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diff --git a/Problem Set 3.page b/Problem Set 3.page index 0090176..9045fa9 100644 --- a/Problem Set 3.page +++ b/Problem Set 3.page @@ -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 $\frac{\partial f}{\partial \theta}$. |