(4-9)  We’ll take a simple example of a superposition of two waves with different wavelengths, travelling in opposite directions (as shown by the opposite signs of the vt term):

                                                                  y  = sin(x-vt)  +  sin(2(x+vt))

This function is a wave , as you can see in the graphs below:  as the time increases, the waveform is translated progressively to the right but it retains its form: