three waves of equal frequency having amplitudes 10,4and 7 micrometre arrive at a given point with successive phase differnce of, - Maths -

Question

three waves of equal frequency having amplitudes 10,4and 7
micrometre arrive at a given point with successive phase differnce
of, pie/2, the amplitude of the resulting wave in micrometre is
given by?

Somnath physics · Accepted Answer

Let the waves be
x1 = 10sin(ωt)
x2 = 4sin(ωt+π/2)
x3 = 7 sin(ωt+π)
Thus,
x2 = 4cos(ωt)
x3 = -7 sin(ωt)
Resultant disturbance by superposition theorem is given by
x = x1 + x2 + x3
x = 10sin(ωt) + 4cos(ωt) - 7 sin(ωt)
= 3 sin(ωt) + 4cos(ωt)
Let R be the resultant amplitude
3 = RcosΦ
4 = RsinΦ
x = RcosΦ sin(ωt) + RsinΦ cos(ωt)
=> x = Rsin(ωt + Φ)
R = √(32 + 42)
= 5 μm

three waves of equal frequency having amplitudes 10,4and 7 micrometre arrive at a given point with successive phase differnce of, pie/2, the amplitude of the resulting wave in micrometre is given by?