Reflections of a harmonic wave

A wave harmonic wave,

$ y= A_i\sin (x-c_1 t+10)$,

travels down a string. For $x <0$ the wave speed is $c_1$ and for $x > 0$ the wave speed is $c_2$. A slower wave speed is indicated by a darker blue line. When the pulse reaches $x=0$ where the wave speed changes, part of the wave is reflected and part of it is transmitted. If $c_1 > c_2$, the reflected wave is inverted. The incident and reflected waves add destructively a $x=0$. If $c_1 < c_2$, the reflected wave is upright and the incident and reflected waves add constructively at $x=0$.

A fixed end can be approximated by maximizing the ratio $c_1/c_2$ . The interference of the right-moving incident wave with the left-moving refelcted wave forms a standing wave with a node at $x=0$. The amplitude of the standing wave has twice the amplitude of the incident. A free end can be approximated by minimizing the ratio $c_1/c_2$ . The interference of the right-moving incident wave with the left-moving reflected wave forms a standing wave with an antinode at $x=0$. If $c_1=c_2$, there is no reflected wave.

The amplitude of the incident wave is $A_i$, the amplitude of the reflected wave is $A_r$, and the amplitude of the transmitted wave is $A_t$.