For example, the first order Laplacian measures the difference between the local and average values of a quantity in an infinitesimal neighborhood of the point in question. So, for a flexible string, the acceleration is proportional to the Laplacian (wave equation).
For plates, the acceleration is proportional to the double Laplacian. Also, the double Laplacian occurs in the modification of the wave equation for a stiff string. Therefore, what is the double Laplacian a measure of? Thanks.
Dear David Mark Koenig ;
We have:
Applications of Double Laplace Transform to Boundary Value Problems DOI: 10.9790/5728-0925760
https://physics.stackexchange.com/questions/282457/help-to-understand-the-use-of-a-double-laplacian-in-this-vibrating-disk-problem
The order of the laplacian measures indicate the energy storing elements in the system. The single order says single energy storing element or single variable.
From acceleration point of view double laplacian indicates two energy storing element. From physical interpretation point of view it can able to predict behavior of position and velocity, means there are two variables.
By use of laplacian operator one can find the stability and system response easily.
I appreciate Dr. Shah's response but I do not think it sufficiently addresses the physical interpretation. Please see the attached document.
I would start with the derivation of the equation of motion for beams or plates. If we start with the simple example of a string, the tension of the string results in a linear restoring force in the transverse dimension. For beams and plates, there is a restoring force proportional to the beam curvature resulting from the fact that the plate has finite thickness and is therefore stiff. Stiff structures resisting bending, while things like (ideal) strings do not resist bending.
P.S. The transition from string-like behavior to beam-like behavior can be found in the piano. The upper register strings are very thin and behave more like ideal strings, whereas the lower register strings are quite thick and resist curvature. The stiffness of low-register strings introduces the double Laplacian into the equation of motion and causes the overtones to not be integer multiples of the fundamental.
Derek, thank you for the explanation, especially the reference to piano strings - your point was hammered home (pun intended). So, is it safe to say that the Laplacian measures the force resulting from a displacement of a flexible structure like a "limp" string or a membrane while the double Laplacian measures the force resulting from the bending of (displacement of) an elastic structure like a plate or a "stiff" string?
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