Question

Basic problem necessary for future understanding of Schrodinger's Equation:


A wave has the form y=Acos(2πx/λ+π/3)when x<0.

For x > 0, the wavelength is λ/2. By applying continuity conditions at x = 0, find the amplitude (in terms of A) and phase of the wave in the region x > 0. Sketch the wave,

showing both x < 0 and x>0.

Answer 1

This is much more a mathematical problem:

Create a condition such that:
\[
y(x)=
\begin{cases}
A\cos\left(\frac{2\pi x}{\lambda}+\frac{\pi}{3}\right)\text{ when } x<0\\
A\cos\left(\frac{\pi x}{\lambda}+\frac{\pi}{3}\right)\text{ when } x>0
\end{cases}
\]Is continuous at \(x=0\)

Answer 2

And find the amplitude, etc, etc.

Although, I'm not 100% sure of the phase, but I believe the above to be the case.

Answer 3

Be mindful of arithmetic ^_^


\[y(x)= \left\{ \begin{array}{l}
A\cos\Big(\frac{2 \pi x}{\lambda} + \frac{\pi}{3}\Big) \text{ when} \ x<0
\\ A \cos \Big( \frac{4 \pi x}{\lambda} + \delta\Big) \text{ when} \ x>0
\end{array} \right.\]