# Can attenuation be factored in terms of frequency and material parameter?

Consider an attenuation coefficient $\mu(x,\omega)$ which depends on material (that is, on the space variable(s) $x$) and frequency variable $\omega$.

The attenuation can be of electromagnetic radiation or of sound, for example.

Is it reasonable to make the approximation $\mu(x,\omega) = \alpha(\omega)\beta(x)$, where $\alpha$ is independent of the material and $\beta$ is independent of the frequency?

Context: I have a nice proof for a physical model, the derivation of which requires making this factorization assumption. Thus, I am interested in how reasonable an assumption this is, and under which conditions it is reasonable.

## 1 Answer

It is fine to make such an assumption mathematically and see where it leads. But I cannot think of a physical reason why it must be true. If we went to first principles we may be able to derive a proof that it is possible in some cases (doesn't violate any physical law but may not always be true) or that it impossible (violates a physical law). Clearly I did not go that far in this answer.

But I can say from personal research experience that is is not true for attenuation of high frequency acoustics in air. The attenuation coefficients are a fairly complicated function of frequency and "material properties", which includes chemical composition (amount of N2, O2, CO2, H2O, etc), and thermal properties of the medium (Temperature, relative humidity, pressure). The functional form of the equation has been published and if you search for "Atmospheric Attenuation of Ultrasound in Air" you will likely find the paper very easily, free code exists for evaluating these coefficients too. Long story short, the function is not separable or factorable. However, it may be in some weak limiting case, or the log form of the function may be separable.

Hence for this reason I'd say carry on with the idea keeping in mind that it is not generally true.