[UPSC CSE 2023] Waves (Q9, 20M)

Write equation for damped harmonic oscillations and obtain expression for logarithmic decrement. In a damped harmonic motion, the first amplitude is 10\text{ cm}, which reduces to 2\text{ cm} after 50 oscillations, each of period 4\text{ seconds}. Determine the logarithmic decrement. Also, calculate the number of oscillations in which the amplitude decreases to 25\%.