So, I was presented in class the fact that (at least in Ultrasound), attenuation coefficients $\mu$ can be expressed in units $\text{dBcm}^{-1}\text{MHz}^{-1}$ as opposed to the form I was used to; simply $\text{cm}^{-1}$.
For a sanity check, I decided I wanted to work out a formula to convert from one to the other.
For sake of simplicity, let's denote the $\text{dB}$ form as $\mu_{dB}$ and the original (in $\text{cm}^{-1}$) as $\mu$.
I know for a fact that:
$$\frac{I(x)}{I_0} = e^{-\mu x}$$
and also (provided, though I wouldn't mind knowing where/how it was derived?)
$$\frac{I(x)}{I_0} \text{[dB]} = fx\mu_{dB}$$ Where $f$ is frequency
If I'm not mistaken, the conversion from the original units to $\text{dB}$ is:
$$10\text{log}_{10}\big(\frac{I(x)}{I_0}\big)$$
So, substituting in, and equating, I ended up with something looking like this:
$$10\text{log}_{10}\big(e^{-\mu x}\big)=fx\mu_{dB}$$
Which, after rearranging, and changing log base, I end up with
$$\mu_{dB} = -\frac{10}{f\text{ln}(10)} \mu$$
The issue I have here, is that this minus sign has appeared, which (I don't think) should be correct?- Can the Attenuation coefficient be negative?
I feel like I'm missing something stupidly obvious (i.e. can I do this in the first place, or is it just its own thing?), but I just can't see it... Any help here as well would be appreciated! Cheers, all!
