Wave on a string meeting a boundary between areas of different densities

[99wattr89]

This is the problem I'm working on: http://i.imgur.com/PBMFG.png

I'm very behind with normal modes and waves, and I need to figure out how to do this sort of question in time for my exams, so I'm hoping that you guys will be able to help me see how this can be answered.

I've answered the first part, deriving he wave equation, but for the second part I'm feeling very lost. Can someone give me a hint or nudge in the right direction for how to get started with it?

 

[99wattr89]

Can anyone help?

 

[NewtonianAlch]

I certainly don't know how to answer that, but seeing those second O.D.E's makes me think you might have better luck not posting this in introductory physics.

 

[99wattr89]

I certainly don't know how to answer that, but seeing those second O.D.E's makes me think you might have better luck not posting this in introductory physics.

Oh, I see! Thanks for the advice, I'll try advanced physics.

 

[asteriatic]

I can understand your struggle with this problem and I am happy to offer some guidance. To begin, it is important to understand the concept of waves and their behavior when meeting a boundary between areas of different densities.

First, let's review the wave equation, which describes the motion of a wave on a string:

d^2y/dt^2 = (T/μ) * d^2y/dx^2

where d^2y/dt^2 is the acceleration of a small element of the string, T is the tension in the string, μ is the linear mass density of the string, and d^2y/dx^2 is the curvature of the string.

In the second part of the problem, we are given a wave on a string that meets a boundary between two areas of different densities. This means that the linear mass density (μ) will change at the boundary. To solve this problem, we need to use the boundary conditions for waves at a boundary, which are:

1. The displacement (y) must be continuous at the boundary.
2. The tension (T) must be continuous at the boundary.

Using these conditions, we can set up equations for the wave on either side of the boundary and then solve for the unknown variables.

I would suggest starting by drawing a diagram of the wave on the string and labeling the different areas of densities. Then, use the wave equation and the boundary conditions to set up equations for both sides of the boundary. From there, you can solve for the unknown variables and determine the behavior of the wave at the boundary.

I hope this helps to nudge you in the right direction. Remember to always review the concepts and equations related to waves and practice solving similar problems to improve your understanding. Best of luck on your exams!