This is my question:
- Compute the characteristic curves of the following wave equation $$ \frac{\partial^{2} u}{\partial t^{2}}-a^{2} \frac{\partial^{2} u}{\partial x^{2}}=0 $$ and draw them on an $x-t$ coordinate system.
I couldn't figure out what "charachteristic curves" means. I found solutions about wave quation but couldn't found anything about this "characteristic curve". I will appreciate any help.
Thanks for your care, (Stay home and keep dealing with math )
The wave equation has the following general solution (in your notation)
$$u(x,t) = A\sin(kx-\omega t)$$
Here, $k$ would denote the wavenumber in $m^{-1}$, and omega the wave frequency in $rad/s$, such that
$$a = w/k$$
Now, a characteristic for a wave is the plane along which there is a constant deflection wrt both space and time, i.e, it would correspond to the following family of planes
$$x-at = c$$
Where c is a constant. The physical intuition of defining these being how you would define a wavefront. If you study acoustics or any wave propagation, a wave is drawn as it's characteristic planes. This allows you to visualise the geometry of the wave, and is especially useful for boundary conditions and reflection/refraction