How Do You Model Small Vertical Vibrations of a Homogeneous String?

[jc2009]

Write the BVP for the small vertical vibrations of a homogeneous string. Assume that the wave's speed is c.

Problem: take L = 2[Pi] , c= 2,
SUppose that the right end is fixed while the left end is allowed to move vertically. At the left end, the tangent line at any time t is horizontal. Suppose that the string is set into motion from rest with an initial position given by the function f(x) = sin3x
Write the BVP

Solution:
u(x,0) = sin3x
$$u_{t}(x,0) = 0$$
u(2[pi], t) = 0
u(0,t) = f(x) since the left end is allowed to move vertically ( is this right? )

 

[gtoguy97]

The Boundary Value Problem is: Find u(x,t) such that u(x,0) = sin3x u_{t}(x,0) = 0 u(2[pi],t) = 0 u(0,t) = f(x) Subject to the wave equation u_{tt} = c^2u_{xx}