How do we justify the generality of the wave equation?

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Discussion Overview

The discussion centers around the generality of the wave equation, particularly in relation to its derivation from the principles of mechanics applied to an oscillating string. Participants explore the conditions under which the wave equation is valid and question its applicability to waves of larger amplitudes.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant notes that the derivation of the wave equation relies on the small angle approximation, which limits its applicability to waves of small amplitude relative to wavelength.
  • Another participant argues that the wave equation has proven to be effective in modeling various wave-like phenomena, regardless of the specific derivation process.
  • A different viewpoint suggests that the linearity assumption inherent in the small angle approximation restricts the wave equation's applicability to linear media, indicating that while the wave equation is generally applicable, there are limits to its use in non-linear contexts.
  • One participant introduces the idea that pulses can propagate on a string at a specific velocity without needing to be described as functions, referencing a proof by P.G. Tait that does not rely on calculus.

Areas of Agreement / Disagreement

Participants express differing views on the generality of the wave equation, with some emphasizing its limitations due to the assumptions made during its derivation, while others highlight its successful application across various physical phenomena. The discussion remains unresolved regarding the extent of its generality.

Contextual Notes

The discussion highlights limitations related to the assumptions made in the derivation of the wave equation, particularly concerning the small angle approximation and linearity. There is also mention of the need to identify when the wave equation ceases to be applicable.

Sturk200
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We derived the wave equation by applying f=ma to an element of an oscillating string, yielding

∂2y/∂x2 = 1/v2 ⋅ ∂2y/∂t2.

In order to get this result it was necessary to assume that the string in question was nearly horizontal, so that the angle formed by the tension vector and the horizontal axis satisfied the small angle approximation sinα=α. It follows that the equation applies only to waves of very small amplitude relative to wavelength.

However, it is claimed, by my textbook for instance, that this equation is a general result that describes all wave phenomena. How can the derivation be generalized to waves with larger amplitudes if it requires the assumption that the string is nearly horizontal?
 
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At the end of the day, it just happen that the wave equation has solutions that model extremely well a host of physical phenomena (those phenomenon that we call "wave-like" :-) ) no matter how or what "inspired" us to find it (to find the wave equations).

Just like Poisson Equation is extremely useful to Gravitation and Electrostatics.

Sometimes we discover these equations by means of studying basic examples, sometimes deriving them from other assumptions,...

But at the end of the day, what is important is that they work, they are useful (they represent correctly certain features of certain physical phenomena).
 
The sinα=α approximation is essentially a linearity assumption. Which really means, the formula you arrive at will hold for all media that are linear. Which is a constraint that a lot of media fulfill (e.g. EM field). It's really just that physical strings stop being linear at some point. So, the wave equation is very general indeed, but you have to know at which point it is no longer applicable.
 
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