Two waves travel down the same string . These waves have the same velocity , frequency `f` and wavelength but having different phase constants `phi_(1)` and `phi_(2) (lt phi_(1))` and amplitudes `A_(1)` and `A_(2) ( lt A_(1))`. Mark the correct statement(s) for the resultant wave which is produced due to superposition of these two waves.

Options

A. The amplitude of the resultant waves is `A = A_(1) + A_(2)`
B. The amplitude of the resultant waves is `A_(1) - A_(2) to A_(1) + A_(2)`
C. The frequency of the resultant waves is `f`
D. The frequency of the resultant waves is `f//2`

Explanation

To solve the problem regarding the superposition of two waves traveling down the same string, we will analyze the given conditions and derive the characteristics of the resultant wave.

### Step-by-Step Solution:

1. **Understanding the Given Parameters:**
- We have two waves with the same frequency \( f \), velocity, and wavelength.
- The amplitudes are \( A_1 \) and \( A_2 \) (with \( A_2 < A_1 \)).
- The phase constants are \( \phi_1 \) and \( \phi_2 \) (with \( \phi_2 < \phi_1 \)).

2. **Resultant Wave Frequency:**
- When two waves of the same frequency interfere, the frequency of the resultant wave remains the same as that of the individual waves. Therefore, the frequency of the resultant wave \( f_R = f \).

3. **Resultant Amplitude Calculation:**
- The amplitude of the resultant wave \( A_R \) can be calculated using the formula:
\[ A_R = \sqrt{A_1^2 + A_2^2 + 2 A_1 A_2 \cos(\phi_2 - \phi_1)} \]
- Here, \( \phi_2 - \phi_1 \) is the phase difference between the two waves.

4. **Amplitude Range:**
- The resultant amplitude \( A_R \) will lie between \( |A_1 - A_2| \) and \( A_1 + A_2 \).
- This means:
\[ |A_1 - A_2| < A_R < A_1 + A_2 \]
- Since \( A_2 < A_1 \), we can write:
\[ A_1 - A_2 < A_R < A_1 + A_2 \]

5. **Conclusion on Statements:**
- The statements regarding the resultant wave can be evaluated based on the above findings:
- The frequency of the resultant wave is the same as the individual waves.
- The amplitude of the resultant wave lies between \( A_1 - A_2 \) and \( A_1 + A_2 \).

### Summary of Correct Statements:
- The frequency of the resultant wave is \( f \).
- The amplitude of the resultant wave lies between \( A_1 - A_2 \) and \( A_1 + A_2 \).