Chapter 12, Problem 84GP

(a)

To determine

To Find: Beat frequency heard by the observer.

Explanation of Solution

Introduction:

The beat frequency depends on the rate at which the volume is heard from high to low.

When both speakers are moving towards the observer at the same speed, that means that the frequency will have the same Doppler shift. So, there will be no beats hear by the observer in this case.

Conclusion:

When the observer listens from position A due to the Doppler shift, there will be no beats.

(b)

To determine

To Find: Beat frequency heard by the observer.

Answer to Problem 84GP

Beat frequency which is heard by the observer when he is between the speakers is $12.37\text{ Hz}$ .

Explanation of Solution

Given:

Velocity of train $v_t=10\text{ m / s}$ .

The velocity of sound $20°\text{C, }{v}_s=343\text{ m / s}$ .

Stationary frequency $f=212\text{ Hz}$ .

Formula used:

The beat frequency is calculated by the following formula,

  $\begin{array}{l}{f}_{\text{Beat}}=f_{\text{toward}}-f_{\text{away}}\\ f_{\text{toward}}=f\times \left(\frac{1}{1-\frac{v_{\text{train}}}{v_{\text{sound}}}}\right)\\ f_{\text{away}}=f\times \left(\frac{1}{1+\frac{v_{\text{train}}}{v_{\text{sound}}}}\right)\end{array}$ where,

  $f$ =frequency emitted by the source uniformly.

Calculation:

Substitute the given values in the above equation,

Beat frequency:

  $\begin{array}{l}\text{= }f\times \left(\frac{1}{1-\frac{v_{\text{train}}}{v_{\text{sound}}}}\right)-f\times \left(\frac{1}{1+\frac{v_{\text{train}}}{v_{\text{sound}}}}\right)\\ \text{= }f\times \left(\frac{v_s}{v_s-v_t}\right)-f\times \left(\frac{v_s}{v_s+v_t}\right)\\ \text{= }212\times \left(\frac{343}{343-10}-\frac{343}{343+10}\right)\\ \text{= }212\times \left(\frac{343}{333}-\frac{343}{353}\right)\\ \text{= }212\times 0.058=12.37\text{ Hz}\end{array}$

Conclusion:

Thus, the Beat frequency which is heard by the observer when he is between the speakers is $12.37\text{ Hz}$ .

(c)

To determine

To Find: Beat frequency which is heard by the observer.

Explanation of Solution

Introduction:

The beat frequency depends on the rate at which the volume is heard from high to low.

When the observer hears the speakers after it passes him that means that speakers are going away from the observer. So there will be no beats in this case also.

Conclusion:

When the observer hears the speaker after passes him than, there will be no beats in this case also.