Does the wave equation only apply to waves with low frequency?

[skatefast08]

Wouldn't the wave equation apply to only waves at low frequency?

I read in this article: https://www.animations.physics.unsw.edu.au/jw/wave_equation_speed.htm, that the equation for a wave that was analyzed as a PDE, had to be a wave with only small deformation, meaning that it would have to be a wave that has only low frequency, correct? because, if the wave had a very high frequency, the same wave equation in the article wouldn't conform, because some of the angles from an infinitley small portion of that wave would be too large and hence the wave equation wouldn't work, right? I hope you understand my question. I just wanted to understand this in a more intuitive way. Thanks.

 

[wwfeldman]

I believe this is the line that led to your question:

"Our analysis only applies for small deformations, for which the string is a linear medium, and we neglect the gravitational force on the string (which in any case is constant). "

note " ...small deformations for which the string is a linear medium..." i.e. the string is (conceptually) a spring that is not pushed past its elastic limit

it is the amplitude, not the frequency, that has to be small so you do not pass the elastic limit.

 

[c_mitra]

the equation for a wave that was analyzed as a PDE, had to be a wave with only small deformation, meaning that it would have to be a wave that has only low frequency, correct?

Unfortunately, no.

In the specific example you cited (vibrations of a string), the deformation refers to the amplitude. But the question will arise naturally, what is small anyway?

As long as the amplitude is small, the vibration is simple harmonic and the result will be a single frequency. As the amplitude gets larger, restoring force will no longer be proportional to the displacement and there will harmonics (why only harmonics??) present.

The deformation is small as long as you can accept the Hooke's law (displacement and force are proportional).

Sometimes we consider a distortion to be small if we can treat and account for it as a perturbation (but some people use terms like large perturbations too!).

Molecules are made of atoms joined together with chemical bonds. These chemical bonds act like springs and each individual bonds can (and do) vibrate with very high frequency. When the vibration amplitude becomes large, the molecule breaks down.