Chapter 12, Problem 51P

(a)

To determine

To compare: The shift in frequency for the given cases.

To identify: Whether the two frequencies exactly equal or close to each other.

Answer to Problem 51P

Frequency of source, $f^1=2091.46\text{ Hz}\text{.}$

Frequency of observer, $f^1=2087$ Hz.

  $f^1{}_{observer}<f^1{}_{source.}$

The two frequencies are closer to each other.

Explanation of Solution

Given:

Shift frequency $\left(f\right)$ $=2000\text{ Hz}.$

Source moving with a speed $\left(v\right)$ $=15\text{ m/s,}$

A person moving with a speed $\left(v\right)$ = 15 m/s

Formula used:

Write the expression of the receiving end.

When source is moving.

  $f^1=\frac{f}{\left(1-\frac{v_{observer}}{v}\right)}$ Hz

When observer is moving.

  $f^1=f\left(1+\frac{v_{observer}}{v}\right)\text{ Hz}$

  $f^1$ =frequency of observer or source, Hz

  $v$ = speed of wave =343 m/s

  $v_{observer}$ =speed of observer, m/s

Calculation:

If the source is moving toward the person and the frequency of the shift is 2000 Hz.

When the source is moving,

  $f^1=\frac{f}{\left(1-\frac{v_{engine}}{v}\right)}$

  $f^1=\frac{2000}{\left(1-\frac{15}{343}\right)}\text{ Hz}\text{.}$

  $f^1=2091.46\text{ Hz}\text{.}$

If an observer is moving,

  $f^1=f\left(1+\frac{v_{observer}}{v}\right)\text{ Hz}$

  $f^1=2000\left(1+\frac{15}{343}\right)\text{ Hz}$

  $f^1=2087$ Hz.

Conclusion:

The frequency shifts are not exactly same but they are closer to each other. $f^1{}_{observer}<f^1{}_{source.}$

(b)

To determine

To compare: The shift in frequency for the given cases.

To identify: Whether the two frequencies exactly equal or close to each other.

Answer to Problem 51P

  $f^1=3554.4\text{ Hz}\text{.}$ (Source)

  $f^1=2874.63\text{ Hz}$ (Observer)

  $f^1{}_{observer}<f^1{}_{source.}$

Both the frequencies are neither same nor close to each other.

Explanation of Solution

Given:

Shift frequency $\left(f\right)$ $=2000\text{ Hz}.$

Source moving with a speed $\left(v\right)$ $=150\text{ m/s,}$

A person moving with a speed $\left(v\right)$ $=150\text{ m/s,}$

Formula used:

Write the expression of the receiving end.

When source is moving.

  $f^1=\frac{f}{\left(1-\frac{v_{observer}}{v}\right)}$ Hz

When observer is moving.

  $f^1=f\left(1+\frac{v_{observer}}{v}\right)\text{ Hz}$

  $f^1=$ frequency of observer or source, Hz

  $v$ = speed of wave =343 m/s

  $v_{observer}$ =speed of observer, m/s

Calculation:

The frequency of shift $=2000\text{ Hz}$ .

When the source is moving,

  $f^1=\frac{f}{\left(1-\frac{v_{source}}{v}\right)}$

  $f^1=\frac{2000}{\left(1-\frac{150}{343}\right)}\text{ Hz}\text{.}$

  $f^1=3554.4\text{ Hz}\text{.}$

When the observer moving

  $f^1=f\left(1+\frac{v_{observer}}{v}\right)\text{ Hz}$

  $f^1=2000\left(1+\frac{150}{343}\right)\text{ Hz}$

  $f^1=2874.63\text{ Hz}$ .

Conclusion:

The frequency shifts are different $f^1{}_{observer}<f^1{}_{source.}$

(c)

To determine

To compare: The shift in frequency for the given cases.

To identify: Whether the two frequencies exactly equal or close to each other.

To explain: The conclusion about the asymmetry of the Doppler formula.

Answer to Problem 51P

  $f^1=15953.48\text{ Hz}\text{.}$ (Source)

  $f^1=3749.27\text{ Hz}$ (observer)

  $f^1{}_{observer}<f^1{}_{source.}$

The two frequencies are different from each other.

The Doppler shift is not the same when the observer is moving towards the source or the source is moving towards the observer.

Explanation of Solution

Given:

Shift frequency $\left(f\right) =2000\text{ Hz}$

Source moving with a speed $\left(v\right) =300\text{ m/s,}$

A person moving with a speed $\left(v\right) =300\text{ m/s,}$

Formula used:

Write the expression of the receiving end.

When source is moving.

  $f^1=\frac{f}{\left(1-\frac{v_{observer}}{v}\right)}$ Hz

When observer is moving.

  $f^1=f\left(1+\frac{v_{observer}}{v}\right)\text{ Hz}$

  $f^1=$ frequency of observer or source, Hz

  $v$ = speed of wave =343 m/s

  $v_{observer}$ =speed of observer, m/s

Calculation:

If a source is moving toward the person, the frequency of shift is 2000 Hz.

When the source moving,

  $f^1=\frac{f}{\left(1-\frac{v_{source}}{v}\right)}$

  $f^1=\frac{2000}{\left(1-\frac{300}{343}\right)}\text{ Hz}\text{.}$

  $f^1=15953.48\text{ Hz}\text{.}$

When the observer moving,

  $f^1=f\left(1+\frac{v_{observer}}{v}\right)\text{ Hz}$

  $f^1=2000\left(1+\frac{300}{343}\right)\text{ Hz}$

  $f^1=3749.27\text{ Hz}$.

The frequency of source and observer has a large difference.

The frequency shifts are slightly different, $f^1{}_{observer}<f^1{}_{source.}$ The Doppler shift is not the same when the observer is moving towards the source or the source is moving towards the observer.