- #1

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## Main Question or Discussion Point

The Beer-Lambert law gives the intensity of monochromatic light as a function of depth ##z## in the form of an exponential attenuation:

$$I(z)=I_{0}e^{-\gamma z},$$

where ##\gamma## is the wavelength-dependent attenuation coefficient.

However, if two different wavelengths are present simultaneously, we will have two different coefficients to consider: ##\gamma_{\lambda_{1}}## and ##\gamma_{\lambda_{2}}##. Would it be mathematically correct to combine the two attenuation coefficients into one using the weighted geometric mean?

I mean, if ##p_1## and ##p_2## are the fractions of the light beam which is of a given wavelength, using geometric mean (instead of the arithmetic formula) we would have:

$$I(z)=I_{o} e^{-\bar{\gamma}z},$$

where:

$$\bar{\gamma}=\left(\gamma_{\lambda_{1}} \times\gamma_{\lambda_{2}}\right)^{1 / (p_{1}+p_{2})}.$$

Is the use of the geometric mean valid here?

$$I(z)=I_{0}e^{-\gamma z},$$

where ##\gamma## is the wavelength-dependent attenuation coefficient.

However, if two different wavelengths are present simultaneously, we will have two different coefficients to consider: ##\gamma_{\lambda_{1}}## and ##\gamma_{\lambda_{2}}##. Would it be mathematically correct to combine the two attenuation coefficients into one using the weighted geometric mean?

I mean, if ##p_1## and ##p_2## are the fractions of the light beam which is of a given wavelength, using geometric mean (instead of the arithmetic formula) we would have:

$$I(z)=I_{o} e^{-\bar{\gamma}z},$$

where:

$$\bar{\gamma}=\left(\gamma_{\lambda_{1}} \times\gamma_{\lambda_{2}}\right)^{1 / (p_{1}+p_{2})}.$$

Is the use of the geometric mean valid here?