How would I calculate the intersection of a plane wave surface and a curve?
Note that I am asking about a plane wave surface intersecting a curve in a plane, not a simple sin wave equation intersecting another curve.
Thank you in advance.
Let the sinusoidal plane wave have a node at the Y axis and the amplitude changes in the Z axis with the zero nodes of the wave at the X Y plane. Looking at the X Y plane, the wave would travel in the X direction with wavelength $L$. I want to be able to look at the intersection points with the curve at different times, essentially freezing the wave at $T=0$ or $T=1$ for example. The curve would be in the X Y plane only. I'm looking for the points where the wave intersects the curve, and necessarily where the intersections are at the wave zero points (meaning when the frozen wave intersects the curve and is necessarily zero at the X Y plane). The specific curve is arbitrary but we could use a parabola: $y = x^2$
The plane wave equation would be some form of this: $\psi_1(x, t) = e^{i(k_1x - \omega_1t)}$
Or this: http://hyperphysics.phy-astr.gsu.edu/hbase/Waves/imgwav/weq.gif
