Which of the following statements are correct ?

Options

A. The decrease in the speed of sound at high altitudes is due to a fall in pressure .
B. The standing wave on a string under a tension , fixed at its ends , does not have well - defined nodes .
C. The phenomenon of beats is not observable in the case of visible light waves.
D. The apparent frequency is `f_(1)` when a source of sound approached a stationary observer with a speed `u` and is `f_(2)` when the observer approaches the same stationary source with the same speed . Then `f_(2) lt f_(1) , if u lt v`, where `v` is the speed of sound.

Explanation

To solve the question regarding which statements are correct, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Statement 1: "Decrease in the speed of sound at high altitudes is due to fall in the pressure."**
- The speed of sound \( v \) is given by the formula:
\[ v = \sqrt{\frac{B}{\rho}} \]
where \( B \) is the bulk modulus and \( \rho \) is the density of the medium.
- At high altitudes, the pressure decreases, which affects the bulk modulus. Since the bulk modulus is related to pressure, a decrease in pressure can lead to a decrease in the speed of sound.
- Therefore, this statement is **correct**.

2. **Statement 2: "The standing wave on a string under tension fixed at both ends does not have well-defined nodes."**
- In a standing wave on a string fixed at both ends, nodes are points where the displacement is always zero. These nodes are well-defined because they occur at fixed positions along the string.
- Therefore, this statement is **incorrect**.

3. **Statement 3: "The phenomena of beat is not observable in case of visible light waves."**
- Beats occur due to the interference of two waves of slightly different frequencies. While the phenomenon can theoretically occur with light waves, the high frequencies of visible light make it practically difficult to observe beats.
- Thus, this statement is **correct**.

4. **Statement 4: "Apparent frequency \( f_1 \) when the source approaches the observer with a speed of \( u \)."**
- When the source approaches the observer, the apparent frequency \( f_1 \) can be calculated using the formula:
\[ f_1 = f_s \frac{v}{v - u} \]
where \( f_s \) is the source frequency, \( v \) is the speed of sound, and \( u \) is the speed of the source.
- If the observer approaches the source, the apparent frequency \( f_2 \) is given by:
\[ f_2 = f_s \frac{v + u}{v} \]
- To compare \( f_1 \) and \( f_2 \), we can substitute \( u < v \) and analyze the expressions. It can be shown that \( f_1 > f_2 \) when \( u < v \).
- Therefore, this statement is **correct**.

### Conclusion:
- The correct statements are: **1, 3, and 4**.
- The incorrect statement is: **2**.