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


Calculation method used for the reference solution
--------------------------------------------------------


+-------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                   |:math:`f(t)`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
+ .. image:: images/100021AA0000170C00002D94868D3395584CE4D3.svg                                                    + contact force in                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|    :width: 297                                                                                                    | :math:`A`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
+    :height: 588                                                                                                   +;                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             +
|                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+                                                                                                                   + :math:`V(z,t)` speed; :math:`U(z,t)` displacement;; :math:`{\tau }_{0}=\frac{{U}_{0}}{{V}_{0}}`;; :math:`{\tau }_{1}={\tau }_{0}+\frac{L}{{c}_{p}}`; Any shock duration; :math:`\tau -{\tau }_{0}=\frac{\mathrm{2L}}{{c}_{p}}` :math:`{c}_{p}=\sqrt{\frac{E(1-\nu )}{\rho (1+\nu )(1-2\nu )}}` :math:`S={a}^{2}` section.                                                                                                                                                                                                                                                                                                                                                                                    +
|                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+                                                                                                                   + .. image:: images/100002BC0000271000002710CD2F459F1DBDD5B0.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               +
|                                                                                                                   |    :width: 297                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+                                                                                                                   +    :height: 588                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+                                                                                                                   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+                                                                                                                   + for point :math:`A`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+                                                                                                                   + .. image:: images/100000B0000004720000003573C4257F10B00D20.svg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               +
|                                                                                                                   |    :width: 297                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
+                                                                                                                   +    :height: 588                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+                                                                                                                   +                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              +
|                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+                                                                                                                   + for point :math:`B`                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          +
|                                                                                                                   |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
+-------------------------------------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+



Benchmark results
----------------------


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


1. R.J. GIBERT, "Vibrations of structures", Summer School for Numerical Analysis, 1988, (Edition EYROLLES).