I understand how the wave function is derived. I also understand how to check if a function serves as a solution to the wave equation:
If a function is solution to this wave equation, what does this mean? Is the function a wave? How do I get an intuitive grasp on this? Essentially, when I look at a function, and see that it is a solution to the wave equation, what insights does it give me into what the function represents?
Best Answer
You are confusing mathematics with physics.
In physics wave are an observed effect in many frameworks, i.e. sinusoidal behavior. When searching for a mathematical model the relevant equation must have sinusoidal solutions. This does not mean that all solutions are realized in nature , they may be, and often are. But to call it a wave equation it is only necessary that it has sinusoidal solutions.
A wave is an observed behavior varying sinusoidally, so a non sinusoidal solution cannot be used to describe a physical wave. It might describe exponential decay , which is a different type of a physical observation than a physical wave.