$(1)[3$ points$]$ Consider a long, horizontal rope that is being kept at a fixed tension. You hold one end of the rope and move it up and down periodically at a certain frequency such that a periodic wave travels along the rope. For each statement below, check the appropriate box to indicate whether the statement is correct or incorrect.

$($a$)$ If you move the rope up and down at twice the frequency, the wave will have half the original wavelength.

$($b$)$ Increasing the tension would decrease the speed of the wave.

$($c$)$ Using a heavier rope would result in a faster wave $($assuming the length, tension, and frequency remain the same$).$

Correct Incorrect

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