A sound wave is produced by a vibrating metallic string stretched between ends. Four statements are given below. Some of them are correct.
(P) Sound wave is produced inside the string.
(Q) Sound wave in the string is transverse.
(R) Wavelength of the sound wave in surrounding air is equal to the wavelength of the transverse wave on the string.
(S) Loudness of sound is proportional to the square of the amplitude of the vibrating string.
Choose the correct option.

Options

A. P
B. R and S
C. P and Q
D. S

Explanation

To solve the problem, we need to analyze each statement one by one to determine which are correct regarding sound waves produced by a vibrating metallic string.

**Step 1: Analyze Statement P**
- **Statement P**: "Sound wave is produced inside the string."
- **Analysis**: Sound waves are produced in the medium (air, in this case) surrounding the string, not inside the string itself. The vibrations of the string create pressure variations in the air, which are perceived as sound. Therefore, this statement is **incorrect**.

**Step 2: Analyze Statement Q**
- **Statement Q**: "Sound wave in the string is transverse."
- **Analysis**: The sound waves produced by the vibrating string are longitudinal waves, not transverse. In a longitudinal wave, the particle displacement is parallel to the direction of wave propagation. Thus, this statement is also **incorrect**.

**Step 3: Analyze Statement R**
- **Statement R**: "Wavelength of the sound wave in surrounding air is equal to the wavelength of the transverse wave on the string."
- **Analysis**: The wavelength of the sound wave in air is not necessarily equal to the wavelength of the transverse wave on the string. The two types of waves (longitudinal in air and transverse in the string) have different properties and can have different wavelengths. Therefore, this statement is **incorrect**.

**Step 4: Analyze Statement S**
- **Statement S**: "Loudness of sound is proportional to the square of the amplitude of the vibrating string."
- **Analysis**: This statement is correct. The loudness of sound is related to the intensity of the sound wave, which is proportional to the square of the amplitude of the wave. Thus, this statement is **correct**.

**Final Conclusion**:
- The only correct statement is **S**. Therefore, the correct option is that only statement S is true.

### Summary of the Correct Statements:
- P: Incorrect
- Q: Incorrect
- R: Incorrect
- S: Correct