Question

Assertion : Sinusoidal waves are created in a thick rope by moving the hand holding the rope up and down. If the hand is moved up and down with greater frequency, the wave reaches the other end more quickly.
Reason : Wave velocity is product of frequency and wavelength for a sinusoidal wave.
(A) Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
(B) Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
(C) Assertion is correct but Reason is incorrect
(D) Assertion is incorrect but Reason is correct

Answer 1

To answer this question, we need to use the formula for the velocity of the wave in a stretched string. From there we can observe the factors on which the velocity depends.

Formula used: The formulae used for solving this question are given by
$v = \sqrt {\dfrac{T}{\mu }}$ , here $v$ is the velocity of wave produced in a string having mass per unit length of $\mu$ , and which is stretched under the force of tension of $T$ .
$v = \lambda f$ , here $v$ is the velocity of a wave having a wavelength of $\lambda$ and a frequency of $f$ .

Complete step-by-step solution
We know that the velocity of a wave is defined as the distance covered by a crest or a trough of the wave per unit time. The velocity of a wave is the property of the medium through which the wave is travelling. It is given by the expression
$v = \sqrt {\dfrac{T}{\mu }}$
Now, we see that the mass per unit length of the rope is a constant for a given rope. Also, if the force of tension in the given thick rope is kept the same, than the velocity of the wave will remain constant. So even if the wave is created by moving the hand up and down with more frequency, then also the velocity of the wave will be a constant and will not increase.
Since the velocity of the wave is constant, so the wave will take the same amount of time to reach the other end for any frequency of the moving hands.
Therefore, the assertion is incorrect.
Now, we know that the velocity of a wave can also be given by
$v = \lambda f$
As can be clearly observed from the above relation that the velocity of the wave is equal to the product of frequency and wavelength.
Therefore the Reason is correct.
Thus, Assertion is incorrect but Reason is correct.
Hence, the correct answer is option D.

Note
We might wonder what about the effect of increasing the frequency of the wave. The answer comes from the second formula for the wave velocity. Since the wave velocity is a constant, and is equal to the product of the frequency and the wavelength, so the wavelength will be inversely proportional to the frequency. So on increasing the frequency, the wavelength of the wave will decrease.