Question

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?

Answer 1

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