Question

Show that a sinusoidal wave propagating to the left along x-axis is a solution of the differential wave equation.

Answer 1

The wave equation is
`(d^(2)u )/(dt^(2) )` =
`c^(2)`
`(d^(2)u )/(dx^(2) )`

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
`(d^(2)u )/(dt^(2) )` =
`b^(2)`*( -Acos(ax + bt) - B*sin(ax + bt))

and
`c^(2)`
`(d^(2)u )/(dx^(2) )`=
`ac^(2)`*( -Acos(ax + bt) - B*sin(ax + bt))

then if a*c = b, this is a solution of the wave equation.