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Notice that the wave equation's partial derivatives below are factored into first order PDE operators like a second degree polynomial. Then, these PDE operators are applied one at a time to get the wave equation for u.
Wave Propagation Free Space: Boundary Conditions Germany residence permit application asks to bring Aufenthaltstitel Analyses of Associations and Predictive Models in Random Forest
Consider the following wave equation: $$\\begin{align} u_{tt}&=u_{xx},\\quad x\\in(0,\\pi),\\quad t>0,\\\\ u(x,0)&=0,\\quad u_t(x,0)=0,\\\\ \\color{red}{u ...
But hence they are just a construction out of the general solution mathematician agreed on just considering the general solution - or in this case the fundamental set of solutions - as the one solution to the differential equation so that they have not to write down a list with about $10$ entries every single time they solve an equation.
The wave equation with the sign flipped can be viewed as the wave equation with imaginary velocity, or equivalently in imaginary time. $\endgroup$ – Ian Commented Jun 3, 2015 at 16:23
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$\begingroup$ @pluton, to be precise, the boundary condition you are considering, strictly speaking, is not the standard Robin condition: this has the following form $$\left[\partial_x u(x,t)+\alpha u(x,t)\right]_{x=0}=0 \iff \partial_x u(0,t)=\color{red}{-\alpha} u(0,t).$$ In this form, the condition $\alpha>0$ implies existence and uniqueness of a solution for the associated equation.
$\begingroup$ Check the laplace method for solving the wave equation. It is well-suited for problems with different boundary conditions and more robust than the d'alembert solution $\endgroup$ – user32882
Limitations of D'Alembert solution of wave equation. 1. d'Alembert solution to wave equation. 2.
About the eigenvalues, in all the exercises I've done, $\lambda_n$ is computed in the spatial equation and then substituted into the temporal one. $\endgroup$ – Serge Commented Aug 3, 2012 at 19:07