Suppose that the function W(x, t) is defined by W(x, t) = f (x + ct), where f (s) is any function (of one variable s, where s = x +ct)

f can’t actually be any function at all, it has to be reasonably nice, but we won’t worry about that detail for now. and c is a constant.

a. Show that This partial differential equation is (a part of) the famous wave equation, which is important for a wide range of applications.

∂W

∂t

= c

∂W

∂x

.

b. To see how this looks in practice, let’s choose a specific function for f . Let f (s) =

s 2

1 + s 2

and plot f (x + ct) as a function of x for various fixed t, and for c = 1. What do you notice about the solution?

You’ve just shown that a solution to the wave equation Answer: you should notice that the function f moves along the x axis without changing its shape. is a shape that moves along the horizontal axis without changing shape. This is called a travelling wave.

c. Now redo the plots for c = 2. What do you notice has changed? What does c control?