrt{5^2 + [\frac{4}{2} ]^2 + 0.6}$

        $L_2 = 5.64 \ m$  

Generally the path difference between the speakers is mathematically represented as

        $pD = L_2 - L_1 = \frac{n * \lambda}{2}$

Here $\lambda$ is the wavelength which is mathematically represented as

         $\lambda = \frac{v}{f}$

=>    $L_2 - L_1 = \frac{n * \frac{v}{f}}{2}$

=>    $L_2 - L_1 = \frac{n * v}{2f}$  

=>    $L_2 - L_1 = \frac{n * v}{2f}$  

Here n is the order of the maxima with  value of  n =  1  this because we are considering two adjacent waves

=>    $5.64 - 5.39 = \frac{1 * v}{2*700}$      

=>    $v = 350 \ m/s$