PDE Wave Equation/boundary condition question

  • Thread starter walter9459
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  • #1
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Homework Statement


I need to visualize the wave equation with the following initial conditions:
u(x,0) = -4 + x 4<= x <= 5
6 - x 5 <= x <= 6
0 elsewhere
du/dt(x,0) = 0

subject to the following boundary conditions:
u|x=0 = 0




Homework Equations


I'm not sure I understand the boundary conditions as they written can someone help me?


The Attempt at a Solution


Not sure where to start with the boundary conditions.
 

Answers and Replies

  • #2
LCKurtz
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This represents a semi-infinite string tied down at the origin and given a triangular displacement between 4 and 6. So at time t = 0 the string looks like this:

It is on the x axis from (0,0) to (4,0) then goes on a slope 1 straight line to (5,1), then on a slope -1 back down to (6,0), then out the x axis.

The du/dt(x,0) = 0 statement says the string is released from rest, not given an initial velocity.

Th boundary condition u|x=0 = 0 would be better written as u(0,t) = 0 for t >0 which says the string is tied down at the end.

Have you studied D'Alembert's solution and how to handle semi-infinite strings with it? That's where you need to look.
 
  • #3
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im sorry if im doing wrong by posting in a another thread,but i have a similar problem...
its an infinite string which is straight ( U(x,0)= 0) and has input velocity defined by some arbitrary function g(x)...
from this conditions i have to derive

http://wiki.fizika.org/w/images/math/e/5/6/e567728fec46f565bb53595c63035ec9.png
(sorry,dont know Latex)

where c is (obviously) its velocity...
im having problem with boundary conditions...well,everything tends to cancel out :biggrin:
 
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  • #4
LCKurtz
Science Advisor
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Gold Member
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im sorry if im doing wrong by posting in a another thread,but i have a similar problem...
its an infinite string which is straight ( U(x,0)= 0) and has input velocity defined by some arbitrary function g(x)...
from this conditions i have to derive

http://wiki.fizika.org/w/images/math/e/5/6/e567728fec46f565bb53595c63035ec9.png
(sorry,dont know Latex)

where c is (obviously) its velocity...
im having problem with boundary conditions...well,everything tends to cancel out :biggrin:

If you start with the the 1-dimensional wave equation for the vibrating string:

[tex] u_{tt} = c^2 u_{xx}[/tex]

have you studied the D'Alembert substitution:

[tex] r = x + ct,\, s = x - ct[/tex]

yet? This leads to the equation:

[tex]u_{rs} = 0 [/tex]

which leads to the solution you are seeking. Perhaps if you show what you have done so far I would know where to help you.
 
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