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Two identical loudspeakers separated by distance d emit 200Hz sound waves along the x-axis. As you walk along the axis, away from the speakers, you don't hear anything even though both speakers are on.What are the three lowest possible values for d? Assume a sound speed of 340m/s.

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Answer:

The three lowest possible values for d are 0.85 m, 2.55 m and 4.25 m.

Explanation:

Given that,

Distance = d

Frequency = 200 Hz

Speed of sound = 340 m/s

We need to calculate the wave length

Using formula of frequency

[tex]f= \dfrac{v}{\lambda}[/tex]

[tex]\lambda=\dfrac{v}{f}[/tex]

Put the value into the formula

[tex]\lambda=\dfrac{340}{200}[/tex]

[tex]\lambda=1.7\ m[/tex]

We need to calculate the three lowest possible values for d

Using formula of destructive interference

[tex]\Delta\phi=2\pi\dfrac{\Delta x}{\lambda}[/tex]

[tex]2\pi\dfrac{\Delta x}{\lambda}=(m+\dfrac{1}{2})2\pi[/tex]

Where, [tex]\Delta x[/tex] = distance

Put the value into the formula

For m = 0,

[tex]\dfrac{\Delta x}{1.7}=(0+\dfrac{1}{2})[/tex]

[tex]\Delta x=\dfrac{1.7}{2}[/tex]

[tex]\Delta x=0.85\ m[/tex]

For m =1 ,

[tex]\dfrac{\Delta x}{1.7}=(1+\dfrac{1}{2})[/tex]

[tex]\Delta x=\dfrac{1.7\times3}{2}[/tex]

[tex]\Delta x=2.55\ m[/tex]

For m=2,

[tex]\dfrac{\Delta x}{1.7}=(2+\dfrac{1}{2})[/tex]

[tex]\Delta x=\dfrac{1.7\times5}{2}[/tex]

[tex]\Delta x=4.25\ m[/tex]

Hence, The three lowest possible values for d are 0.85 m, 2.55 m and 4.25 m.

oladayoademosu

The three lowest possible values for d are 0.85 m, 2.55 m and 4.25 m.

Destructive interference

Since we are looking for the points of no sound, these are points of destructive interference. So, the path difference, ΔL which is the distance between the two speakers, d is

ΔL = d = (n + 1/2)λ where

  • n = integer and
  • λ = wavelength = v/f where
  • v = speed of sound = 340 m/s and
  • f = frequency of sound waves = 200 Hz

So, d = (n + 1/2)v/f

d = (n + 1/2)340m/s ÷ 200 Hz

d = (n + 1/2)1.7 m

The lowest possible values of d

The lowest possible values of d are when n = 0, 1 and 2.

So,

When n = 0

d = (n + 1/2)1.7 m

d = (0 + 1/2)1.7 m

d = (1/2)1.7 m

d = 0.85 m

When n = 1

d = (n + 1/2)1.7 m

d = (1 + 1/2)1.7 m

d = (3/2)1.7 m

d = (1.5)1.7 m

d = 2.55 m

When n = 2

d = (n + 1/2)1.7 m

d = (2 + 1/2)1.7 m

d = (5/2)1.7 m

d = 2.5 × 1.7 m

d = 4.25 m

So, the three lowest possible values for d are 0.85 m, 2.55 m and 4.25 m.

Learn more about destructive interference here:

https://brainly.com/question/19869031

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