Two identical but separate strings, with the same tension, carry sinusoidal waves with the same frequency. Wave A has an amplitude that is twice that of wave B and transmits energy at a rate that is ........that of wave B.

Options

A. half
B. twice
C. one-fourth
D. four times

Answer

To solve the problem, we need to analyze the relationship between the amplitude of the waves and the energy transmitted by them. The key points to consider are:

1. **Understanding Wave Properties**: The energy transmitted by a wave on a string is related to its amplitude and frequency. The average power (energy per unit time) transmitted by a wave on a string can be expressed as:
\[ P = k \cdot A^2 \]
where \( P \) is the average power, \( A \) is the amplitude, and \( k \) is a constant that depends on other factors like mass per unit length and wave velocity.

2. **Identifying Given Information**:
- Wave A has an amplitude \( A_A = 2x \).
- Wave B has an amplitude \( A_B = x \).
- Both waves have the same frequency and tension in the strings.

3. **Calculating Power for Each Wave**:
- For wave A:
\[ P_A = k \cdot (A_A)^2 = k \cdot (2x)^2 = k \cdot 4x^2 = 4k \cdot x^2 \]
- For wave B:
\[ P_B = k \cdot (A_B)^2 = k \cdot (x)^2 = k \cdot x^2 \]

4. **Relating the Powers**:
- Now, we can express the power of wave A in terms of the power of wave B:
\[ P_A = 4 \cdot P_B \]

5. **Conclusion**:
- This means that wave A transmits energy at a rate that is 4 times that of wave B.

Thus, the final answer is that wave A transmits energy at a rate that is **4 times** that of wave B.