To analyze the problem, we can use the relationship between the speed of a wave (\(v\)), its frequency (\(f\)), and its wavelength (\(\lambda\)). This relationship is expressed as:
\[ v = f \cdot \lambda \]
When a wave travels from one medium (the first type of rope) to another medium (the second type of rope), the frequency of the wave remains constant. However, the wavelength changes as stated in the problem.
Let's denote the initial wavelength in the first rope as \(\lambda_1\) and the initial speed as \(v_1\). After the transfer to the second rope, the new wavelength is:
\[ \lambda_2 = \frac{1}{4} \lambda_1 \]
Let’s denote the new speed in the second rope as \(v_2\). Since the frequency remains constant, we can express the speeds as follows:
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In the first medium: \[ v_1 = f \cdot \lambda_1 \]
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In the second medium: \[ v_2 = f \cdot \lambda_2 = f \cdot \left(\frac{1}{4} \lambda_1\right) \]
Now we can substitute for \(\lambda_1\):
\[ v_2 = f \cdot \left(\frac{1}{4} \lambda_1\right) = \frac{1}{4}(f \cdot \lambda_1) = \frac{1}{4} v_1 \]
This indicates that the speed of the wave becomes one-fourth of its original speed after the transfer.
So, the correct response is:
The speed of the wave becomes one-fourth its original speed after the transfer.
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