Fn = ((2*n - 1)*Vs)/(4*H)

Fn - is the natural frequency of the corresponding mode

n - is the mode number

Vs - is the average shear wave velocity

H - is the depth of the soil column

Vs = H/(sum (Hi/Vsi))

Hi - is the thickness of the each i - layer over bedrock

Vsi - is the shear wave velocity corresponding of the i - layer of the Hi thickness over bedrock

sum - is the sum of the fractions (Hi/Vsi)

In the relationship of the Mr..Seyed Amin Mousavi  above, there is a mistake

you can use KRAMER 's book( the geotechnical earthquake engineering). Also, there is a program named SHAKE that you can model the soil profile with boundaries and then calculate the natural frequencies.

frequency of the nth mode of a soil column can be estimated as

fn=Vs.(2n-1)/4/H

in which n is the mode number, Vs is the average shear wave speed, and H is depth of the soil column. 

If you need these frequencies to calculate damping matrix of the soil column, use n=1 and n=4 or 5, that is use f1 and f4 rather than f1 and f2 (2nd mode). This formulation is very very rough and has been proposed by Kramer in 1996. Note that this damping belongs to the inherent damping of the linear soil. Damping of the soil due to its nonlinear hysteretic behavior should be obtained based on its degradation curves (shear modulus and damping vs. shear strain). This latter technique is used in a classical equivalent linear analysis. You can also do a nonlinear analysis in time domain through either total stress or effective stress techniques. All of these analyses can be easily done through DEEPSOIL software which is free to download. 

In response to your question, I give you two suggestions, I hope to help you

(1) to soil, different from the structure, "the natural  frequency of soil"  is an unstrict appellation, we often call it  "dominant frequency of site", because the dynamic characteristics and parameters of soil are dependent on strain level.

(2) if you need to get first and second natural frequency for caculating the Rayleigh damping coefficient in FEM analysis,  you could establish the FEM model with Dynamic Artificial Boundary, then calculate the F1 and F2 according to the idea of FEM in structure analysis. The F1 and F2 is usually the dominant frequency of site in the elastic region(small strain level ,such as 10-6). In noninear analysis of soil model, all  change.

Fn = ((2*n - 1)*Vs)/(4*H)

Fn - is the natural frequency of the corresponding mode

n - is the mode number

Vs - is the average shear wave velocity

H - is the depth of the soil column

Vs = H/(sum (Hi/Vsi))

Hi - is the thickness of the each i - layer over bedrock

Vsi - is the shear wave velocity corresponding of the i - layer of the Hi thickness over bedrock

sum - is the sum of the fractions (Hi/Vsi)

In the relationship of the Mr..Seyed Amin Mousavi  above, there is a mistake

Hello

In addition to the above answers, may I suggest also a rather complete (but not the only one, of course) book: https://www.researchgate.net/profile/A_Pecker/publications?pubType=book&ev=brs_pubs_book

Good luck!

Hi All. Thanks a lot.

in FEM analysis for example with Abaqus . i have to chose boundary in vertical border and bottom . which boundary element i have to select , finite or infinite , for frequency analysis.

Hi

I have the same problem. did you find a way? can you share it with me?

For first frequency, quarter wave length methods is simple to use for uniform deposits. For layered medium, it is a bit tricky, as each layer acts as a filter. Analogy is MDOF system. If one can estimate equivalent Vs for the deposit, quarter wave length method may give approximate solution, with some errors. If the layers are differing in engineering properties, say, Vs, and density, significantly, this method may not be suitable. In such cases, reverse engineering, in terms of simple one dimensional wave propagation technique can be used, say, with DeepSoil or similar codes.

Recall the equation for resonance frequency of a single layer of thickness H and shear-wave velocity Vs, as, f = Vs/(4*H).

This is a quarter-wavelength condition, as it states that resonance will occur for a period for which the wavelength (VS/f) is 4 times the layer thickness. For higher modes, one can use, a factor of (2*n - 1), as explained by others, in the previous answers.