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Governing equations of envelope surface created by nearly bichromatic waves propagating on an elastic plate and their stability

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Abstract

Plates are common structural elements of most engineering structures, including aerospace, automotive, and civil engineering structures. The study of plates from theoretical perspective as well as experimental viewpoint is fundamental to understanding of the behavior of such structures. The dynamic characteristics of plates, such as natural vibrations, transient responses for the external forces and so on, are especially of importance in actual environments. In this paper, we consider natural vibrations of an elastic plate and the propagation of a wavepacket on it. We derive the two-dimensional equations that govern the spatial and temporal evolution of the amplitude of a wavepacket and discuss its features. We especially consider wavenumber-based nearly bichromatic waves and direction-based nearly bichromatic waves on an elastic plate. The former waves are defined by the waves that almost concentrate the energy in two wavenumbers, which are very closely approached each other. The latter waves are defined by the waves that almost concentrate the energy in two propagation directions and two propagation directions are very close each other. The fact that the solution of the governing equation for wavenumber-based nearly bichromatic waves is stable is also shown.

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Abbreviations

A:

slowly varying amplitude in space and/or time of a wavepacket

Cl :

the function space consisting of functionsf(x) which is lth differentiable at any point ofx and the lth derivative off is continuous

D:

plate flexural rigidity

E:

elastic modulus

G:

direction-based spectrum of a wavepacket or Green’s function

H:

value used in boundary conditions

L:

linear operator defined by Equations (A.19) and (A.20) or value used in boundary conditions

M:

positive integer

N:

positive integer

O:

function of order

Q:

function defined by Equation (A.18) or Fourier coefficient

R:

real space

S:

function used by the method of separation of variables in Equation (A.3)

U:

function used by the method of separation of variables in Equation (A.3)

a:

cosθ0

b:

sinθ0

c:

Fourier coefficient

cc:

complex conjugate

f:

function of initial condition

h:

plate thickness

i:

imaginary unit or unit vector alongx axis

j:

unit vector alongy axis

k:

wave number

m:

integer

n:

integer

t:

time

u:

function

v:

function

v:

velocity of envelope surface

w:

plane traveling waves

x, y:

rectangular coordinate system

Ψ:

function used by the method of separation of variables in Equation (A.9)

Φ:

function used by the method of separation of variables in Equation (A.9)

α:

coefficient in Equation (A.34)

β:

coefficient in Equation (A.34)

γ:

coefficient in Equations (A.10) and (A.11)

δ:

Delta function

ε:

order of time and spacial scales

θ:

propagation direction

λ:

separation constant

μ:

separation constant

ν:

Poisson’s ratio

ρ:

mass density

ω:

angular frequency

δD :

boundary of rectangle

∇4 :

biharmonic operator

0:

dominant value

1,2:

number of function or value

m:

number of coefficient

n:

number of function or coefficient

’:

differentiation with respect to wave numberk

∼:

modified value

∧:

modified value

*:

adjoint operator

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Authors and Affiliations

  1. Department of Electronic and Computer Engineering, Musashi Institute of Technology, Tamazutsumi, Setagaya, 158-8557, Tokyo, Japan

    Ben T. Nohara

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  1. Ben T. Nohara
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Nohara, B.T. Governing equations of envelope surface created by nearly bichromatic waves propagating on an elastic plate and their stability. Japan J. Indust. Appl. Math. 22, 87 (2005). https://doi.org/10.1007/BF03167478

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  • DOI: https://doi.org/10.1007/BF03167478

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