Question

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

Answer 1

Answer:

The wave equation is $\frac{d^{2}u }{dt^{2} }$ = $c^{2}$ $\frac{d^{2}u }{dx^{2} }$

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 $\frac{d^{2}u }{dt^{2} }$ = $b^{2}$*( -Acos(ax + bt) - B*sin(ax + bt))

and $c^{2}$ $\frac{d^{2}u }{dx^{2} }$= $ac^{2}$*( -Acos(ax + bt) - B*sin(ax + bt))

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