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# Discussing the physical behaviour of PDE's Watch

1. Hi All

I have to discuss the physical behaviour of a partial differential system:
The system is defined by

d^2v/dx^2=1/2d^2v/dt^2.

over 0<x<2, t>0 for the boundary conditions

v(0,t)=v(2,t)=0

and the initial condition

v(x,0)=sin(pi*x)-3sin(2*pi*x)

dv/dt(x,0)=0

I have no idea what to do when it asks me to describe the physical behaviour, of the Partial Differential System shown.
If needs be I have solved the system, but the question states describe the system given, not the solution.

Can anybody help?
2. The equation you've solved is the one-dimensional wave equation, which governs how a wave changes in time in, for example, a vibrating string.

The initial condition tells you the initial shape of the wave. The boundary condition can be interpreted physically - have a good think about what it would mean for a string, and why it would mean that the string is fixed at both ends.

What I've got is the following:

"The system is linear and homogenous for both boundary conditions. The equation we have solved is the one-dimensional wave equation, it governs how a wave changes over time. The initial conditions show us that the wave is initially shaped as the curve defined by sin (πx) -3sin (2πx). The boundary conditions show that for time greater than zero the value of x is between 0 and 2, we can see that the wave is 0 when x is 2 and 0, showing us that this is when the wave repeats"

Is that along the right lines?
I think that the reason the end points are fixed is because v(x,0)=v(x,2)=0
that's what i was trying to mention at the end of the paragraph, but i'm not sure if what i've written is true, does that mean it repeats? or does it move backwards on itself? or does it just end?
4. (Original post by Hazbuk)

What I've got is the following:

"The system is linear and homogenous for both boundary conditions. The equation we have solved is the one-dimensional wave equation, it governs how a wave changes over time. The initial conditions show us that the wave is initially shaped as the curve defined by sin (πx) -3sin (2πx). The boundary conditions show that for time greater than zero the value of x is between 0 and 2, we can see that the wave is 0 when x is 2 and 0, showing us that this is when the wave repeats"

Is that along the right lines?
I think that the reason the end points are fixed is because v(x,0)=v(x,2)=0
that's what i was trying to mention at the end of the paragraph, but i'm not sure if what i've written is true, does that mean it repeats? or does it move backwards on itself? or does it just end?
Good, looks like you've understood pretty well.

The boundary condition probably would be interpreted as fixed end-points, such as a string being held still at both ends.

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Updated: April 16, 2013
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