Two traveling waves are generated on the same taut string. Individually, the two

Question

Two traveling waves are generated on the same taut string. Individually, the two traveling waves can be described by the following two equations:
y1 (x,t)=(1.49 cm)sin(k1x+(0.348 rad/s)t+Q1)
y2 (x,t)=(5.78 cm)sin(k2x-(9.35 rad/s)t+Q2)
If both of the above traveling waves exist on the string at the same time, what is the maximum positive displacement that a point on the string can ever have?
What are the smallest positive values of the unknown phase constants (in radians) such that the above displacement occurs at the origion x=0 and time t=1.68s.
Q1= (radians) Q2= (radians)

Explanation / Answer

This solution is based on the observation that two sinewaves of equal amplitude but differing phase have a maximum absolute sum where they intersect; this is shown by the fact that their slopes are equal in terms of |dy/dx| at the intersection, and as you move away, the decreasing function has a steeper slope than the increasing one. Any time that happens, you have a maximum sum. So we can say in this case that each amplitude = 0.725A. Then half the phase difference ??/2 = arccos(0.725) = 0.759762 rad; ?? = 1.5195 rad = 87.06 deg = 0.2418?.

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