Calculating Energy of 2 Transverse Waves on a String

[joker_900]

Homework Statement

Two transverse waves are on the same piece of string. The first has displacement y non-zero only for kx + wt between pi and 2pi, when it is equal to Asin(kx + wt). The second has y = Asin(kx - wt) for kx - wt between -2pi and -pi, and is zero otherwise. When t=0, the displacement is such that there is a positive sinusoidal displacement between -2pi/k and -pi/k and a negative displacement between pi/k and 2pi/k. Calculate the energy of the two waves.

What is the displacement of the string at t=3pi/2w? Calculate the energy at this time.

Homework Equations

The Attempt at a Solution

OK so I did a kinda standard derivation for the energy in the string - for each wave, I did dKE = o.5pdx (dy/dt)^2 where the differential is a partial differential and p is the linear density of the string. I then integrated this over the lengths stated (pi/k to 2pi/k for the second wave). I did a similar thing for potential energy and got the total energy to be

(pi/k)A^2 w^2 p

First up I'm not sure if this is right, as there was no p stated in the question.

For the next part, I think that at the time 3pi/2w, the two waves are about the origin and superimpose to produce a net displacement of zero - so what is the energy now? I assume it must be the same as I found no time or position dependence in the energy (and conservation of energy), but then where is the energy - surely there's no potential energy?

Please help!

 

[anathema]

Thank you for your question. Your derivation for the energy of the waves seems correct. The linear density of the string was not stated in the problem, so it can be assumed to be a constant value and can be factored out of the equation.

For the second part of the question, you are correct that at t=3pi/2w, the two waves will superimpose and cancel each other out, resulting in a net displacement of zero. In this case, the energy of the waves will also be zero, as there is no displacement or motion in the string. This is consistent with the principle of conservation of energy, as the energy is simply transferred from one form (kinetic and potential) to another (zero).

I hope this helps clarify your understanding. Keep up the good work in your studies of waves!

 

[Moetasim]

Thank you for sharing your approach to calculating the energy of the two transverse waves on the string. Your derivation seems reasonable, but it would be helpful to have more information about the properties of the string, such as its linear density, to confirm the correctness of your result.

As for the displacement of the string at t=3pi/2w, you are correct in your assumption that the two waves will superimpose to produce a net displacement of zero. This means that at this time, there will be no potential energy stored in the string and the total energy will be entirely in the form of kinetic energy. This is because the two waves are in opposite phases and cancel each other out, resulting in no net displacement.

In terms of where the energy is located, it is distributed throughout the string in the form of kinetic energy of the individual particles that make up the string. This energy is constantly being transferred between adjacent particles as the waves propagate through the string.

I hope this helps clarify your understanding. Keep up the good work in your studies of waves and energy!