# Pa^2/Hz to dB/Hz conversion

I have PSD data from an computational aeroacoustics analysis, which is in the units of Pa^2/Hz. I want to convert this to dB/hz; would it be correct if I use PSD[in dB/Hz] = 10log10(PSD[in Pa^2/Hz]) or is there another way of doing this that I am not aware of?

This depends a bit on what you are after. As Laurent Duval has pointed out, you need can only take the logarithm of a unitless quantity so you need to do some form of normalization

1. If you want Sound Pressure Level (dbSPL) you would normalize to the standard reference pressure of 20 uPa (20e-6 Pa)
2. You could also divide by the free field acoustic impedance of air, which would give you the intensity in W/m^2.
3. The most common units for a PSD would be W/Hz. To calculate that you'd have to integrate over the entire volume flow and so you need to make an assumption for the shape of your sound wave. That's pretty easy for spherical and plane waves, but difficult for more complex shapes.
• Thank you! Just to clarify, if I do the first form of normalization that you mentioned, the units would be dB and not dB/hz? – clarice Sep 1 '17 at 12:49

Usually, a logarithm ought to be computed on a dimensionless number. Logarithm can be defined as an integral for the function $$x\mapsto \frac{1}{x}$$, with $$x$$ being unit-less. Some discussions can be found in:

You can check for an extended discussion: What is the logarithm of a kilometer? Is it a dimensionless number? A classical example is the signal-to-noise ratio, computing the logarithm of a dimensionless quantity, the power of the signal divided by the power of the noise. So often, one computes a ratio over some (field dependent) reference pressure $$P_0$$:

$$10 \log P^2/P_0^2$$

$$P_0$$ could be 1 Pa (if Pa mean Pascal in your example), but this depends on the field of application, as detailed by @Hilmar. Then, the Hz mention could be left out, to keep dimensions correct. So basically the formula is nice, except the parentheses:

$$\textrm{PSD} [\textrm{in dB}]/\textrm{Hz} = 10\log_{10}(\textrm{pressure normalized PSD})/\textrm{Hz}$$

assuming in the PSD that the data is already squared.