Fourier… 3

Illustration of Wave equation

Image via Wikipedia

Fourier again.

A few years ago, I was with some students to read a textbook about audio engineering.  I did not like the book in a few ways.  One of the reasons is about treatment of wave equation.

1D wave equation looks something like this:

\displaystyle \dfrac{\partial^2 u}{\partial t^2} = c^2 \dfrac{\partial^2 u}{\partial x^2}.

After presenting the equation, the book immediately declares its solution is given by

\displaystyle u(x, t) = f(x -ct) + g(x + ct),

for arbitrary function f and g, without any derivation, explanation no nothing.  The solution comes down like an oracle.  I still don’t like the way it is given.

This solution is usually called “d’Alembert’s Solution“.  And in case his name chimes in, we can see a slightly better derivation.

The derivation goes like following.  Introducing new variables

\displaystyle \xi = x -ct,

\displaystyle \eta = x + ct,

the PDE becomes

\displaystyle \dfrac{\partial^2 u}{\partial \xi \partial \eta} = 0.

Integrating this with respect to \xi gives

\displaystyle \dfrac{\partial u}{\partial \eta} = G(\eta),

(G: arbitrary function of \eta, independent of \xi, since differentiating G with respect to \xi gives 0).

Then integrating above with respect to \eta gives

\displaystyle u = f(\xi) + g(\eta),

(f: arbitrary function of \xi, and g(\eta) = \int G(\eta) d \eta).  By substituting the variables back, we reach the solution

\displaystyle u(x, t) = f(x -ct) + g(x + ct).

Okay.  There is a bit of explanation above.  It is better, only very slightly though, than nothing.  But still \xi and \eta come down out of blue sky.  It is unclear how this change of variables invented.

I recently saw Fourier comes for rescue, which is like


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3 Responses to Fourier… 3

  1. Pingback: Fourier Series: before I begin… | cartesian product

  2. worldofaircraftdesign says:

    I understand you wonder about even first step of introducing those variables…where are they from? Those questions puzzled me most of the time during my engineering studies. Here is my answer. It’s either intuition, art, or try and error by the person who was solving it. Why is it so? Look, there can be also a question where the differential equation comes from at the first place. The answer, more or less, is directly from ZFC axioms. In order to define PDE you do not need any physical process. In math there are assumed truths, starting points, which are the ZFC axioms. You have to start with something! Similar when you try to give solution to PDE you have to guess a function! That’s your starting point. There is nothing else there. It either works or not. If it works then, good, let’s remember and USE the solution. If it does not, then, have to try again with different functions or steps.

    It is similar with proofs. I am sure you remember those initial steps for more complex proofs. Where those starting steps come from? The answer is there is no derivation for them. Those first steps in a proof are guessed, tested, obtained by intuition, etc. Proofs can not be constructed from straightforward derivations from somewhere.

  3. Pingback: Fourier… 4 | 0909

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