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I have some questions concerning the wave equation:

$${\partial^2 y \over \partial x^2} = {1\over c^2}{\partial^2 y \over \partial t^2}$$

Firstly, does the method of separation of variables give all the solutions? I presume not, since we will only get solutions of the form $y(x,t)=X(x)T(t)$ and I don't think all other solutions can be formed from the superposition of these types of waves.

Secondly, on the topic of superposition. Let's say I have a finite string length $L$ and I put an arbitrary wave on the string. If I allow this wave to take any form, can we form that wave from the superposition of stationary waves only or do we need travelling waves? (i.e. is there a wave that cannot be formed only from stationary waves on a finite (or infinite for that matter) string?

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  • $\begingroup$ Since there are many "generalized" wave equations, please write down the specific one you are talking about. $\endgroup$
    – ACuriousMind ♦
    Mar 20, 2015 at 18:39
  • $\begingroup$ @ACuriousMind $\frac{\partial^2 y}{\partial t^2}=c^2\frac{\partial^2 y}{\partial x^2}$ $\endgroup$
    – Quantum spaghettification
    Mar 20, 2015 at 18:45
  • $\begingroup$ a function $f(x-ct)$ should also solve it... $\endgroup$
    – danimal
    Mar 20, 2015 at 18:54
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    $\begingroup$ I think Wikipedia pretty much answers your question. $\endgroup$
    – ACuriousMind ♦
    Mar 20, 2015 at 18:57
  • $\begingroup$ @ACuriousMind the first part, yes, thanks for pointing this out. But I don't think it answers the second part. $\endgroup$
    – Quantum spaghettification
    Mar 20, 2015 at 19:02

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The answer to the first part of my question is on wiki, so I will not answer that here.

After some research I have come up with the following answer to the second part of my questions. There are 4 types of waves we need to consider:

  1. Travelling waves (also known as progressive waves). (TW)
  2. Compound waves (CW)
  3. Stationary waves (SW)
  4. Infinite (harmonic) plane waves. (IPW)

The IPW are basically the 'normal modes' of the infinite string. They are the only type that can be superimposed to form any TW. IPW can also be superimposed to form the other two types of wave (SW and CW) (in a given region where a finite string lies). SW's can be superimposed to form CW only but not TW. All of this information can be summarised on the following diagram.

enter image description here Where an arrow pointing from wave A to wave B indicates the wave B can be made from the superposition of wave A.

Sources:

  1. http://encyclopedia2.thefreedictionary.com/compound+wave
  2. https://books.google.co.uk/books?id=Geqi-oR8lR4C&pg=PA95&dq=superposition+of+standing+waves&hl=en&sa=X&ei=WfEPVdCDIq7B7AbpvYGADA&ved=0CCcQ6AEwAQ#v=onepage&q=superposition%20of%20standing%20waves&f=false

3.http://www.colorado.edu/physics/EducationIssues/baily/courses/BailySP12_GroupVelocity.pdf

  1. Vibrations and Waves by A.P. French pg 205
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