Question

# Two sinusoidal waves of the same frequency are to be sent in the same direction along...

Two sinusoidal waves of the same frequency are to be sent in the same direction along a taut string. One wave has an amplitude of 6.2 mm, the other 8.5 mm. (a) What phase difference φ1 between the two waves results in the smallest amplitude of the resultant wave? (b) What is that smallest amplitude? (c) What phase difference φ2 results in the largest amplitude of the resultant wave? (d) What is that largest amplitude? (e) What is the resultant amplitude if the phase angle is (φ1 - φ2)/2?

let
A1 = 6.2 mm
A2 = 8.5 mm

a) when the waves are out of phase the resultant amplitude will be smallest.

so, phi1 = 180 degrees (or) pi radians

b) smallest amplitue, A_smallest = A2 - A1

= 8.5 - 6.2

= 2.3 mm

c) when the waves are in phase the resultant amplitude will be largest.

so, phi2 = 0 degrees (or) 0 radians

d) largest amplitue, A_largest = A1 + A2

= 6.2 + 8.5

= 14.7 mm

e) If phase diffrence, phi = (phi1 - phi2)/2

= (180 - 0)/2

= 90 degrees (or) pi/2 radians

so, resultant amplitude, A = sqrt(A1^2 + A2^2 + 2*A1*A2*cos(phi))

= sqrt(6.2^2 + 8.5^2 + 2*6.2*8.5*cos(90))

= 10.5 mm

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