Energy lost per oscillation is proportional to remaining energy, producing exponential decay.

Evidence

Thus the amount of energy lost per period δE = (dT/m)E is proportional to the energy remaining, the hallmark of exponential decay. Following what we have seen earlier, the fraction of energy lost in one period appears as the factor in the exponential multiplying the time, measured in periods. — Eric J. Heller, Why You Hear What You Hear- An…, 964