(4-8)  ①  If we modify figure 4-vii to show a wave moving to the left, we get: 

Following the same line of thought as we did before, we can see that after a time t1 the leftward movement of the waveform brings a point that was initially at x+vt1 into line with x.  Hence, the equation for the wave is:

                                        y = f(x + vt) 

②  Following the same steps as we did for the wave moving to the right, we get:

And hence, comparing the partial derivatives we get, this time: