Put

u = c t - x

and assume

f

has continuous second derivatives. The chain rule yields

y_{x} = - f^{'} (u), y_{xx} = f^{''} (u), y_{t} = c f^{'} (u), y_{tt} = c^{2} f^{''} (u) .

Thus the wave equation is satisfied. A point of fixed wave phase

u = u_{0}

has

x = c t - u_{0}

, so for

c > 0

this travelling wave moves in the

positive x direction at speed c

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