[UPSC CSE 2026] Waves (Q1, 10M)

A particle executes simple harmonic motion of amplitude A and angular frequency \omega. A damping force proportional to velocity acts on the particle. Derive the expression for the displacement of the particle as a function of time and discuss the effect of damping on amplitude and frequency. Also, calculate the time at which the amplitude reduces to half its initial value, if the damping coefficient is b and mass is m.