Explaining the Ripple Effect in a Circular Wave Pulse

[Mindscrape]

Here is the problem:
If a pebbel is tossed into a pond, a circular wave pulse propagates outward from the disturbance. If you look closely you will see a fine structure in the pulse consisting of surface ripples moving inward though the circular disturbance. Explain this effect in terms of group and phase velocity if the phase velocity of ripples is given by $$v_p = \sqrt{2 \pi S/ \lambda \rho}$$, Where S is the surface tension and p is the density of the liquid.

I am not really sure where to start. Should I find the envelope velocity and compare it to the high frequency velocity. I know it will have something to do with superposition, but Fourier analysis seems like it is the wrong approach.

 

[Nate810]

Can someone help me out here?Answer:The explanation for this effect can be found by considering the group velocity and phase velocity of the wave pulse. The group velocity of a wave is the velocity of the envelope of the wave, and it is given by v_g = \frac{d\omega}{dk}, where \omega is the angular frequency and k is the wavenumber. In contrast, the phase velocity of a wave is the velocity of the individual ripples and it is given by v_p = \sqrt{\frac{2\pi S}{\lambda \rho}}, where S is the surface tension and \rho is the density of the liquid. In this case, since the phase velocity of the ripples is given, we can calculate the group velocity of the wave pulse. We can see that since the group velocity is the velocity of the envelope of the wave, it is slower than the phase velocity. This means that the individual ripples will move inward through the circular disturbance at a faster rate than the wave itself is propagating outward. This is why you can see the fine structure in the pulse consisting of the surface ripples moving inward.

 

[kyle8921]

I can provide an explanation for the Ripple Effect in a Circular Wave Pulse. When a pebble is tossed into a pond, it creates a disturbance in the water, causing a circular wave pulse to propagate outward. This pulse consists of two types of waves - surface ripples and a circular disturbance. The surface ripples are caused by the surface tension and density of the liquid, while the circular disturbance is caused by the energy from the pebble.

The surface ripples have a different velocity compared to the circular disturbance. This is because the velocity of the surface ripples is determined by the phase velocity equation v_p = \sqrt{2 \pi S/ \lambda \rho}, where S is the surface tension and \rho is the density of the liquid. On the other hand, the circular disturbance has a constant velocity that is determined by the energy of the pebble.

Now, let's look at how these two types of waves combine to create the Ripple Effect. As the circular disturbance moves outward, it creates a disturbance in the water surface, causing the surface ripples to form. These ripples then move inward towards the center of the circular disturbance. This movement is due to the fact that the phase velocity of the surface ripples is greater than the velocity of the circular disturbance. This difference in velocity creates an interference pattern, where the surface ripples appear to move inward through the circular disturbance.

Furthermore, the group velocity of the surface ripples is also affected by the phase velocity equation. The group velocity is the velocity at which the overall shape of the wave pulse moves. In this case, the group velocity is determined by the envelope velocity, which is the average of the high frequency velocity and the low frequency velocity. The high frequency velocity is determined by the phase velocity equation, while the low frequency velocity is determined by the velocity of the circular disturbance. This means that the group velocity of the surface ripples will be somewhere between the phase velocity and the velocity of the circular disturbance.

In summary, the Ripple Effect in a Circular Wave Pulse can be explained by the different velocities of the surface ripples and the circular disturbance. The surface ripples, which are caused by the surface tension and density of the liquid, have a greater phase velocity compared to the constant velocity of the circular disturbance. This creates an interference pattern, where the ripples appear to move inward through the circular disturbance. Additionally, the group velocity of the surface ripples is determined by the average of