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Question

# Write the d’Alembert solution for the problem $u_{t t}=c^{2} u_{x x}$ for $-\infty0$ and $u(x, 0)=f(x), u_{t}(x, 0)=g(x)$ for $-\infty. $c=1, f(x)=x^{2}, g(x)=-x$

Solution

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We are solving the equation (wave equation with $c^2=1$)

$\dfrac{\partial^2 u}{\partial t^2}= \dfrac{\partial^2 u}{\partial x^2}$

usin the d'Alambert method. We employ the substitution

$\xi=x-t$ \ and \ $\eta=x+t$

expressing $x$ and $t$ in terms of $\xi$ and $\eta$ that is

$x=\dfrac{1}{2}(\eta+\xi)$ \ and \ $t=\dfrac{1}{2}(\eta-\xi)$

We use this to express the derivative in terms of $\xi$ and $\eta$ to get (after some arduous computation)

$\dfrac{\partial^2 u}{\partial x^2}=\dfrac{\partial^2 u}{\partial \xi^2}+2\dfrac{\partial^2 u}{\partial \xi \partial \eta}+\dfrac{\partial^2 u}{\partial \eta^2}$

and

$\dfrac{\partial^2 u}{\partial t^2}= \dfrac{\partial^2 u}{\partial \xi^2}-2\dfrac{\partial^2 u}{\partial \xi \partial \eta}+\dfrac{\partial^2 u}{\partial \eta^2}$

Insertin this into our initial equation we get

$\dfrac{\partial^2 u}{\partial \xi^2}-2\dfrac{\partial^2 u}{\partial \xi \partial \eta}+\dfrac{\partial^2 u}{\partial \eta^2}=\dfrac{\partial^2 u}{\partial \xi^2}+2\dfrac{\partial^2 u}{\partial \xi \partial \eta}+\dfrac{\partial^2 u}{\partial \eta^2}$

we cancel (what's the same) from the left and right

$\dfrac{\partial^2 u}{\partial \xi \partial \eta}=0$

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