There are two wires of linear mass densities `lambda and 2lambda` and the tension in them are T and 2T, respectvely . Which wire is supposed to be the denser medium for transverse wave propagation ?

Options

A. `1^("st")` wire
B. `2^("nd")` wire
C. both are equally denser
D. nothing can be said in general

Answer

To determine which wire is the denser medium for transverse wave propagation, we will analyze the given information about the two wires.

### Step 1: Understand the relationship between tension, linear mass density, and wave velocity.
The velocity of a transverse wave in a wire is given by the formula:
\[ V = \sqrt{\frac{T}{\mu}} \]
where \( V \) is the wave velocity, \( T \) is the tension in the wire, and \( \mu \) is the linear mass density (mass per unit length) of the wire.

### Step 2: Analyze the first wire.
For the first wire:
- Linear mass density \( \mu_1 = \lambda \)
- Tension \( T_1 = T \)

Using the wave velocity formula:
\[ V_1 = \sqrt{\frac{T_1}{\mu_1}} = \sqrt{\frac{T}{\lambda}} \]

### Step 3: Analyze the second wire.
For the second wire:
- Linear mass density \( \mu_2 = 2\lambda \)
- Tension \( T_2 = 2T \)

Using the wave velocity formula:
\[ V_2 = \sqrt{\frac{T_2}{\mu_2}} = \sqrt{\frac{2T}{2\lambda}} = \sqrt{\frac{T}{\lambda}} \]

### Step 4: Compare the velocities of both wires.
From the calculations:
- For the first wire: \( V_1 = \sqrt{\frac{T}{\lambda}} \)
- For the second wire: \( V_2 = \sqrt{\frac{T}{\lambda}} \)

Since \( V_1 = V_2 \), both wires have the same velocity of transverse wave propagation.

### Step 5: Determine which wire is denser.
The linear mass density of the first wire is \( \lambda \) and for the second wire, it is \( 2\lambda \). Since the second wire has a higher linear mass density, it is denser.

### Conclusion:
Even though both wires allow transverse waves to propagate at the same velocity, the second wire (with linear mass density \( 2\lambda \)) is the denser medium.

### Final Answer:
The second wire (with linear mass density \( 2\lambda \)) is the denser medium for transverse wave propagation.

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