To understand how the speed of the wave changes when it moves from one medium (the first type of rope) to another (the second type of rope), we can use the wave speed formula:
\[ v = f \lambda \]
where:
- \( v \) is the wave speed,
- \( f \) is the frequency of the wave, and
- \( \lambda \) is the wavelength.
When a wave travels from one medium to another, its frequency \( f \) remains constant (since it is determined by the source of the wave), but the wavelength \( \lambda \) can change.
In this case, when the wave transfers from the first type of rope to the second type of rope, the wavelength becomes one-fourth of what it was before. If we let the original wavelength be \( \lambda_1 \) and the new wavelength be \( \lambda_2 \), we have:
\[ \lambda_2 = \frac{1}{4} \lambda_1 \]
Since the frequency remains constant, we can set up the relationship for the speeds in both media:
\[ v_1 = f \lambda_1 \quad \text{(speed in the first rope)} \] \[ v_2 = f \lambda_2 = f \left(\frac{1}{4} \lambda_1\right) = \frac{f \lambda_1}{4} = \frac{v_1}{4} \]
Thus, the speed of the wave in the second rope \( v_2 \) is one-fourth of the speed in the first rope \( v_1 \).
Considering the provided responses, the correct choice is:
The speed of the wave becomes one-fourth its original speed after the transfer.