Benchmark solution
=====================


Calculation method
-----------------

We use the coherence function proposed by Luco and Wong (1986) [:ref:`bib1 <bib1>`]:

where :math:`>` refers to the distance between two points :math:`>` and :math:`>` on the foundation, :math:`>` is the frequency and :math:`>` the apparent surface propagation speed of the wave :math:`>`. The parameter :math:`>` may vary from :math:`>` to :math:`>` depending on the case but is generally taken to be equal to :math:`>`.


Reference quantities and results
-----------------------------------

Covariance coefficients obtained by Mita and Luco for :math:`>` [:ref:`bib2 <bib2>`]:


.. csv-table::

    ":math:`>` "," :math:`>` "," :math:`>`"
    "1.0", "0.732", "0.730"
    "2.0", "0.402", "0.416"
    "3.0", "0.251", "0.270"


a0 refers to the non-dimensional frequency :math:`>`

For :math:`>`, we get the case without spatial variability, for this case we know the solution (analytical). Since the foundation is rigid without weight, the response to a unit excitation is equal to :math:`>`, regardless of the calculation frequency.


Uncertainties about the solution
----------------------------

No uncertainties.


Bibliographical references
---------------------------

[:ref:`bib1 <bib1>`] Luco J.E and Wong H.L.: *Response of a rigid foundation to a spatially random ground motion*. Earthquake Engrg. Structure. Dyn. 14, 1986, pp.891-908.

[:ref:`bib2 <bib2>`] Luco J.E and Mita A.: *Response of a circular foundation to spatially random ground motion*. J. Engrg.Mech. 113, 1987, pp.1-15.