Understanding Standing Waves for SAT II Physics

[panli19]

Homework Statement

So I am studying SAT II physics by myself and I don't really understand how standing waves form. I understand that they are formed by the interference of the two traveling waves which results in complete destructive interference at some points, and complete constructive interference at others. What I don't understand is how can two waves be completely constructive and completely destructive at the same time. For two waves to be completely constructive, crusts need to meet crusts and troughs need to meet troughs. For them to be completely destructive, everything must be out of phase. How can two waves be in and out of phase at the same time

2. I also don't understand this statement in the Princeton review book"while every point on the string had the same amplitude as the traveling wave went by, each point on a string supporting a standing wave has an individual amplitude.

 

[graphene]

1. Constructive and destructive interferences happen at different times.
check the figures in http://en.wikipedia.org/wiki/Standing_wave
and the animation in http://www.physicsclassroom.com/mmedia/waves/swf.cfm.

At the instant the two waves lie exactly over each other, there is constructive interference. After a little time, (as they travel in opposite direction), one wave would 'complement' the other & you get destructive interference.

2. The final wave is $$y = 2 y_0 \sin(kx) \cos(\omega t)$$ or $$y = A \cos(\omega t)$$ where A is a function of x. The amplitude of the standing wave is thus different at different points.
While certain points vibrate from y = -A to y = A, certain other points don't vibrate at all.

This contrast the situation in case of a traveling wave as each point would vibrate from y = -A to y = A.

 

[panli19]

thanks a lot