Question

Question: A handclap on stage in an amphitheater sends out sound waves that scatters from terraces of width W 0.75 mfigure. The sound returns to the stage as a periodic series of pulses, one from each terrace; the parade of pulses sounds like a played note (a)Assuming that all the rays in figure are horizontal , find the frequency at which the pulses return(that is ,the frequency of the perceived note .(b)If the widthof the terraces were smaller would the frequency be higher or lower?

Short answer

Answer

1. The frequency at which the pulse return is $f=2.3\times 1{\text{0}}^{\text{2}}\text{Hz}$.
2. If the width of the terraces were smaller, the change in frequency would be higher.

Step-by-step solution

Step 1: Given

1. The sound in air has velocity v = 343 m/s
2. Width w = 0.75 m

Determining the concept

There would be a difference of for the distance to travel by the subsequent sound wave. Using this extra distance, find the frequency of the pulses. Also, using the same concept, compare the frequency of the pulses if the width of the terrace was smaller.

The frequency of the pulse,

$\mathbf{f}\mathbf{=}\frac{\mathbf{1}}{\mathbf{\Delta t}}$

The velocity of the pulse,

$\mathbf{v}\mathbf{=}\frac{\mathbf{x}}{\mathbf{\Delta t}}$

Where, v is velocity,tis time, xis displacement and f is frequency.

(a) Determining the frequency at which the pulses return

String of pulse returning to the stage. Extra distance covered by the pulse as compared to its previous pulse is x = 2W .so,

$∆t=\frac{2w}{v}$

And frequency is,

$$\begin{gathered} f=1/∆t \\ =\frac{v}{2\text{w}} \\ =\frac{343m/s}{2\times 0.75\text{m}} \\ ==2.3\times 1{\text{0}}^{\text{2}}\text{Hz} \end{gathered}$$

Hence, frequency at which the pulses return is$f=2.3\times 1{\text{0}}^{\text{2}}\text{Hz}$ .

(b) Determining if the width of the terraces were smaller, then what would be the change in the frequency 

Fromtheabove equation,thefrequency is inversely proportional to the width,

$f\propto \frac{1}{\text{w}}$

i.e. ifthewidth is smaller then frequency is higher.

Hence, if the width of the terraces were smaller, the change in frequency would be higher.