Question

The distance between three consecutive crests of waves is $60cm$. If the waves are produced at the rate of $180/\min$, calculate (i) wavelength (ii) time period (iii) wave velocity.

Answer 1

Using the definition of the wavelength, we can determine its value from the given distance. Also, from the given rate of wave production we can determine the time period. Finally, using the formula for the wave velocity in terms of wavelength and the time period we can determine its value.
Formula used: The formula used to solve this question is given by
$v = \dfrac{\lambda }{T}$, here $v$ is the velocity of a wave having a wavelength of $\lambda$ and a time period of $T$.

Complete step-by-step solution:
(i) We know that the distance between two consecutive crests or two consecutive troughs of a wave is equal to the wavelength of the wave. So the distance between the three consecutive crests of the wave must be equal to the twice of the wavelength. According to the question, we are given the distance between three consecutive to be $60cm$. So if the wavelength of the given wave is equal to $\lambda$, then we can say that
$2\lambda = 60cm$
$\Rightarrow \lambda = 30cm$
We know that $1cm = 0.01m$. So we get
$\lambda = 0.3m$..............(1)
Hence, the wavelength of the given wave is equal to $0.3m$.
(ii) According to the question, the waves are produced at the rate of $180/\min$. This means that $180$ waves are produced in a minute. Since there are $60s$ in a minute, so the number of waves produced in one second becomes
$n = \dfrac{{180}}{{60}}{s^{ - 1}}$
$\Rightarrow n = 3{s^{ - 1}}$
Now, we know that the frequency of a wave is equal to the number of cycles produced in a second. Since one cycle corresponds to one wave, we can say that the frequency is equal to the number of waves produced in one second. So the above number is nothing but the frequency of the given wave. So we have
$f = 3{s^{ - 1}}$
We know that the time period is equal to the inverse of the frequency, that is,
$T = \dfrac{1}{f}$
$\Rightarrow T = \dfrac{1}{3}s$ …………………………..(2)
Hence, the time period of the given wave is equal to $\dfrac{1}{3}s$.
(iii) We know that the velocity of a wave is equal to the ratio of the wavelength and the time period, that is,
$v = \dfrac{\lambda }{T}$
Substituting (1) and (2) in the above equation, we get
$v = \dfrac{{0.3}}{{1/3}}$
$\Rightarrow v = 0.9m{s^{ - 1}}$

Hence, the velocity of the given wave is equal to $0.9m{s^{ - 1}}$.

Note: Do not take the distance between three consecutive crests to be equal to thrice the wavelength of the wave. Remember, two consecutive crests or troughs correspond to one wave. So three consecutive crests corresponds to two waves, and hence twice the wavelength.