Let’s repeat the scalar wave equation from the last section:

I’m going to define a new symbol: , which admittedly seems pretty arbitrary, but it’s my book and I can make up symbols if I want. It will make sense later. So if I use my new symbol in the scalar wave equation it becomes

We’re going to solve this equation using the time-honored technique of just writing down the answer and then proving that it works. The solution is any twice-differentiable function , as long as and are grouped together as follows:

What I mean by ‘twice differentiable’ is that I can take the derivative of twice without getting into trouble with infinities, so in principle things like step functions are out. Other than that restriction, can be any function, so we could have (for example)

or

amongst many other choices. Now, I’ll prove that this solution actually satisfies the scalar wave equation, by substituting it into the scalar wave equation. The derivative of our function is , and the second derivative of is . In our case , so we can see that

and, by ye olde chain rule,

similarly

and

So we’re pretty much done proving this solution, because we can see that

so

Hopefully you can see that there is another possible solution to the wave equation; namely,

In the next section we’ll explore the properties of these solutions.