Reflections of a wave pulse

A wave pulse with the form,

$\large y= A_i\exp(-(x-c_1 t+10)^2)$,

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 can be achieved by making the mass per unit length of the string larger. A darker line indicates a slower speed. 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. Otherwise the reflected wave is upright.

A fixed end can be approximated by maximizing the ratio $c_1/c_2$ . A free end can be approximated by minimizing the ratio $c_1/c_2$ . If $c_1=c_2$, there is no reflected wave.

$y$

$x$

Ai = 1 [m]

c1 = 3 [m/s]

c2 = 7 [m/s]

$t=$

The amplitudes of the reflected pulse and the transmitted pulse are,

$$A_r = A_i\frac{c_2-c_1}{c_1+c_2},$$ $$A_t = A_i\frac{2c_2}{c_1+c_2}.$$

For the string to remain continuous, $A_i+A_r = A_t$.