The Relation between the Loudness of a Sound and Its Physical Stimulus

Experiments with many types of sounds have shown that the loudness of a sound is a function of its energy frequency spectrum and its level above the threshold of hearing and that if this relationship be represented as follows: $L=\frac{10}{3}{log}_{10}{\left[\Sigma \stackrel{i=k}{i=1\mathrm{}}{\left(W_i{P}_i\right)}^{\frac{2}{r}}\right]}^{r^2}$ sounds whose calculated values $L$ are equal will appear equally loud to the average normal ear. $P_i$ is the r.m.s. pressure of the ith component of the sound wave. The weight and root factors $W$ and $r$, respectively, are functions of the sensation level, which is synonymous with the term loudness as formerly used and is defined as follows: $S=10\mathrm{}{log}_{10}\left[\frac{\Sigma \stackrel{k}{1}P_i^{}{}_{}{}^2}{\Sigma \stackrel{k}{1}P_{0i}^{}{}_{}{}^2}\right]$ where $P_{0i}$ is the r.m.s. pressure of the ith component when the complex sound is at the threshold of hearing. In case the components in a narrow band of frequencies $\Delta n$ are not resolved their energy must be integrated to obtain the energy of the equivalent single component. The root factor $r$ is inversely proportional to the ratio of the minimum perceptible increase in energy to the total energy. For intensities near the threshold, the weight factors are equal to the reciprocals of the minimum audible pressures. Curves are given showing the values of $W$ for various frequencies at various sensation levels, also the values of $r$ as a function of $S$. As the intensity is increased the weight factors give greater weight to the lower frequencies; hence, even though the amplitude of the sound wave be increased without distortion, the ear will perceive both an increase and a distortion. This effect is due to the non-linearity of the ear.