Chain rule of differentials

dwsmith

Well-known member
Take $$U(\eta) = u(x - ct)$$ and the wave equation $$u_{tt} - u_{xx} = \sin(u)$$. Then making the transformation, we have
$(1 - c^2)U_{\eta\eta} = \sin(u).$
My question is the chain rule on the differential.
$U_{\eta} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial\eta} + \frac{\partial u}{\partial t} \frac{\partial t}{\partial\eta}$
but this doesn't seem to work out correctly.

dwsmith

Well-known member
Re: chain rule of differentials

If I take $$\eta = x - ct$$, then
$\partial_x = \partial_{\eta}\frac{\partial\eta}{\partial x} = \partial_{\eta}$
and
$\partial_t = \partial_{\eta}\frac{\partial\eta}{\partial t} = -c\partial_{\eta}$
Therefore, the transformation yields
$u_{\eta\eta}(c^2 - 1) = \sin(u).$
How do am I suppose to get $$1 - c^2$$?