The loudness of a sound can be measured in terms of watts of sonic energy P. Usually, however, it is measured with decibels.

Question

The loudness of a sound can be measured in terms of watts of sonic energy P. Usually, however, it is measured with decibels. The relationship between the decibel level L and the watts of sonic energy P is given by L=(10/ln10)ln(P/P(sub o)) when P(sub 0) is an extremely small constant (10^-12, although you do not need to know that to solve this problem.)

A jackhammer emits 1 watt of sonic energy when it is running. When it is turned off, it will take a moment to completely stop; as it slows down, it becomes quieter. Half a second after the jackhammer is turned off, it is emitting 0.25 watts of sonic energy, and that amount is decreasing by 0.75 watts/second. How is the decibel level changing at that moment?

Answer 1

The equation for the level in dB is given as

L(P(t)) = (10 / ln(10)) * ln(P(t) / P_0)

Notice that the units cancel out and we have no dimensions for the level (only dB scale).

Taking the time derivative of this function, we find

dL/dt = (10/ln(10)) * (dP/dt) / P(t)

where the values are given for the moment when t=0.5 seconds.

Plugging in values, we find

dL/dt = (10/ln(10)) * (-0.75 Watts / sec) / (0.25 Watts) = -13.0288 dB / seconds.

Note the units are as expected, and the value is negative since it is decreasing in time.