What are the key differences between linear and non-linear waves?

[quid]

Could someone please tell me the fundamental differences between linear and non-linear waves?

 

[toombs]

Usually, linear means if f(x, t) and g(x, t) are both solutions, then so is Af + g. I think though sometimes in the context of wave mechanics and optics "linear" just means non-dispersive. Can I get some backup on that? Non-dispersive meaning the wave speed is not a function of wavelength and phase speed equals group speed. It is possible to have a linear equation that produces dispersive waves. Example: the free particle Schrödinger equation: $-\frac{\hbar^2}{2m} \psi_{xx} = i\hbar \psi_t$

 

[quid]

Right, Thanks toombs

 

[Bobbywhy]

Here is a comprehensive discussion of both linear and nonlinear waves:

http://www.scholarpedia.org/article/Linear_and_nonlinear_waves

 

[pmacias]

Linear waves are those that follow the principle of superposition, meaning that the overall wave is the sum of its individual parts. This means that the amplitude, frequency, and wavelength of the wave remain constant as it propagates through a medium. Examples of linear waves include sound waves, water waves, and electromagnetic waves.

On the other hand, non-linear waves do not follow the principle of superposition and can exhibit behaviors such as changing amplitude, frequency, and wavelength as they propagate. These waves are often more complex and can arise in systems with non-linearities, such as in fluid dynamics or plasma physics. Non-linear waves can also exhibit phenomena such as dispersion, where different frequencies travel at different speeds, and solitons, where the wave maintains its shape as it propagates.

In summary, the fundamental difference between linear and non-linear waves lies in their behavior as they propagate through a medium. Linear waves maintain their properties, while non-linear waves can exhibit changes and complex behaviors. Understanding the differences between these types of waves is crucial in various fields, such as acoustics, optics, and geophysics.