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authorluccul <luccul@gmail.com>2010-07-06 04:54:05 +0000
committerbnewbold <bnewbold@adelie.robocracy.org>2010-07-06 04:54:05 +0000
commit2c81d19e7558f7ab6f007bbc80223db67e6d7864 (patch)
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downloadafterklein-wiki-2c81d19e7558f7ab6f007bbc80223db67e6d7864.tar.gz
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@@ -23,7 +23,7 @@ $$ \tau \frac{\partial u}{\partial t} - \lambda^2 \frac{\partial^2 u}{\partial x
Use Fourier series to solve this equation in the case of a circular wire. How does the solution depend on the magnitudes of the positive constants $\kappa$ and $\tau$?
6. The wave equation is a partial differential equation that models the propogation of disturbances in a medium (for example, the vibrations of a metal object that has been struck by a hammer). In the case of a one-dimensional object it is given by:
-$$ \frac{\partial^2 u}{\partial t^2} = \frac{\partial^2 u}{\partial x^2} $$
+$$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} $$
Use Fourier series to solve the wave equation in the case of a vibrating ring. Interpret the solution as a superposition of two waves travelling with a certain velocity around the ring (but in opposite directions). At what velocity do they travel?
7. Write the Cauchy-Riemann equations in polar coordinates, i.e. express them as a relationship between $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial \theta}$.