Question

A bat emitting an ultrasonic wave of frequency 4.5 × 10⁴ Hz flies at a speed of 6 m s⁻¹ between two parallel walls. Find the fractional heard by the bat and the beat frequencies heard by the bat and the beat frequency between the two. The speed of sound is 330 m s⁻¹.

Answer 1

Given:
Speed of bat v = 6 ms⁻¹
Frequency of ultrasonic wave f = 4.5 × 10⁴ Hz
Velocity of bird $v_s$ = 6 ms⁻¹
Let us assume that the bat is flying between the walls X and Y.
Apparent frequency received by the wall Y is

$$\begin{gathered} f{\text{'}}_{}=\left(\frac{v}{v-v_s}\right)\times f_0 \\ \Rightarrow f\text{'}=\left(\frac{330}{330-6}\right)\times 4.5\times {10}^4 \\ \Rightarrow f\text{'}=4.58\times {10}^4\mathrm{Hz} \end{gathered}$$

Now, the apparent frequency received by the bat after reflection from the wall Y is given by:

$$\begin{gathered} f"=\left(\frac{v+v_s}{v}\right)\times f_x \\ \Rightarrow f"=\left(\frac{330+6}{330}\right)\times 4.58\times {10}^4 \\ \Rightarrow f"=4.66\times {10}^4\mathrm{Hz} \end{gathered}$$

Frequency of ultrasonic wave received by wall X:

$$\begin{gathered} n\text{'}=\frac{330}{330+6}\times 4.5\times {10}^4 \\ =4.41\times {10}^4\mathrm{Hz} \end{gathered}$$

The frequency of the ultrasonic wave received by the bat after reflection from the wall X is

$$\begin{gathered} n"=\frac{v-v_s}{v}\times n\text{'} \\ =\frac{330-6}{330}\times 4.41\times {10}^4 \\ =4.33\times {10}^4\mathrm{Hz} \end{gathered}$$

Beat frequency heard by the bat is

$$\begin{gathered} =4.66\times {10}^4-4.33\times {10}^4 \\ =3300\mathrm{Hz}. \end{gathered}$$