To determine how the speed of the wave changes as it moves from one type of rope to another, we can use the relationship between wave speed (\(v\)), frequency (\(f\)), and wavelength (\(\lambda\)). The relationship is given by:
\[ v = f \cdot \lambda \]
When the wave travels from the first rope to the second rope, the speed changes in relation to changes in wavelength and frequency. According to the problem, when the wave moves to the second rope, the wavelength (\(\lambda\)) becomes one-fourth of its original wavelength (\(\lambda_1\)). Thus:
\[ \lambda_2 = \frac{1}{4} \lambda_1 \]
We also know that the frequency of the wave must remain constant when it transitions between different media (assuming no energy is lost).
Given that the frequency remains the same, we can denote the initial wave characteristics as:
\[ v_1 = f \cdot \lambda_1 \]
And after the transition, we have:
\[ v_2 = f \cdot \lambda_2 = f \cdot \left(\frac{1}{4} \lambda_1\right) = \frac{1}{4} (f \cdot \lambda_1) = \frac{1}{4} v_1 \]
This means that the speed of the wave in the second rope (\(v_2\)) becomes one-fourth of its original speed (\(v_1\)).
Therefore, the correct response is:
The speed of the wave becomes one-fourth its original speed after the transfer.