Boundary Conditions for Waves in Joined Strings with a Knot of Mass

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Homework Help Overview

The discussion revolves around boundary conditions for waves in two strings joined by a knot of mass. The participants are exploring the implications of tension and continuity at the boundary where the strings meet.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the application of Newton's laws to the knot and the need for continuity in the strings. There are attempts to formulate the conditions mathematically, particularly concerning the derivatives of the wave functions at the boundary.

Discussion Status

The conversation is ongoing, with some participants providing guidance on how to approach the formulation of the boundary conditions. There is recognition of the complexity introduced by the mass of the knot, and multiple interpretations of the mathematical relationships are being explored.

Contextual Notes

Participants are grappling with the specifics of the mathematical formulation, including the role of the knot's mass and the assumptions regarding angles and tensions in the strings.

mathlete
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I'm given the fact that two strings under tension T are joined by a knot of mass m... I'm supposed to find the appropriate boundary conditions. I know that the tensions are the same in both ropes and that the boundary will be continuous. I know the "trick" in this problem is knowing the derivatives don't equal because of the knot of mass m, but I don't really know how to formulate it. Can anyone provide a nudge in the right direction?
 
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You can make progress by making of use of Newton's laws for the massive knot while remembering that the string is continuous. In other words, the boundary conditions have to insure that the massive knot, which is a point but with non-infinitesimal mass, stays with the string.
 
I understand that part, but I'm not sure how to formulate it... I get df/dz (from the left) - df/dz (from the right) = m/T but I don't think that's right.
 
mathlete said:
I understand that part, but I'm not sure how to formulate it... I get df/dz (from the left) - df/dz (from the right) = m/T but I don't think that's right.
Close.
1. Should =-mg/T.
2. You need sin\theta. (Unless you can make a small angle approx.)df/dz is tan\theta.
 
Last edited:

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