Units of "mean free time" of a wave/particle [duplicate]

Question

Suppose you have a wave $\phi$ with a complex angular frequency:

$$\phi = e^{j\omega t} = e^{j(\omega_R+j\omega_I)t} = e^{j\omega_Rt} e^{-\omega_It} = e^{\frac{-t}{\tau}} e^{j\omega_Rt} $$

Here, $\tau = 1/\omega_I$ is the time necessary for the amplitude to decay with $1/e$.

Now my question: what is the unit of $\tau$?

1. I would think that the unit is seconds. In this case, the exponent becomes dimensionless and this suits with the definition time necessary for the amplitude to decay with $1/e$.

2. $\tau$ is the inverse of the imaginary part of the radial frequency. So one can argue that the unit can be second/rad.

What is the unit of $\tau$ and do you need to multiply or divide by $2\pi$ in the calculation of $\tau$ (to get rid of the radians)?

Answer 1

Radian, the unit of angle, is dimensionless because it's defined on length divided by length basis(Arc length / radius).

Hence, either will work.

Answer 2

Thanks for the answers!

However, I am still wondering if you need to multiply with $2 \pi$ or not. For example, the precession period is $T=\frac{2\pi}{\omega_R}$, here you need to multiply with $2 \pi$. Maybe the mean lifetime is then $\tau=\frac{2\pi}{\omega_I}$?

Answer 3

As angle is defined as the ratio of arc length to the radius of a circle, it is actually dimensionless. Radian doesn't have any physical meaning as a unit, as far as I know.

It must be also multiplied by $2\pi$ since as you pointed out it is divided by $\omega=2\pi f$, not $f$.

So it would be multiplied by $2\pi$ with unit of seconds.