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Fundamentals Of Physics
Found in: Page 512

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Short Answer

A point source that is stationary on an x-axis emits a sinusoidal sound wave at a frequency of 686 Hz and speed343 ms. The wave travels radially outward from the source, causing air molecules to oscillate radially inward and outward. Let us define a wavefront as a line that connects points where the air molecules have the maximum, radially outward displacement. At any given instant, the wavefronts are concentric circles that are centered on the source. (a) Along x, what is the adjacent wavefront separation? Next, the source moves along at a speed of110 ms. Along x, what are the wavefront separations (b) in front of and (c) behind the source?

  1. Adjacent wave front separation is0.5 m
  2. Wave front separation in front of source is 0.34 m
  3. Wave front separation behind the source is 0.66 m
See the step by step solution

Step by Step Solution

Step 1: Given data:

  1. Frequency of wave is f=686 Hz
  2. Speed of wave c=343 ms

Step 2: Determine the concept of the Doppler Effect

Using Doppler’s Effect, we can get the wavefront separation.

Formula:

The wavelength shift in the interference patternΔλ=λ0(±v)c(i)

Here,

λ0=wavelenght of source, c=speed of light, v=speed of source

The frequency of a wave oscillation, f=vλ (ii)

Step 3: a) Calculate the adjacent wave front separation

With given values in equation (ii), we can easily find the wave front separation by using the formula as:

λ0=vf=343686=0.5 m

Hence, the value of adjacent wave front separation is 0.5 m

Step 4: b) Calculate wave front separation in front of the source

In front of the source, the shift in wavelength using equation (i) is given as:

Δλfront=(110)(0.5)343 =0.16 m

The negative sign shows that the shift is opposite in direction to the speed of the sound.

The wave front separation is given as:

0.50.16=0.34 m

Hence, the wave front separation is0.34 m

Step 5: c) Calculatewave front separation behind the source

Behind the source, the shift in wavelength using equation (i) is given as:

The shift is in the direction of the sound:

Δλfront=+(110)(0.5)343=0.16 m

The wave front separation is given as:

0.5+0.16=0.66 m

Hence, the wave front separation is 0.66 m

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