Question

The distance between the two consecutive crests in a wave train produced in a string is \[5\,{\text{cm}}\]. If two complete waves pass through any point per second. The velocity of the wave is
A. \[15\,{\text{cm}} \cdot {{\text{s}}^{ - 1}}\]
B. \[10\,{\text{cm}} \cdot {{\text{s}}^{ - 1}}\]
C. \[2.5\,{\text{cm}} \cdot {{\text{s}}^{ - 1}}\]
D. \[5\,{\text{cm}} \cdot {{\text{s}}^{ - 1}}\]

Answer 1

Hint: 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 the wavelength of the wave. Determine the values of the wavelength and frequency of the given wave and calculate the velocity of the given wave train by using these values.

Formula used:
The velocity of a wave is given by
\[v = n\lambda \] …… (1)
Here, \[v\] is the velocity of the wave, \[n\] is the frequency of the wave and \[\lambda \] is the wavelength of the wave.

Complete step by step answer:
The distance between two consecutive crests or two consecutive troughs in a wave is known as the wavelength of the wave.

The distance between the two consecutive crests in a wave train produced in a string is \[5\,{\text{cm}}\]. Hence, the wavelength \[\lambda \] of the wave train produced in the string is \[5\,{\text{cm}}\].
\[\lambda = 5\,{\text{cm}}\]

One cycle of a wave is one complete.
The frequency of a wave is the number of the wave cycles (complete waves) passing in one second.
We have given that two complete waves pass in one second. Hence, the frequency \[n\] of the given wave is \[2\,{{\text{s}}^{ - 1}}\].
\[n = 2\,{{\text{s}}^{ - 1}}\]

We can determine the velocity of the given wave train using the formula given in equation (1).
Substitute \[2\,{{\text{s}}^{ - 1}}\] for \[n\] and \[5\,{\text{cm}}\] for \[\lambda \] in equation (1).
\[v = \left( {2\,{{\text{s}}^{ - 1}}} \right)\left( {5\,{\text{cm}}} \right)\]
\[ \Rightarrow v = 10\,{\text{cm}} \cdot {{\text{s}}^{ - 1}}\]
Therefore, the velocity of the wave is \[10\,{\text{cm}} \cdot {{\text{s}}^{ - 1}}\].

So, the correct answer is option (B).

Note:
The students may convert the unit of the wavelength of the given wave train from centimeter to meter i.e. in the SI system of units as we always do while doing calculations. But there is need of the conversion of the unit of wavelength as the ultimate unit of velocity given in the options is in \[{\text{cm}} \cdot {{\text{s}}^{ - 1}}\].