GSO000 Problem 2

In this problem, we calculate the Doppler effect due to the motion of the sound source and observer. Let the rightward direction be the positive direction.

1. Frequency of Direct Sound $f_1$

The sound source moves at velocity $+V_s$ and the observer moves at velocity $+V_o$ . The direct sound travels rightward (velocity $+V$ ) from the source to the observer.

By the Doppler effect formula, the frequency $f_1$ of the direct sound heard by the observer is:

f_1 = \frac{V - V_o}{V - V_s} f_0

2. Frequency of Reflected Sound $f_2$

First, find the frequency $f_w$ received by the stationary wall. The sound wave travels rightward and the wall's velocity is zero.

f_w = \frac{V - 0}{V - V_s} f_0 = \frac{V}{V - V_s} f_0

Next, the wall acts as a new sound source (at frequency $f_w$ ), reflecting the sound wave to the left. The reflected sound velocity is $-V$ . The observer moves rightward at velocity $+V_o$ , moving toward the reflected source (the wall).

f_2 = \frac{V + V_o}{V} f_w = \frac{V + V_o}{V} \left( \frac{V}{V - V_s} f_0 \right) = \frac{V + V_o}{V - V_s} f_0

3. Beat Frequency $\Delta f$

The beat frequency is given by the absolute value of the difference between the two frequencies.

\Delta f = |f_2 - f_1| = \frac{V + V_o}{V - V_s} f_0 - \frac{V - V_o}{V - V_s} f_0 = \frac{2V_o}{V - V_s} f_0

Substituting numerical values:

\Delta f = \frac{2 \times 5}{340 - 20} \times 640 = \frac{10}{320} \times 640 = \frac{1}{32} \times 640 = 20\text{ Hz}

Answer: 20

GSO000 Problem 2

Beats from a Chasing Sound Source and a Fleeing Observer

Problem Statement

Constraints

Input Format

Solution

1. Frequency of Direct Sound $f_1$

2. Frequency of Reflected Sound $f_2$

3. Beat Frequency $\Delta f$

Beats from a Chasing Sound Source and a Fleeing Observer

Problem Statement

Constraints

Input Format

Solution

1. Frequency of Direct Sound f1f_1f1​

2. Frequency of Reflected Sound f2f_2f2​

3. Beat Frequency Δf\Delta fΔf

1. Frequency of Direct Sound $f_1$

2. Frequency of Reflected Sound $f_2$

3. Beat Frequency $\Delta f$