Question

In a ripple tank when one pulse is sent every tenth of a second, the distance between consecutive pulses is30mm. In the same depth of water pulses are produced at half second intervals. What is the new distance between consecutive pulses?
\[
A. 0.67mm \\
B. 6.0mm \\
C. 150mm \\
D. 600mm \\
 \]

Answer 1

Hint: The frequency and wavelength of any wave \[\left( {in{\text{ }}this{\text{ }}case,{\text{ }}the{\text{ }}ripple} \right)\] gives the speed of the wave,
\[v = f\lambda \]
Under the same depth (that means essentially the same density and pressure, as well as temperature), the velocity of the wave will remain the same.
$ {v_{initial}} = {v_{final}} $

Complete step by step answer:
The wave speed in a medium depends on the frequency and the wavelength of the wave. We are given that initially, one pulse is sent every tenth of a second, the distance between consecutive pulses is30mm. In the second case, at the same depth of water pulses are produced at half second intervals, we need to find out the distance between the consecutive pulses, that is, the wavelength of the wave.

From above, we’ve.
\[
  {f_1} = \dfrac{1}{{{T_1}}} = \dfrac{1}{{1/10}} = 10{s^{ - 1}} \\
  {\lambda _1} = 30mm \\
  {f_2} = \dfrac{1}{{{T_2}}} = \dfrac{1}{{1/2}} = 2{s^{ - 1}} \\
 \]

As we know that, at the same depth under water velocity of sound will be same,
\[{v_{initial}} = {v_{final}}\]
\[ = > {f_1}{\lambda _1} = {f_2}{\lambda _2}\]
\[ = > {\lambda _2} = {f_1}{\lambda _1}/{f_2}\]
Putting the given values in the above equation we’ve,
\[
   = > {\lambda _2} = \dfrac{{10 \times 30}}{2} \\
   = > {\lambda _2} = 150 \\
 \]

So, the distance between consecutive ripples is found to be \[150{\text{ }}mm\] when the pulses are created at half second interval

So, the correct answer is “Option C”.

Note:
The wave works in such a way that the increase in frequency \[\left( {decrease{\text{ }}in{\text{ }}time{\text{ }}period} \right)\] and increase in wavelength will lead to increase in wave speed.
Notice that the wavelength has increased from \[30mm{\text{ }}to{\text{ }}150{\text{ }}mm\] due to decrease in frequency from \[10{\text{ }}Hz{\text{ }}to{\text{ }}2{\text{ }}Hz\], as the wave speed has to remain constant.