Chapter 12, Problem 50P

(a)

To determine

To calculate: The frequency that can be detected for the given values.

Answer to Problem 50P

  $f^1=1709.48\text{ Hz}\text{.}$

Explanation of Solution

Given:

Fire engine’s siren frequency, $\left(f\right)$

  $=1550\text{ Hz},$

Fire engine moves with a speed $\left(v\right)$

  $=32\text{ m/s,}$

Formula used:

Write the expression of apparent frequency.

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

Where,

  $f$ = frequency of engine, Hz

  $v_{\text{engine}}$ = Speed of the fire engine, m/s

  $v$ = Ideal speed = 343 m/s

Calculation:

The actual frequency of a siren $=1550\text{ Hz}\text{.}$

Consider negative x- direction, the actual fire engine siren’s speed is $v_{\text{engine}}=-32\text{ m/s}$

The source frequency $\left(v_s\right)=0$ and $v=343\text{ m/s}$

Apparent frequency,

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

  $n^1=\frac{1550}{\left(1-\frac{32}{343}\right)}\text{ Hz}\text{.}$

  $f^1=1709.48\text{ Hz}\text{.}$

Conclusion:

The frequency that can be detected is $1709.48\text{ Hz}$ when a fire engine approaches a person with a speed of 32 m/s.

(b)

To determine

To calculate: The frequency that can be detected for the given value.

Answer to Problem 50P

  $f^1=1422.018\text{ Hz}\text{.}$

Explanation of Solution

Given:

Fire engine’s siren frequency $\left(n\right)$

  $=1550\text{ Hz,}$

A wave moves with a speed $\left(v\right)$

  $=32\text{ m/s,}$

Formula used:

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

  $f$ = frequency of engine, Hz

  $v_{engine}$ = Speed of the fire engine, m/s

  $v$ = Ideal speed = 343 m/s

  $f^1$ = frequency that can be detected

Calculation:

Consider negative x -direction, the actual fire engine siren’s speed is $v_{engine}=-32\text{ m/s}$

The source frequency $\left(v_s\right)=0$ and $v=343\text{ m/s}$

The source frequency ( v_s ) = 0

Apparent frequency,

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

  $f^1=\frac{1550}{\left(1+\frac{32}{343}\right)}\text{ Hz}\text{.}$

  $f^1=1422.018\text{ Hz}\text{.}$

Conclusion:

The frequency that can be detected is $f^1=1422.018\text{ Hz}$ when a fire engine moves with a speed of 32 m/s away from the person.