Creating ripples in water is so much fun!

Allow me to first borrow the above illustration from the Internet to probe into the water ripple scenario a bit deeper. When a single ripple is created, we must appreciate the fact that the energy of the wave source is constant and that as the ripple grows outwards, the energy of the circular water wave spreads outwards. This would imply that there would be increasingly less and less energy per unit length of the circumference of the circular water wave as it continues to grow outwards. Therefore with this understanding, we can now proceed to find the amplitude. Earlier, we have already established the understanding that the energy of the wave spreads throughout the circumference of the water wave which if we translate into mathematical terms, it will produce the following result $E\propto \frac { 1 }{ 2\pi R }$ where the energy of the wave is inversely proportional to the circumference of the wave. Hence, by using this relationship, we can easily see that $\frac { { E }_{ 0 } }{ { E }_{ 1 } } =\frac { { R }_{ 1 } }{ { R }_{ 0 } } =\frac { 10 }{ 5 } =2$ . After which, we can make use of the idea in classical wave theory that the energy of the wave, $E$ is proportional to the square of the amplitude of the wave, $A^2$ . Hence, we now can translate this into our mathematical expression as follows: $\frac { { E }_{ 0 } }{ { E }_{ 1 } } ={ \left( \frac { { A }_{ 0 } }{ { A }_{ 1 } } \right) }^{ 2 }$ , which would give ${ \left( \frac { { 4 } }{ { A }_{ 1 } } \right) }^{ 2 }=\frac { 2 }{ 1 }$ . With a little bit of manipulation, we will get ${ A }_{ 1 }=\sqrt { 8 }$ . Note that $R$ is the radius of the circular wave and $R_{ 0 }$ is the distance of the wave from the source at a distance of $5 cm$ and $R_{ 1 }$ is the distance of the wave at $10 cm$ .