Maximum Frequency Variation Received Between Oscillating Transmitter and Receiver

To find the maximum variation in the frequency as measured by the receiver, we first determine the maximum speeds of the transmitter ($T$) and the receiver ($R$).

Step 1: Maximum Speed of the Pendulums

Both the transmitter and the receiver are identical simple pendulums suspended from strings of length $L = 8\text{ m}$. They are released from rest from an initial angular amplitude $\theta_0$, where $\cos\theta_0 = 0.9$.

Using the principle of conservation of mechanical energy for either pendulum: $m g L (1 - \cos\theta_0) = \frac{1}{2} m v_{\text{max}}^2$

Solving for $v_{\text{max}}$: $v_{\text{max}} = \sqrt{2 g L (1 - \cos\theta_0)}$

Given:

• $g = 10\text{ m/s}^2$
• $L = 8\text{ m}$
• $\cos\theta_0 = 0.9 \implies 1 - \cos\theta_0 = 0.1$

$v_{\text{max}} = \sqrt{2 \times 10 \times 8 \times 0.1} = \sqrt{16} = 4\text{ m/s}$

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Step 2: Doppler Effect Calculations

Since both pendulums are released simultaneously in opposite directions from their respective equilibrium positions, they pass through their lowest points simultaneously:

1. Moving towards each other: Both reach their maximum speed $v_{\text{max}} = 4\text{ m/s}$ towards one another, yielding the maximum frequency $f_{\text{max}}$.
2. Moving away from each other: Half a period later, both reach their maximum speed $v_{\text{max}} = 4\text{ m/s}$ directed away from one another, yielding the minimum frequency $f_{\text{min}}$.

Given parameters:

• Natural frequency, $f_0 = 660\text{ Hz}$
• Speed of sound in air, $v = 330\text{ m/s}$

Maximum Observed Frequency ($f_{\text{max}}$):

$f_{\text{max}} = f_0 \left( \frac{v + v_{\text{max}}}{v - v_{\text{max}}} \right) = 660 \times \left( \frac{330 + 4}{330 - 4} \right) = 660 \times \frac{334}{326} \approx 676.20\text{ Hz}$

Minimum Observed Frequency ($f_{\text{min}}$):

$f_{\text{min}} = f_0 \left( \frac{v - v_{\text{max}}}{v + v_{\text{max}}} \right) = 660 \times \left( \frac{330 - 4}{330 + 4} \right) = 660 \times \frac{326}{334} \approx 644.19\text{ Hz}$

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Step 3: Maximum Variation in Frequency

The maximum variation in frequency ($\Delta f$) is the total range between the maximum and minimum measured frequencies:

$\Delta f = f_{\text{max}} - f_{\text{min}}$

$\Delta f = f_0 \left( \frac{v + v_{\text{max}}}{v - v_{\text{max}}} - \frac{v - v_{\text{max}}}{v + v_{\text{max}}} \right) = f_0 \left( \frac{4 v v_{\text{max}}}{v^2 - v_{\text{max}}^2} \right)$

Substituting the numerical values: $\Delta f = 660 \times \frac{4 \times 330 \times 4}{330^2 - 4^2} = 660 \times \frac{5280}{108884} \approx 32.00\text{ Hz}$

Using the approximation $v \gg v_{\text{max}}$: $\Delta f \approx f_0 \left( \frac{4 v_{\text{max}}}{v} \right) = 660 \times \frac{4 \times 4}{330} = 32\text{ Hz}$

The maximum variation in the frequency measured by the receiver is 32.