Chapter 12, Problem 49P

(a)

To determine

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

Answer to Problem 49P

  $n^1=1685.56\text{ Hz}\text{.}$

Explanation of Solution

Given:

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

  $=1550\text{ Hz},$

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

  $=30\text{ m/s},$

Formula used:

Write the expression for apparent frequency.

  $n^1=n\frac{v-v_0}{v-v_s}\text{ Hz}\text{.}$

Where,

  $n^1$ = apparent frequency, Hz

  $n$ = Siren frequency, Hz

  $v$ = velocity of light, m/s

  $v_s$ = Speed of source, m/s

  $v_o$ = Speed of observer, m/s

Calculation:

Calculations are based on the person, moving towards or away from the fire engine.

In this case, considering a person moving towards the fire engine and the fire engine actual frequency of a siren is $1550\text{ Hz}.$

So, consider negative x- direction, $v_o=-30\text{ m/s}$ and the actual fire engine siren’s speed is

  $v=343\text{ m/s}$ in the air.

The source frequency $\left(v_s\right)=0$

Apparent frequency,

  $n^1=n\frac{v-v_0}{v-v_s}\text{ Hz}\text{.}$

  $n^1=1550\frac{(343-(-30)}{(343-0)}\text{ Hz}\text{.}$

  $n^1=1685.56\text{ Hz}\text{.}$

Conclusion:

Therefore, frequency that can be detected is $1685.56\text{ Hz}$ when a person moves with a speed of 30 m/s towards the engine.

(b)

To determine

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

Answer to Problem 49P

  $n^1=1414.43\text{ Hz}\text{.}$

Explanation of Solution

Given:

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

  $=1550\text{ Hz},$

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

  $=30\text{ m/s},$

Formula used:

Write the apparent frequency.

  $n^1=n\frac{v-v_0}{v-v_s}\text{ Hz}\text{.}$

  $n^1$ = apparent frequency, Hz

  $n$ = Siren frequency, Hz

  $v$ = velocity of light, m/s

  $v_s$ = Speed of source, m/s

  $v_o$ = Speed of observer, m/s

Calculation:

The calculations are based on the person, moving towards or away from the fire engine.

In this case, considering a person moving towards the fire engine and the fire engine actual frequency of a siren is $1550\text{ Hz}\text{.}$

So, consider positive x -direction, $v_o=30\text{ m/s}$ and the actual fire engine siren’s speed is $v=343\text{ m/s}$ in the air.

The source frequency ( v_s ) = 0

Apparent frequency,

  $n^1=n\frac{v-v_0}{v-v_s}\text{ Hz}\text{.}$

  $n^1=\text{ (}1550\text{ Hz)}\frac{\text{(}343\text{ m/s)}-( 30\text{ m/s})}{\text{(}343\text{ m/s)}-(0)}\text{ Hz}\text{.}$

From above expression,

  $n^1=1414.43\text{ Hz}\text{.}$

Conclusion:

Therefore, the frequency that can be detected is $1414.43\text{ Hz}$ when a person moves with a speed of 30 m/s away from the engine.