The displacement equations of two interfering waves are given by $$y_{1}=10 \sin \left(\omega t+\frac{\pi}{3}\right) \mathrm{cm}$$ $$y_{2}=5[\sin (\omega t)+\sqrt{3} \cos \omega t] \mathrm{cm}$$ respectively. The amplitude of the resultant wave is ____ cm.

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Waves & Sound

The displacement equations of two interfering waves are given by

y_{1} = 10 sin (ω t + \frac{π}{3}) cm

y_{2} = 5 [sin (ω t) + 3 cos ω t] cm

respectively. The amplitude of the resultant wave is ____ cm.

Answer: 20

The second wave can be recast as a single sinusoid. Writing

y_{2} = 5 sin ω t + 53 cos ω t = A sin (ω t + ϕ)

, its amplitude is

A = 5^{2} + (53)^{2} = 25 + 75 = 10 cm

, and its phase satisfies

tan ϕ = \frac{5 3}{5} = 3

, so

ϕ = \frac{π}{3}

. Both waves then have amplitude

10 cm

and the same phase

\frac{π}{3}

, so they are exactly in step and add directly, giving a resultant amplitude of

20 cm

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Integer answer