Chapter 2: Q34E (page 540)

Small speakers A and B are driven in phase at 725 Hz by the same audio oscillator. Both speakers start out 4.50 m from the listener, but speaker A is slowly moved away (Fig. E16.34). (a) At what distance d will the sound from the speakers first produce destructive interference at the listener’s location? (b) If A is moved even farther away than in part (a), at what distance d will the speakers next produce destructive interference at the listener’s

Short Answer

Expert verified

The range is 0.518m,0.303m,0.086m,0.126m

Step by step solution

Concept of the destructive interference

The path difference is given as $δr = (n + \frac{1}{2}) λ$ where, $r_{1}$ is the path length of the first speaker and $r_{2}$ is the path length of the second speaker

Rearrangement of position of the minimums with respect to distance from first speaker

The path difference is given by

$r_{2} - r_{1} = (n + \frac{1}{2}) λ 1.25 - r_{2} - r_{1} = (n + \frac{1}{2}) λ r_{1} = \frac{1}{2} (1.25 - (n + \frac{1}{2}) λ)$

The wavelength of the sound wave is given as

$λ = \frac{v}{f} = \frac{344}{8 \times 10^{2}} = 0.429 m$

STEP 3 Plug in wavelength into the r1 and find all minimums that give r1 = [0,1.25 m] because we look for points between the speakers only:

Equation is given by $r_{1} = \frac{1}{2} (1.25 - (n + \frac{1}{2}) λ)$

For n=0

role="math" localid="1664341848808" $r_{1} = \frac{1}{2} (1.25 - (0 + \frac{1}{2}) 0.429) = 0.518 m$

For n = 1

$r_{1} = \frac{1}{2} (1.25 - (1 + \frac{1}{2}) 0.429) = 0.303 m$

For n = 2

$r_{1} = \frac{1}{2} (1.25 - (2 + \frac{1}{2}) 0.429) = 0.089 m$

For n = 3

$r_{1} = \frac{1}{2} (1.25 - (3 + \frac{1}{2}) 0.429) = 0.126 m$

Therefore, the range is 0.518m,0.303m,0.086m,0.126m

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Short Answer

Step by step solution

Concept of the destructive interference

Rearrangement of position of the minimums with respect to distance from first speaker

STEP 3 Plug in wavelength into the r1 and find all minimums that give r1 = [0,1.25 m] because we look for points between the speakers only:

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