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A mass that has velocity and a force f1, can create a sound and even break a wall upon impact. However if the same force f1 is applied to the wall without any motion (velocity) then the wall will not break and no sound can be heard. Why is this? I suspect that since the mass is in motion it is traveling as a wave and upon impact this wave is what generates a vibration and sound. I also suspect that the vibration caused by this impact causes the molecules around the impact area to vibrate at its harmonic frequency (causing the molecules to break apart). Is any of this true? Or am I off base? What equations exists that describes the vibrations upon impact?? Thanks!!

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The strength of materials is a complicated subject, but the wall very well could break under a static force of the same magnitude as the impact force. Do you understand that the impact force of an object against a wall can be many orders of magnitude greater than the weight of the object? See en.wikipedia.org/wiki/Strength_of_materials and en.wikipedia.org/wiki/Impact_%28mechanics%29 for relevant discussion. –  Michael Brown Jul 17 '13 at 2:15
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One does not need to call on the wave nature of matter to explain this. Most of a perfectly good classical answer is:

  1. Impacts are extremely swift decelerations: thus even small masses can beget extremely high forces, so it is hard to apply similar forces "slowly" and "evenly" to the wall in both cases (i.e. it is hard to realize the same $f1$ in both the steady load and impact cases in your notation);

  2. Impacts initially involve very small contact areas. Think of a sphere touching a plane: the contact area in ideal geometry is nought and only becomes nonzero after the wall has been dinted. Therefore, the forces arising in the deceleration are being supported by many fewer molecular bonds than they would be if you applied the same force over an appreciable area.

  3. The time variation of the force matters - if a force is applied very quickly to the matter inside the wall and a shock wave arises - and very high forces arise at the wavefront so you are right in saying this is an acoustic phenomenon. This is quite different from the case where one begins with a small force and slowly increases it: the matter has time to re-arrange itself elastically and spread the load between may molecular layers, so no shock wave results. The formation of waves depends strongly on the material properties, which is partly why some things are brittle, others not.

What equations exist? To my knowledge, problems of this kind are dealt with practically (and very dryly) by engineering rules of thumb and engineering standards specifying impact tests (Charpy test and so forth). For a more mathematical treatment: hopefully you'll also get an answer from someone who studies this kind of acoustics or who analyses the effect of explosives. These are the kinds of people I would go to first to try to get an answer.

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Thanks very much! Very helpful. –  Robert Rosario Jul 17 '13 at 15:37
    
@RobertRosario I added "explosions" and "acoustics" tag to your question - someone who studies blasts or other shock waves could help with good, physical mathematical models. The engineering specifications will help you with working out things like what materials will withstand impacts and so forth, but they will tell you nothing of the underlying real description. –  WetSavannaAnimal aka Rod Vance Jul 18 '13 at 1:50
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