Two waves u= Asin (kx-Wt) and v= Asin (kx+wt) are added. The result is y= 2Asin

Question

Two waves u= Asin (kx-Wt) and v= Asin (kx+wt) are added. The result is y= 2Asin kxcos wt.
-fix time at zero where are the nodes?
-Fixed tome so angle= 90 degree. what is y? where are the nodes?
-If x is fixed, will that point oscillate? suppose kx=n pi.
-All the above apply to two infinitely long traveling waves moving in opp direction
-In actual string, waves reflect off the fixed ends. why are they inveterted at reflection? multiple reflection set up standing waves.
-If the spring is pluckked with a triangular wave form, many frequencies and phases are in that wave form. What happens to the energy in these frequencies when the standing wave is set up?

Explanation / Answer

So, if we're looking at the superposition of two waves at t=0, then we just set the t in cos wt equal to zero. Now u+v = 2Asin(kx). Now we want to find the nodes of this wave, so basically where u+v = 0
0 = u+v = 2Asin(kx)
0 = sin(kx)
kx = 0, , 2, 3 ....

Therefore x = n/k, where n = 0, 1, 2, 3 ...

b) cos(90) = 0, so y = 0. The string/wave is not moving, so the whole line is a node

c) If x is fixed at kx = n, that point will not oscillate because it's at a node, so it's not moving.

d) The pulse is reflected inverted on fixed ends because when it reaches the end, it will exert an upward force on the support which holds the end fixed. Newton's third law says equal and opposite forces, so the support will exert a downward force on the string, causing the pulse to be reflected downward.

e) Depends on how the wave forms are superimposed. When two opposing triangular wave forms overlap, they will destructively interfere. The energy is still there in the form of kinetic energy because every part of the rope is still moving. If the pulses constructively interfere, then the string at that point has the energy of the two different pulses. In other words, energy is still conserved in all situations

Hope this helps, let me know if anything's unclear

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