A great MIT applet
A great MIT applet that may help you to understand Remark 4.8
http://mathlets.org/mathlets/wave-equation/
The "left wave" and the "right wave" in the applet are exactly and that appear in Remark 4.8.
Another interesting thing is the see how the "harmonics" (i.e. the Fourier coefficients) change when you "pluck" the string on the side or in the middle.