Thus the wave equation is satisfied for all x and t; i.e.,

(∂²y/∂t²) −C²(∂²y/∂x²) = 0

For t=0

y(x, 0) = ½[f(x) + f(x)] + (1/2C)∫_{x}^{x}g(z)dz
which reduces to
y(x,0) = f(x)

Also for t=0

∂y/∂t)_{t=0} = ½[f'(x)C − f'(x)C] + (1/2C)[g(x)C + g(x)C]
which reduces to
∂y/∂t)_{t=0} = g(x)

Thus the initial conditions are satisfied.

The Wave Equation and Sinusoidal Solutions

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.