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The d'Alembert Solution to the Wave Equation with Initial Conditions |
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Consider the wave equation
Let f(x) equal y(x, 0) and g(x) equal ((∂y/∂t)_{t=0}. Then the d'Alembert solution to the wave equation with the given initial conditions (at t=0) is
The partial derivatives with respect to t are
On the other hand
Thus the wave equation is satisfied for all x and t; i.e.,
For t=0
Also for t=0
Thus the initial conditions are satisfied.
Wave equations are thought to involve sinusoidal solutions but the wave equation is just part of an initial values problem. However the d'Alembert solution shows that a sinusoidal profile will be propagated, as would any initial profile, but a sinusoidal solution will not be created when it does not exist under the inital conditions.
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