A heavy uniform rope is held vertically by clamping it to a rigid support at the upper end. A wave of a certain frequency is set up at the lower end. The wave will

Options

A. travel with same velocity along the string
B. travel with decreasing velocity along the string
C. travel with increasing velocity along the string
D. travel with increasing frequency along the string

Answer

To solve the problem step by step, we will analyze the situation of a heavy uniform rope held vertically and how a wave propagates through it.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a heavy uniform rope that is clamped at the top and has a wave generated at the lower end.

**Hint**: Visualize the rope as hanging vertically, with the bottom free to vibrate.

2. **Identifying Key Parameters**:
- The wave speed in a rope is given by the formula:
\[ V = \sqrt{\frac{T}{\mu}} \]
where \( V \) is the wave speed, \( T \) is the tension in the rope, and \( \mu \) is the mass per unit length of the rope.

**Hint**: Remember that tension in the rope varies with height due to the weight of the rope below the point of interest.

3. **Calculating Tension**:
- The tension \( T \) at a height \( h \) from the bottom of the rope is equal to the weight of the rope below that point:
\[ T = \text{Weight of rope below} = \rho \cdot A \cdot h \cdot g \]
where \( \rho \) is the density of the rope, \( A \) is the cross-sectional area, and \( g \) is the acceleration due to gravity.

**Hint**: Think about how the weight of the rope changes as you move up the rope.

4. **Substituting Tension into the Wave Speed Formula**:
- Substitute the expression for tension into the wave speed formula:
\[ V = \sqrt{\frac{\rho \cdot A \cdot h \cdot g}{\mu}} \]

**Hint**: Recognize that \( \mu \) (mass per unit length) is constant for a uniform rope.

5. **Analyzing the Relationship**:
- Since \( \mu \) is constant, we can see that the wave speed \( V \) is directly proportional to the square root of \( h \):
\[ V \propto \sqrt{h} \]
- As \( h \) increases (moving up the rope), the wave speed \( V \) also increases.

**Hint**: Consider what happens to the speed of the wave as you move higher up the rope.

6. **Conclusion**:
- Therefore, the wave will travel with increasing velocity along the string as you move from the bottom to the top of the rope.

**Final Answer**: The correct option is **C**: travel with increasing velocity along the string.