Mathematical modeling - how to calculate the displacement at any point in a membrane - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:03:08Z http://mathoverflow.net/feeds/question/11925 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11925/mathematical-modeling-how-to-calculate-the-displacement-at-any-point-in-a-membr Mathematical modeling - how to calculate the displacement at any point in a membrane Aristaeus 2010-01-15T21:32:31Z 2010-09-04T20:22:16Z <p>I'm trying to calculate the displacement of a wall at any point due to a point source of vibration. The vibration is considered to be directly perpendicular to the surface of the wall, for calculation purposes.</p> <p>Input: random force applied to some point (call it white noise)</p> <p>Output: Displacement at any other part of the wall (located (x,y) from the original point)</p> <p>I assume that the displacement will at a simple level be related to some K(x,y). The relationship is probably K/(x^2 + y^2), I just need to figure out what goes into K</p> <p>As far as I can say, the stiffness of the material will increase the speed and distance with which displacement waves travel, and decrease the total displacement.</p> <p>Also, things like standing waves with the borders of the wall, vibrational characteristics of the material, and others come into play. I'm not really sure how to model all these things.</p> <p>Can anyone give me some more input into what I should be considering, or maybe some actual equations that can help me relate the material properties of the wall to the displacement at a specific location?</p> <p>Thanks.</p> http://mathoverflow.net/questions/11925/mathematical-modeling-how-to-calculate-the-displacement-at-any-point-in-a-membr/36320#36320 Answer by Aleksey Pichugin for Mathematical modeling - how to calculate the displacement at any point in a membrane Aleksey Pichugin 2010-08-21T19:29:27Z 2010-08-21T19:29:27Z <p>The question implied by the title is not the same as the question actually asked. I'll answer both and you'll have to decide which one you actually need.</p> <p>The membrane is basically a 2D wave equation. Your source would be an inhomogeneous term in it. You seem to be interested in solving a Cauchy problem for this equation. For a point source and infinite membrane this amounts to the standard Green function, which is probably described in every textbook on partial differential equations. For simple geometries like circle or rectangle the infinite series solutions are available (you'll find it in Graff's book "Wave motion in elastic solids"). For more complicated shapes the answer is more cumbersome and generally would require application of a numerical method.</p> <p>The actual description of what you would like to solve is different and seems to imply the interest in the elastic problem. Then, assuming that the wall is of infinite thickness, your Green's function will be the solution of so-called "Lamb's problem". For wall of infinite extent you may easily obtain an integral representation of the solution. For the finite walls it is a largely numerical exercise all the way.</p>