Boundary Problem: Massless String with Knot of Mass 'm' at z=0

In summary, the conversation is about a problem involving a wave incident on a boundary with a mass of 'm' at z=0 and a massless string on the other side. The first condition is that the string must be continuous, and the second condition is that T*[df(+) - df(-)] = m*d^2f/dz^2. The person discussing the problem has tried letting df(+) = 0 and letting the transmitted wave be imaginary, but neither solution has worked. They are seeking help to understand why their solutions are not giving the correct answer.
  • #1
Thorazine
So I have a wave incident on a boundary problem. The incidence is normal and the boundary is a knot of mass 'm' at z = 0, with the wave coming from minus infinity. The problem is that the string on the other side is massless, and I can't figure out exactly how that effects the boundary conditions. I know the string must be continuous as the first condition, and:

T*[df(+) - df(-)] = m*d^2f/dz^2

as the second condition. Where df(+) is the first derivative on the positive side of z = 0 and df(-) is the derivative on the negative side.

What I thought to do was let df(+) = 0 because there is no force coming from that side, but that hasn't worked. I also tried letting the transmitted wave be imaginary only because without mass I can't see how a string could vibrate, and that didn't work either. Any help would be much appreciated.
 
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  • #2
Originally posted by Thorazine
What I thought to do was let df(+) = 0 because there is no force coming from that side...

Thorazine, I think this is physically correct. I think df/dz=0 everywhere in the massless part. Could you please explain why it doesn't work?
 
  • #3
I could be wrong but shouldn't the equation be:

m*[df(+) - df(-)]/dz = T*(d^2f/dz^2)


Maybe?
 
  • #4


Originally posted by arcnets
Thorazine, I think this is physically correct. I think df/dz=0 everywhere in the massless part. Could you please explain why it doesn't work?

I know it's not correct because it doesn't give me the right answer, I don't know why it doesn't work. :)
 

1. What is a boundary problem?

A boundary problem is a type of mathematical problem that involves solving equations or systems of equations on the boundary of a given region or surface. In physics, it often refers to finding solutions to equations that describe the behavior of physical systems at the boundaries of their defined domains.

2. What is a massless string?

A massless string is a theoretical concept in physics that refers to a perfectly light and thin string with no mass. In reality, all strings have some mass, but in certain situations, such as in classical mechanics problems, assuming a massless string can simplify the calculations.

3. What is a knot of mass 'm' at z=0?

A knot of mass 'm' at z=0 refers to a physical situation where a string with a knot or mass attached to it is placed at the origin of the z-axis in a three-dimensional Cartesian coordinate system. This is a common scenario in physics problems involving strings and their behavior under different forces or conditions.

4. How is the boundary problem solved for a massless string with a knot of mass 'm' at z=0?

The boundary problem for a massless string with a knot of mass 'm' at z=0 can be solved by applying the appropriate equations and boundary conditions. This may involve using concepts from classical mechanics, such as Newton's laws of motion, and mathematical techniques such as solving differential equations.

5. What are some real-world applications of the boundary problem for a massless string with a knot of mass 'm' at z=0?

The boundary problem for a massless string with a knot of mass 'm' at z=0 has many real-world applications, including in the study of oscillating systems, vibrations in musical instruments, and the behavior of ropes and cables in engineering and construction. It can also have implications in quantum mechanics, where strings are considered fundamental building blocks of the universe.

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