A sound source is moving towards a stationary observer with 1/10 of the speed of sound. The ratio of apparent to real frequency is

Options

A. `10/9`
B. `(10/9)^2`
C. `(11/10)^2`
D. `11/10`

Answer

To solve the problem of finding the ratio of apparent frequency to real frequency when a sound source is moving towards a stationary observer, we can use the Doppler effect formula. Here’s the step-by-step solution:

### Step 1: Understand the Variables
- Let \( v \) be the speed of sound.
- Let \( v_s \) be the speed of the source.
- Let \( n_0 \) be the real frequency emitted by the source.
- Let \( n' \) be the apparent frequency observed.

### Step 2: Given Values
- The speed of the source \( v_s \) is given as \( \frac{1}{10} \) of the speed of sound. Therefore, we can write:
\[ v_s = \frac{1}{10} v \]

### Step 3: Use the Doppler Effect Formula
The formula for the apparent frequency when the source is moving towards a stationary observer is:
\[ n' = \frac{v}{v - v_s} n_0 \]

### Step 4: Substitute the Values
Substituting \( v_s \) into the formula:
\[ n' = \frac{v}{v - \frac{1}{10} v} n_0 \]

### Step 5: Simplify the Denominator
Now, simplify the denominator:
\[ v - \frac{1}{10} v = \frac{10}{10} v - \frac{1}{10} v = \frac{9}{10} v \]

### Step 6: Substitute Back into the Formula
Now substituting this back into the formula for \( n' \):
\[ n' = \frac{v}{\frac{9}{10} v} n_0 \]

### Step 7: Simplify the Expression
This simplifies to:
\[ n' = \frac{10}{9} n_0 \]

### Step 8: Find the Ratio of Apparent to Real Frequency
The ratio of apparent frequency to real frequency is:
\[ \frac{n'}{n_0} = \frac{10}{9} \]

### Conclusion
Thus, the ratio of apparent frequency to real frequency is:
\[ \frac{n'}{n_0} = \frac{10}{9} \]

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