Vibrating String: Interpreting ∂u/∂x(L,t)=0 Boundary Condition

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Homework Help Overview

The discussion revolves around the physical interpretation of the boundary condition ∂u/∂x(L,t)=0 in the context of a vibrating string described by the wave equation. Participants are exploring the implications of this condition within the framework of the problem.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to understand the meaning of the boundary condition, with some suggesting it indicates the string's behavior at the endpoint L. There are differing interpretations regarding whether the string is clamped or allowed to slide.

Discussion Status

The discussion is ongoing, with various interpretations being explored. Some participants have offered insights into the physical meaning of the boundary condition, while others are questioning these interpretations. No consensus has been reached yet.

Contextual Notes

Participants are considering the implications of derivatives in the context of the problem and how they relate to the physical setup of the vibrating string.

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Homework Statement



(∂^2 u)/(∂t^2 )=a^2 (∂^2 u)/(∂x^2 ) x∈(0,l) t>0
u(0,t)=0 ; ∂u/∂x(L,t)=0
u(x,o)=u1; ∂u/∂t(x,o)=x

I can not figure out physical interpretation of boundary condition ∂u/∂x(L,t)=0, what does it mean. Can someone can help me with this ?

Homework Equations





The Attempt at a Solution


 
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The string is clamped at L and 0. So its amplitude is 0 there.
 
That's not true. ∂u/∂x(L,t)=0 means that the string is always perpendicular to the wall at the other endpoint. So you can think a very relaxed string being attached to a pole, sliding along it without friction.
 
Thanks.
 
One way to understand what is it each time, is:
1. Know what derivatives in your problem mean.
2. See how it affects your solution after you bring them in it.
 

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