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 key parameters given are:

- Amplitude of the wave: \( y_m \)
- Speed of the wave: \( V \)
- Linear density of the string: \( \rho \)
- Angular frequency of the wave: \( \omega \)

We will derive the expression for the average power carried by the wave and analyze the options provided.

### Step 1: Understand the formula for power in a wave
The average power \( P \) carried by a sinusoidal wave on a string can be expressed as:
\[ P = \frac{1}{2} \mu \omega^2 A^2 V \]
where:
- \( \mu \) is the linear density of the string (\( \rho \)),
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude of the wave,
- \( V \) is the speed of the wave.

### Step 2: Analyze the effect of doubling frequency
If we double the angular frequency (\( \omega \to 2\omega \)):
\[ P' = \frac{1}{2} \mu (2\omega)^2 A^2 V = \frac{1}{2} \mu (4\omega^2) A^2 V = 4 \left(\frac{1}{2} \mu \omega^2 A^2 V\right) = 4P \]
This shows that doubling the frequency increases the power by a factor of 4.

### Step 3: Analyze the effect of doubling amplitude
If we double the amplitude (\( A \to 2A \)):
\[ P' = \frac{1}{2} \mu \omega^2 (2A)^2 V = \frac{1}{2} \mu \omega^2 (4A^2) V = 4 \left(\frac{1}{2} \mu \omega^2 A^2 V\right) = 4P \]
This shows that doubling the amplitude also increases the power by a factor of 4.

### Step 4: Analyze the effect of doubling both frequency and amplitude
If we double both frequency and amplitude:
\[ P' = \frac{1}{2} \mu (2\omega)^2 (2A)^2 V = \frac{1}{2} \mu (4\omega^2) (4A^2) V = 16P \]
This shows that doubling both frequency and amplitude increases the power by a factor of 16.

### Step 5: Analyze the effect of velocity
The power is directly proportional to the wave speed \( V \):
\[ P \propto V \]
If the speed of the wave increases, the power carried by the wave also increases proportionally.

### Conclusion
Now, we analyze the options:
- **Option A**: Doubling the frequency increases power by a factor of 4 (not correct).
- **Option B**: Doubling the amplitude increases power by a factor of 4 (not correct).
- **Option C**: Doubling the amplitude does not double the power (not correct).
- **Option D**: The rate at which energy is carried is directly proportional to the velocity of the wave (correct).

Thus, the correct conclusion is **Option D**.