How may I show that any function of the form $z = f(x+at) + g(x+at)$ is a solution of the wave equation. $$ \frac{\partial^2 z}{\partial t^2 } = a^2 \frac{\partial^2 z}{\partial x^2 } $$
How can I solve this question? Please help me to answer.
How to do questions of partial differentiation?
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pde
partial-derivative
wave-equation
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1Do you mean $x = f(x+at) + g(x - at)$? – 2017-02-26
1 Answers
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It can be shown directly like this. Suppose: $$ z = f(x+at) + g(x-at) $$ Then the spatial derivatives are $$ \partial_x z = f'(x+at) + g'(x-at) $$ $$ \partial_{xx} z = f''(x+at) + g''(x-at) $$ Thus we get the following using the temporal derivatives \begin{align*} \partial_t z &= f'(x+at)a - g'(x-at)a \\ \partial_{tt} z &= \partial_t a(f'(x+at) - g'(x-at)) \\ &= a(f''(x+at)a + g''(x-at)a) \\ &= a^2(f''(x+at) + g''(x-at)) \\ &= a^2 \partial_{xx} z \end{align*} Hence $z$ satisfies $$ \partial_{tt} z = a^2 \partial_{xx} z $$ As the comments above note, the canonical solution is the one I've put here. But $ z = f(x+at) + g(x+at) = \tilde{f}(x+at) $ is also a solution by simply taking $g\equiv 0$.