Question

A point source that is stationary on an x-axis emits a sinusoidal sound wave at a frequency of$\mathbf{\text{686Hz}}$and speed$\mathbf{\text{343}}\frac{\mathbf{\text{m}}}{\mathbf{\text{s}}}$. 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 alongat a speed of$\mathbf{\text{110}}\frac{\mathbf{\text{m}}}{\mathbf{\text{s}}}$. Along x, what are the wavefront separations (b) in front of and (c) behind the source?

Short answer

1. Adjacent wave front separation is$0.5\text{m}$
2. Wave front separation in front of source is$0.34\text{m}$
3. Wave front separation behind the source is$0.66\text{m}$

Step-by-step solution

Given data:

1. Frequency of wave is$f=686\text{Hz}$
2. Speed of wave$c=343\frac{\text{m}}{\text{s}}$

Determine the concept of the Doppler Effect

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

Formula:

The wavelength shift in the interference pattern$\Delta \lambda ={\lambda }_0\frac{(\pm v)}{c}$(i)

Here,

$$\begin{gathered} {\lambda }_0=\text{wavelenghtofsource}, \\ c=\text{speedoflight,} \\ v=\text{speedofsource} \end{gathered}$$

The frequency of a wave oscillation, $f=\frac{v}{\lambda }$ (ii)

a) Calculate the adjacent wave front separation

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

$\begin{array}{c}{\lambda }_0=\frac{v}{f}\\ =\frac{343}{686}\\ =0.5\text{m}\end{array}$

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

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:

$\begin{array}{c}\Delta {\lambda }_{front}=-\frac{\left(110\right)\left(0.5\right)}{343}\\ =-0.16\text{\hspace{0.17em}m}\end{array}$

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

The wave front separation is given as:

$0.5-0.16=0.34\text{\hspace{0.17em}m}$

Hence, the wave front separation is$0.34\text{\hspace{0.17em}m}$

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:

$\begin{array}{c}\Delta {\lambda }_{front}=+\frac{\left(110\right)\left(0.5\right)}{343}\\ =0.16\text{\hspace{0.17em}m}\end{array}$

The wave front separation is given as:

$0.5+0.16=0.66\text{\hspace{0.17em}m}$

Hence, the wave front separation is$0.66\text{\hspace{0.17em}m}$