Question

Which is true for a wave? (here $n =$ frequency, $T =$ time period)
(A) $nT = 1$
(B) $\dfrac{n}{T} = 2$
(C) $n = T$
(D) None of these

Answer 1

To solve this question, consider the definition of frequency and time period. After knowing what these quantities are, find a relation between them. To obtain a relation between the frequency and the time period, you can consider a particle of a wave. You solve this question to some extent using dimensional analysis and rule out options which do not meet the requirement.

Complete step by step solution:
Frequency is defined as the number of times an event occurs per unit time. The unit of frequency is
hertz ($Hz$). $1Hz$ is equal to the number of times an event occurring per second. From this, you
can conclude that the dimension of frequency is inverse of seconds, that is ${s^{ - 1}}$. In a wave,
frequency can be defined as the number of times a particle passes through a point in one second.

Time period is defined as the time taken by an event to repeat itself. As it is the time taken, the unit
the time period has to be $s$ which is quite obvious. In a wave, the time period can be defined as the
time taken by the particle of the wave to come back again to the same point which it had passed
before. The point to be noted here is that the velocity of the particle should be in the same direction
as it was before. For example, if the particle passes its mean position with its velocity directed
upwards, then the particle should possess the same motion to obtain the time period.
From the above discussion, if you compare the dimensions of frequency and time, you can say that
both are reciprocal of each other. Hence, you have $n = \dfrac{1}{T} \to nT = 1$.
Therefore, the relation $nT = 1$ is true for a wave.

Option (A) is correct.

Note: To solve these kinds of questions, where you have to obtain relation between some
quantities, first look for the definitions of the quantities and try to solve using dimensional analysis.
This method does not always work, but still in a few cases it can help you. Also remember the
definitions of both, frequency and time period.