To analyze what happens when a wave transitions from one medium (the first type of rope) to another (the second type of rope) with a change in wavelength, we can use the wave speed equation:
\[ v = f \lambda \]
where:
- \( v \) is the wave speed,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.
When the wave transitions to the second medium and the wavelength becomes one-fourth of what it was before, we have:
\[ \lambda_2 = \frac{1}{4} \lambda_1 \]
However, the frequency remains constant during the transition between materials because of the continuity of the wave. Thus, we have:
\[ f_1 = f_2 \]
We can express the wave speeds in the two mediums as follows:
\[ v_1 = f_1 \lambda_1 \] \[ v_2 = f_2 \lambda_2 \]
Substituting \( f_2 = f_1 \) and \( \lambda_2 = \frac{1}{4} \lambda_1 \) into the second equation gives:
\[ v_2 = f_1 \left(\frac{1}{4} \lambda_1\right) = \frac{1}{4} f_1 \lambda_1 = \frac{1}{4} v_1 \]
This shows that the wave speed in the second rope (medium) is one-fourth of the wave speed in the first rope. Hence, the correct response is:
The speed of the wave becomes one-fourth its original speed after the transfer.