Question

A man sitting on a beach watching water waves sees 4 waves pass by in 2 seconds, each with a wavelength of $0.5\,{\text{m}}$. Find out the speed of the waves.
A. $0.25\,{\text{m/s}}$
B. $0.5\,{\text{m/s}}$
C. $1.0\,{\text{m/s}}$
D. $2.0\,{\text{m/s}}$
E. $4.0\,{\text{m/s}}$

Answer 1

Use the formula for frequency of the wave. This formula gives the relation between the frequency of the wave, number of waves and time. Also use the formula for the velocity of the wave. This formula gives the relation between the velocity of the wave, frequency of the wave and wavelength of the wave. First calculate the frequency of the wave and then calculate the speed of the wave.

Formulae used:
The frequency $n$ of a wave is given by
$n = \dfrac{N}{t}$ …… (1)
Here, $N$ is the number of complete cycles of the wave and $t$ is the time.
The speed $v$ of a wave is given by
$v = n\lambda$ …… (2)
Here, $n$ is frequency of the wave and $\lambda$ is wavelength of the wave.

Complete step by step answer:
We have given that the man sitting on the beach watches 4 waves in 2 seconds.Hence, the number of complete waves 4 in time 2 seconds.
$N = 4$
$\Rightarrow t = 2\,{\text{s}}$
The wavelength of the wave seen by the man is $0.5\,{\text{m}}$.
$\lambda = 0.5\,{\text{m}}$
Let us first calculate the frequency of the waves seen by the man.
Substitute 2 for $N$ and $2\,{\text{s}}$ for $t$ in equation (1).
$n = \dfrac{4}{{2\,{\text{s}}}}$
$\Rightarrow n = 2\,{{\text{s}}^{ - 1}}$
Hence, the frequency of the waves seen by the man is $2\,{{\text{s}}^{ - 1}}$.
Let us now calculate the speed of the waves seen by the man.
Substitute $2\,{{\text{s}}^{ - 1}}$ for $n$ and $0.5\,{\text{m}}$ for $\lambda$ in equation (2).
$v = \left( {2\,{{\text{s}}^{ - 1}}} \right)\left( {0.5\,{\text{m}}} \right)$
$\therefore v = 1.0\,{\text{m/s}}$
Therefore, the speed of the wave is $1.0\,{\text{m/s}}$.

Hence, the correct option is C.

Note: The students should be careful while determining the frequency and speed of the waves seen by the man. Also the students should keep in mind that the frequency of the wave is shown by different symbols. So, the students should not get confused between these different signs. Any of the variables used to denote the frequency can be used.