String `I` and `II` have identical and linear mass densities, but string `I` is under greater tension than string `II`. The accompanying figure shows four different situations, `A` to `D`, in which standing wave patterns exist on the two strings. In which situation it is possible that strings `I` and `II` are oscillating at the same resonant frequency ?

Options

A.
B.
C.
D.

Explanation

To determine in which situation strings I and II can oscillate at the same resonant frequency, we need to analyze the relationship between tension, frequency, and wavelength for both strings.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The frequency of a wave on a string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where \( f \) is the frequency, \( L \) is the length of the string, \( T \) is the tension, and \( \mu \) is the linear mass density of the string.

2. **Identifying Given Conditions**:
- Both strings I and II have identical linear mass densities (\( \mu \)).
- String I is under greater tension than string II (\( T_I > T_{II} \)).

3. **Analyzing Frequency**:
Since the tension in string I is greater than in string II, the frequency of string I will be greater than that of string II:
\[ f_I > f_{II} \]

4. **Considering Wavelength**:
The frequency is also inversely proportional to the wavelength (\( \lambda \)):
\[ f = \frac{v}{\lambda} \]
where \( v \) is the wave speed. For the same frequency, the wavelengths must satisfy:
\[ \lambda_I > \lambda_{II} \]
This means that for strings I and II to have the same frequency, the wavelength of string I must be greater than that of string II.

5. **Evaluating the Situations (A to D)**:
We need to look at the standing wave patterns in each situation (A to D) to determine where the wavelength of string I is greater than that of string II.

6. **Identifying the Correct Option**:
Upon examining the standing wave patterns:
- In situation C, the number of nodes and antinodes indicates that the wavelength of string I is indeed greater than that of string II, which satisfies our condition for equal frequencies.

7. **Conclusion**:
Therefore, the only situation where strings I and II can oscillate at the same resonant frequency is in **option C**.