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

In summary, the problem involves finding the appropriate boundary conditions for two strings under tension T that are joined by a knot of mass m. The tensions in both ropes are the same and the boundary must ensure that the knot stays with the string. Newton's laws can be used for the massive knot while considering the string's continuity. One possible formulation is df/dz (from the left) - df/dz (from the right) = m/T, but this may not be accurate. Another factor to consider is the angle of the knot, which can be represented by sin\theta and df/dz can be represented by tan\theta.
  • #1
mathlete
151
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
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FAQ: Boundary Conditions for Waves in Joined Strings with a Knot of Mass

What are waves and how are they created?

Waves are disturbances that propagate through a medium, carrying energy from one place to another. They can be created by a variety of sources such as wind, earthquakes, or vibrations.

What are boundary conditions and why are they important in wave behavior?

Boundary conditions refer to the conditions at the interface between two media where a wave is propagating. They are important because they determine how the wave will behave when it encounters a change in the medium, such as a change in density or speed.

What is the difference between transverse and longitudinal waves?

Transverse waves are characterized by the displacement of particles perpendicular to the direction of wave propagation, while longitudinal waves have particles that move parallel to the direction of wave propagation. Examples of transverse waves include light and water waves, while sound waves are examples of longitudinal waves.

How do waves interact with boundaries?

When a wave encounters a boundary between two media, it can be reflected, transmitted, or absorbed depending on the properties of the media and the angle of incidence. The law of reflection and Snell's law are commonly used to describe the behavior of waves at boundaries.

What is the principle of superposition and how does it apply to waves?

The principle of superposition states that when two or more waves meet, the resulting displacement at any point is equal to the sum of the displacements of the individual waves. This applies to both constructive interference, where waves combine to form a larger wave, and destructive interference, where waves cancel each other out.

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