Ate of energy transfer by sinusoidal waves on a string q

[lizzyb]

Q: A two-dimensional water wave spreads in circular wave fonts. Show that the aplitude A at a distance r from the initial disturbance is proportional to $$\frac{1}{\sqrt{r}}$$. (Hint: Consider the energy carried by one outward moving ripple.)

Comments:
Let's consider the energy carried by one outward-moving ripple:
$$E_\lambda = \frac{1}{2} \mu \omega^2 A^2 \lambda$$
and I suppose there is another wave directly across the origin for some particle. But how do I relate this to r?

thanks!

 

[Kurdt]

Consider the energy density of the ripple. That is the energy per unit length. The length of the wave being a circle is $$\pi r$$ and then consider the equation you have given above for energy as it contains the amplitude.

 

[OlderDan]

Q: A two-dimensional water wave spreads in circular wave fonts. Show that the aplitude A at a distance r from the initial disturbance is proportional to $$\frac{1}{\sqrt{r}}$$. (Hint: Consider the energy carried by one outward moving ripple.)

Comments:
Let's consider the energy carried by one outward-moving ripple:
$$E_\lambda = \frac{1}{2} \mu \omega^2 A^2 \lambda$$
and I suppose there is another wave directly across the origin for some particle. But how do I relate this to r?

thanks!

All you really need to know to do this problem is that the energy is proportional to A². Since the wave is spreading out in a circle, the energy is being spread over the curcumference of the wave front.