I know that the one-dimensional wave equation can be written as $$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial ^2 u}{\partial x^2}$$ and has solutions of the form $$ u = F(x+ct) + G(x-ct)$$

I'm having trouble developing a proper intuition about the meaning of the solution, though. I superficially understand that it's the sum of two functions "travelling" in different directions with time, but that doesn't help me be able to really visualize what solutions look like. More specifically, I'd like to develop an intuitive or visual understanding of what solutions to the wave equation have in common, and what separates them from functions that aren't solutions.

  • $\begingroup$ For starters, do you feel you intuitively understand the cases $F = 0$ or $G = 0$? $\endgroup$ – Andrew D. Hwang Jun 5 '16 at 0:35
  • $\begingroup$ A good place to continue is to consider the IVP with $u(0,x)=f(x),u_t(0,x)=g(x)$ and rewrite the solution in terms of $f$ and $g$. This is not very hard to do (some basic linear algebra, now that you have the form of the general solution) but it is helpful. $\endgroup$ – Ian Jun 5 '16 at 0:35
  • $\begingroup$ @AndrewD.Hwang , not really. $\endgroup$ – SquarerootSquirrel Jun 5 '16 at 0:41
  • $\begingroup$ The case $F=0$ or $G=0$ gives solutions to the transport equation. These are easy to visualize because there is no "interaction" going on. $\endgroup$ – Ian Jun 5 '16 at 0:49
  • $\begingroup$ Am I correct in thinking that for F = 0, the solution is literally any function whose waveform moves to the right at c x-units per t-unit? So, a constant function (say, u = 1) would still count as a wave? $\endgroup$ – SquarerootSquirrel Jun 5 '16 at 0:55

In the hope an animation is worth a thousand words: The blue wave $F$ travels left, the red wave $G$ travels right. Their sum is magenta. Both $F$ and $G$ were chosen to be spatially periodic so the loop would be smooth.

Linear superposition of wave moving at constant speed

  • $\begingroup$ What did you use to create this animation? $\endgroup$ – mathematician Jun 5 '16 at 1:59
  • $\begingroup$ This works really well, thanks! Is there a similar visual example of a function you could give that depends on both x and t, but does not satisfy the wave equation? $\endgroup$ – SquarerootSquirrel Jun 5 '16 at 3:01
  • $\begingroup$ @mathematician: ePiX. Squarerootsquirrel: "Anything not of this form," such as a single graph moving to the right but whose speed varies ($u = G(x - v(t))$ for some differentiable function $v$ with $v'$ non-constant), or $u = \sin(xt)$, or.... $\endgroup$ – Andrew D. Hwang Jun 5 '16 at 13:41

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