(4-10)  ①  we start with the general equation we derived for a wave travelling to the right:

Since  y = f(x – vt)  we make the same substitution we did before, q = x-vt,  to get

Since we already know that  ∂y/∂t  =  -v(∂y/∂x), we can substitute this into (b)  to get:

Comparing (a) and (c), we now get:

②  this time we start our general wave equation for a left-travelling wave:

Following the same approach as we did in ①, we recognise that a left-travelling will be of the form y = f(x + vt) and hence make the substitution q = x + vt to obtain:

Comparing (a) and (c), we again get:

By going to the second derivative, we now get this same result for both left- and right-travelling waves. So this looks to be the general wave equation we have been seeking.