A sinusoidal wave with amplitude `y_(m)` is travellling with speed `V` on a stgring with linear density `rho`. The angular frequency of the wave is `omega`. The following conclusions are down. Mark the one which is correct.

Options

A. doubling the frequency doubles the rate at which energy is carried along the string
B. if the amplitude were doubled, the rate at which energy is carried would be halved
C. if the amplitude were doubled, the rate at which energy is carried would be doubled
D. the rate at which energy is carried is directly proportional to the velocity of the wave.

Answer

To solve the problem, we need to analyze the relationship between the various parameters of a sinusoidal wave traveling on a string. The parameters given are amplitude \( y_m \), speed \( V \), linear density \( \rho \), and angular frequency \( \omega \). We need to determine which of the provided conclusions about the wave is correct.

### Step-by-Step Solution:

1. **Understanding the Energy Carried by the Wave**:
The energy carried by a sinusoidal wave is related to its amplitude, frequency, and speed. The average power \( P \) transmitted by a wave can be expressed as:
\[ P = \frac{1}{2} \mu \omega^2 A^2 v \]
where:
- \( \mu \) is the linear density of the string,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude,
- \( v \) is the speed of the wave.

2. **Analyzing the Options**:
We will evaluate each option based on the derived formula for power.

- **Option A**: "Doubling the frequency doubles the rate at which energy is carried along the string."
- Since power is proportional to \( \omega^2 \), if frequency (and thus \( \omega \)) is doubled, the power increases by a factor of \( 2^2 = 4 \). Therefore, this option is **incorrect**.

- **Option B**: "If amplitude is doubled, the energy would be halved."
- Power is proportional to \( A^2 \). If amplitude is doubled, the power increases by a factor of \( 2^2 = 4 \). Thus, this option is also **incorrect**.

- **Option C**: "If I double the amplitude, then the rate of energy doubles."
- Again, since power is proportional to \( A^2 \), doubling the amplitude results in a power increase by a factor of 4, not 2. Therefore, this option is **incorrect**.

- **Option D**: "The rate at which energy is carried is directly proportional to the velocity of the wave."
- From the formula, we see that power \( P \) is directly proportional to the speed \( v \). Thus, this option is **correct**.

3. **Conclusion**:
The only correct statement is **Option D**.