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Boundary Conditions
We have found the two components of the solution, now we can write:
Now we have the solution is, and we can apply some
additional conditions. First, let us set the speed
pf propagation (c) at 10 cm/sec. Since we know that blood flow is
periodic in time and space, we can impose what are known as
periodic boundary conditions: u(x,0) = 2.5 cos (x); u(x,1) = 2.5 cos(x),
and u(0,t) = cos(t), u(10,t) = cos(t). We see that g needs
to have a period of 1 (sec), whereas f has to have period
of 10 (cm). Finally, we are free to add any constant to the solution, because
the wave equation only involves second derivatives, so the constant will not
affect it. Since we would like blood flow to be positive (blood never reverses
direction!), we add 2.5 to the solution. The final solution can be written now:
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