Reflection at a Free End of the String

At , if a string is free there is no transverse force.

Net transverse force is proportional to the slope $\partial y/\partial x$ :

$\displaystyle \frac{\partial y}{\partial x}$	$\displaystyle =$	$\displaystyle \frac{\partial}{\partial x}y_r(t-x/c)+ \frac{\partial}{\partial x} y_l(t+x/c)$
	$\displaystyle =$	$\displaystyle \frac{1}{c}\left[y_l(t+0/c)- y_r(t-0/c)\right]$
	$\displaystyle =$	0
$\displaystyle y_r(t)$	$\displaystyle =$	$\displaystyle y_l(t).$

$\fbox{ \parbox{5in}{The reflected wave is equal to the incident wave with no change of sign.}}$

Reflection from a free boundary:

**Figure:** Damped Vibration, animation courtesy of Dr. Dan Russell, Penn State University.

``Music 206: Digital Waveguides'' by Tamara Smyth, Department of Music, University of California, San Diego (UCSD).