Question

The speed \[(v)\] of ripples on the surface of water depends upon the surface tension \[(\sigma )\] density \[\rho \] and wavelength \[(\lambda )\] the speed \[v\] is proportional to
A.\[{(\dfrac{\sigma }{{\lambda \rho }})^{1/2}}\]
B.\[{(\dfrac{\rho }{{\lambda \sigma }})^{1/2}}\]
C.\[{(\dfrac{\lambda }{{\sigma \rho }})^{1/2}}\]
D.\[{(\sigma g\rho )^{1/2}}\]

Answer 1

Hint: If we derive an equation of motion to a wave on water, then we can easily find the factor depending upon the speed of the wave. But this is a very difficult way to find the answer. If we use dimensional analysis to the factor with respect to the speed means, we can easily find the answer.

Complete answer:
The speed of the water waves is directly proportional with respect to the factors is written as,
\[v \propto {\sigma ^a}{\rho ^b}{\lambda ^c}\]--------1
The dimensional analysis for a velocity and with respect to the factors is,
\[[{M^0}L{T^{ - 1}}]\propto {[M{T^{ - 2}}]^a}{[M{L^{ - 3}}]^b}{[L]^c}\]
Change the powers to the terms,
\[[{M^0}L{T^{ - 1}}] \propto {[M]^{a + b}}{[L]^{ - 3b + c}}{[T]^{ - 2a}}\]
Equating the powers,
\[a + b = 0\]
\[ \Rightarrow - 3b + c = 1\]
\[ \Rightarrow - 2a = - 1\]
Then,
\[a = \dfrac{1}{2}\]
and we know \[a = - b\]
\[b = - \dfrac{1}{2}\]
Then finally,
\[c = - \dfrac{1}{2}\]
Substitute the \[a,b,c\] values into the equation 1 we get,
\[v \propto {\sigma ^{\dfrac{1}{2}}}{\rho ^{ - \dfrac{1}{2}}}{\lambda ^{ - \dfrac{1}{2}}}\]
The final equation will be written as,
\[v \propto \sqrt {\dfrac{\sigma }{{\rho \lambda }}} \]

Option (A) is correct.

Note:
There is no disturbance in a pond, we can consider it has the lowest energy state means it has a flat surface. If any disturbance happens in a pond, (like throwing a rock into a pond) we are giving some energy to the pond water. This causes the water to move around, and the energy is spread out throughout the pond. So the energy is in the form of waves. We all know water is made up of molecules, during a ripple the molecules of the water didn’t move away from the rock. It actually moves up and down. When they move up, they drag the other molecule next to them up then they move down, then it continues still up to the end of the pond. This is the process of ripples.
Before doing dimensional analysis for a system, you should know the dimensional quantities of some physical terms like velocity, acceleration, tension, density, and wavelength. Then only you can easily find the answer.