SPL = 10*log10(Ap*Ap/(1 µPa)2)
The deci in decibel indicates that the logarithm to the base 10 of squared pressure is multiplied by a factor of 10. This factor of 10 applies to quantities that are second order in the acoustic variables; for example, are proportional to the square of pressure, squared particle velocity amplitude, or the product of pressure and particle velocity. Examples of such quantities are acoustic energy density, magnitude of vector acoustic intensity, and acoustic power (see the “Physics of Sound” section of the Glossary). A factor of 20 is used for quantities at first order in the acoustic variables– acoustic pressure and acoustic particle velocity amplitude, for example. With regard to particle velocity, the sound particle velocity amplitude level (SPVL) can be defined in terms of the particle velocity amplitude, Av, as
SPVL = 20*log10(Av/1 m/s) = 10*log10[Av*Av/(1 m/s)2]
It has units of dB re 1 m/s. More care must be taken in dealing with particle velocity because of its vector nature and because the polarization of the motion typically is more complicated than that of simple rectilinear motion.
The original definition of the decibel was given in terms of intensity amplitude ratios. This original definition is repeated in some modern textbooks. However, as indicated above, the decibel now is used in a much broader way, as can be seen in the national and international acoustics standards adopted by the American National Standards Institute (ANSI) and the International Organization for Standardization (ISO). In fact, those textbooks that define the decibel in terms of intensity amplitude ratios often proceed to report quantities in units of dB re 1 µPa or dB re 1 µPa2; these reference values pertain to quantities of pressure and pressure squared, respectively, and are not the units of intensity amplitude (which are W/m2).
Calls for the elimination of the decibel sometimes are heard. The decibel is here to stay, not only because it is part of ANSI and ISO standards, but because it is a valuable way (among others) of reporting acoustical quantities. It was invented and popularized for good reasons by the early pioneers in acoustics. The major reasons for its continued usefulness are given in Chapter 1, such as the fact that sound levels can span a large range of values (large dynamic range) and human perception of loudness appears to be logarithmic in nature. A far better recommendation than the elimination of the decibel is to insist that its reference units always be reported clearly.