The waves travel 'through' each other, entering the picture on one side and exiting on the other. When both travelling waves are acting on the string, standing waves are formed, which move up and down in the same position along the string's length.

This animation shows waves of just one wavelength. A vibrating piano string simultaneously carries waves of many different wavelengths.

At the beginning of the clip there are no standing waves, only travelling waves. At the end of the animation it loops back to the beginning of the clip. On a string, once the standing waves start, they carry on until the motion stops. 

The amplitude of the waves in the animation is much greater than the amplitude of a wave on a piano string. On a real string, the waves travel much faster.

The animation may be slow while downloading. 

The speed it runs depends on your browser and computer.

If it is still too slow after download, you can download it separately by right clicking on the image and choosing "Save picture..."  Save it on your desktop and run it by clicking the icon.