Two waves are propagating on the same very long string. A generator at one end o

Question

Two waves are propagating on the same very long string. A generator at one end of the string creates wave 1, given by the followingequation.
y = (8.0 cm)cos p/2[(4.0m-1)x + (9.0s-1)t]
A generator at the other end of the string creates wave 2, given bythis equation.
y = (8.0 cm)cos p/2[(4.0m-1)x - (9.0s-1)t]
For x > or = 0, what are the locations of the nodes having the smallest(x1), second smallest (x2),and third smallest (x3) values of x.
x1---------m

x2------ m

x3---------m
Repeat for the antinodes with the smallest values of x.
x1-----------m

x2----------m

x3----------m

Explanation / Answer

y1 = 8.0 cos p/2[4.0x + 9.0t]

y2 = 8.0 cos p/2[4.0x - 9.0t]

Resulting Wave can be obtained by adding these two waves

y = y1 + y2

=> y = 8.0 cos p/2[4.0x + 9.0t] + 8.0 cos p/2[4.0x - 9.0t]

=> y = 8.0 { cos p/2[4.0x + 9.0t] + cos p/2[4.0x - 9.0t] }

=> y = 8.0 { 2*cos p/2(4.0x)*cos p/2(9.0t) }             {since cosA + cosB = 2*cos( (A+B)/2 ) * cos( (A-B)/2 )}

Let's take the snapshot at t=0

=> y = 16.0 cos p/2(4.0x)

For nodes y = 0

=> y = 16.0 cos p/2(4.0x) = 0

=> cos p/2(4.0x) = 0

=> p/2(4.0x) = /2 , 3/2 , 5/2

=> x1 , x2 , x3 = /4p , 3/4p , 5/4p

For antinodes |y| = maximum

=> |y| = |16.0 cos p/2(4.0x)| = 16.0

=> p/2(4.0x) = 0 , , 2

=> x1 , x2 , x3 = 0 , /4p , 2/4p

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