In a structural dynamic wave propagation problem, is the displacement at a point a continuous function of time or is it a continuous function of time only after the wave reaches the point?
Dear Mir Aamir Abbas
Numerical solution of structural dynamics problems use finite difference methods for space and for time derivatives. It uses explicit difference schemes in a similar way, that they are used for other numerical wave propagation problems---for example exploration geophysics & earthquake geophysics wave propagation & seismic modelling.
I find interesting the following link:
https://www.nafems.org/blog/posts/focus-on-structural-dynamics/
That depends on the boundary and initial conditions. Even in the simplest case of the scalar linear wave equation there exist discontinuous solutions. However, you are more likely to encounter cases in which the solution is continuous but not continuously differentiable. You can also construct (rather academic) examples in which the solution is differentiable arbitrarily many times.
Hi Mir Aamir Abbas
I believe that it depends a bit.
For unbounded systems, there would not be much of a response before the wave reaches the receiver position. More on this below.
For bounded systems, one must cater for rigid body modes. These have infinite (or a very high) wave speed.
This may at first sound strange, but once we think about it a little, it becomes obvious (or there could simply not be any wave propagation on planet earth which has a high velocity indeed when viewed from the galaxy centrum).
To exemplify, in the case or airbag firing inside cars, there is a pressure increase at the driver's ear before the acoustic wave reaches it. This took the industry about 20 years to figure out and it can be correctly detected when using both pressure transducers and microphones. The former tracks Line Pressure and Sound, while the microphone only tracks sound.
So, for an unbounded system, the 'rigid body' response would still be there, but it would very low and thus something that can be ignored.
Now, let us complicate matters a bit. There is an amplitude dependent portion in wave propagation that causes the wavefront to distort - it is referred to as steepening, and it causes rouge waves, shock waves, etc.
Simply put, waves with a high amplitude travel faster than the linear free wave speed. After a distance, the peak catches up with the trough and the wave collapses and the process starts anew.
So, when %LP = 100*Pulsation/LinePressure = 100*Sound/Ambient is high, things may happen faster than we would otherwise expect. This is generally the case whenever LP is low, e.g. in vacuum systems, on the suction side of machines and so on.
More on the differences of pressure and steepening is found here. https://qringtech.com/2010/09/15/wave-steepening-increase-peak-pressure-piping-pumps/
One may think the issue of steepening to be nerdy beyond the practical, at least I did so when I first came across it, but I find it to be more of a rule than the exception for industrial application whenever liquids with gas dissolved in it is considered.
I am sure one can compound the issue further if one would want to do so.
So, to sum up. For the linear case, I would say, yes, it would be linearly and continuously related as it can be described by the summation of a set of modes. For the non-linear case, the answer would be, no, in particular at the wave breaking point where it simultaneously would exist at multiple locations plus the fact that it (just like me) forgets what it did previous to its latest (col)lapse.
Just my 2 cents
Claes
PS - We have all seen steepening. It is what happens, albeit for other reasons, at the sea/shore interface at beaches.
Hi Claes Richard Fredö. Thanks for your answer. Can you give examples of unbounded and bounded systems? I look forward to further learning.
Hi Mir Aamir Abbas
An unbounded system is one in which for a lossless system, the wave propagates forever without hitting a boundary. Free wave propagation can exist at any frequency in an unbounded system and its node (minimum amplitude) moves with the wave speed. https://en.wikipedia.org/wiki/Unbounded_system
A bounded system, a bounded domain, is one in which energy is contained and/or edge losses can be accounted for. Free waves only exist at natural (aka eigen-) frequencies. The interference of free waves bouncing around in a bounded system forms standing waves, i.e. waves in which the node point is fix. Standing waves are called modes. Vibration does only occur at other frequencies by forced excitation and it can be described as a linear combination of modes. https://en.wikipedia.org/wiki/Domain_(mathematical_analysis)
Mathematical boundary conditions for a bounded domain (the better and more common name) would be Dirichlet, Cauchy, Neumann, Robin. (It has been a long time since I used any of these, hope memory serves me correctly)
https://en.wikipedia.org/wiki/Dirichlet_boundary_condition
https://en.wikipedia.org/wiki/Cauchy_boundary_condition
https://en.wikipedia.org/wiki/Neumann_boundary_condition
https://en.wikipedia.org/wiki/Robin_boundary_condition
The above discounts wave steepening which is nonlinear.
Hope this helps
Claes
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