Wave packets in a thin Timoshenko-like shell with variable parameters
Pavel Aleksandrovich Gladkov
Post-graduate student, Vitebsk State University
Non-stationary waves propagating in an axially symmetric non-homogeneous thin Timoshenko-like shell with
variable thickness, density, Young’s modulus and Poisson’s ratio are studied. A formal asymptotic solution of the
equations governing the wave process is constructed in the form of short localized waves named as the wave
packets (the WP) and running along the longitudinal axis of the shell. The solution represents a superposition of
bend and longitudinal waves.
All variable functions are expended into the series in a neighborhood of the moving center z  q(t ) called the wave
packet center. The solution of a Timoshenko-like set of differential equations is constructed in a form



V   k 2 Vk exp i 1 S ,t ,  ,
k 0
V  u(, t ) w(, t ) (, t ), S  (t )dt 1 2 p(t ) 12 b(t ) 2 ,
where i  - 1 ,   1 2 (  q(t )) ,  is a dimensionless longitudinal co-ordinate.
A small parameter  is introduced for studying the packets of short waves propagating in the longitudinal direction,
with the momentary frequency  1(t ) and the wave length being of the order . The functions
u(, t ), w(, t ), (, t ) are polynomials in  , p(t ) is the wave number, the function b(t ) characterizes the width
of the WP in the longitudinal direction, Imb(t )  0 for any t. It is important to note that the last inequality
guarantees the attenuation of the wave amplitudes within the packet.
The zeroth order approximation problem has a solution if and only if the condition
(t )  q (t ) p(t ) H  [ p(t ), q(t )]
holds. Here
H  ( p, q )   
5 E (C  p 2 )
12 (1  )
is the Hamiltonian function with C a modulus of subgrade reaction, E and  Young’s modulus and Poisson’s
ratio and  is a density. The signs () indicating the availability of two branches (positive and negative) of the
solution. Then the solution of zeroth order approximation problem may be presented as V0  P0 (, t )Y (t ) , where
P0 (, t ) is an unknown polynomial in .
The compatibility condition for the non-homogeneous problem arising in the first order approximation gives the
Hamiltonian system
H
H
.
q 
, p  
p
q
The condition for the existence of a solution of the non-homogeneous problem in the second order approximation
produces the Riccati equation
 2H 2
 2H
 2H
b 
b

2
b

0
 pq
 p2
q 2
and the amplitude equation
h0 (t )
 2 P0
P
P
h1(t ) 0 h2 0 h3 (t )P0 0 ,
2

t

where the h j (t ) are known functions.
In particular, if the Young’s modulus is variable, then the constructed solution permits to investigate high-frequency
free oscillations of the medium in a neighborhood of the so-called “weakest” plane where function E () is
maximum. The qualitative analysis of the Hamiltonian system has also allowed to reveal that the presence of the
“weakest” plane may result in strong localization of the non-stationary wave processes near this plane and also in
the effects of the reflection of the running wave packets, these reflections being accompanied by strong focusing
and growing amplitudes in the packets.