Answer:
The wave equation is [tex]\frac{d^{2}u }{dt^{2} }[/tex] = [tex]c^{2}[/tex] [tex]\frac{d^{2}u }{dx^{2} }[/tex]
a sinusoidal wave can be u = Acos( ax + bt) + B*sin(ax + bt)
where A, a, B and b are real constants. (here you also can add a phase to the arguments of the sin and cosine)
then [tex]\frac{d^{2}u }{dt^{2} }[/tex] = [tex]b^{2}[/tex]*( -Acos(ax + bt) - B*sin(ax + bt))
and [tex]c^{2}[/tex] [tex]\frac{d^{2}u }{dx^{2} }[/tex]= [tex]ac^{2}[/tex]*( -Acos(ax + bt) - B*sin(ax + bt))
then if a*c = b, this is a solution of the wave equation.
