How do we justify the generality of the wave equation?

[Sturk200]

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?

 

[mattt]

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).

 

[rumborak]

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 So, the wave equation is very general indeed, but you have to know at which point it is no longer applicable.

 

[PietKuip]

The interesting thing is that any kind of pulse will propagate on a string with velocity \sqrt(Tension/density). It does not need to be a function. The proof by P.G. Tait is interesting to read. It does not use calculus: http://www.archive.org/stream/encyclopediabrit15newyrich#page/740/mode/2up