Almost periodic oscillations and waves

308 Pages·2009·1.53 MB·English

by  Constantin Corduneanu (auth.)

2 1 Introduction
Ifweconsidertherectilinearmotionofamaterialpointofmassmwithout
friction that is attracted by a fixed point on the line of motion with a force
proportionaltothedistancetothisfixedpoint,thenNewton’slawofdynamics
(ma=f) leads to the following second-order ordinary differential equation:
mx¨(t)=−kx(t). (1.1)
In equation (1.1), x(t) represents the abscissa of the point in motion, while
k > 0 is the proportionality constant. We must assume k > 0 because the
fixed point of attraction is taken as the origin on the line of motion, and only
inthiscaseistheoriginattractingthepointofmassm.Ofcourse,x¨(t)stands
for the acceleration a= dv, where the velocity v = dx =x˙(t).
dt dt
Let us now consider equation (1.1) and denote ω2 = k. Then equation
m
(1.1) becomes
x¨(t)+ω2x(t)=0, (1.2)
whichisahomogeneoussecond-orderdifferentialequationwithconstantcoef-
ficients. It is well known that cosωt and sinωt are linearly independent so-
lutions of equation (1.2). Therefore, the general solution of equation (1.2)
is x(t) = c1cosωt+c2sinωt, with arbitrary constants c1,c2. If one de(cid:2)not(cid:3)es
c1 =Acosα,c2 =−(cid:4)As(cid:5)inα,whichmeansA=(c21+c22)1/2,α=−arctan cc21 ,
c (cid:2)= 0, and α = − π signc , c = 0, these notations allow us to write the
1 2 2 1
general solution of equation (1.2) in the form
x(t)=Acos(ωt+α). (1.3)
Formula (1.3) shows that any motion of the harmonic oscillator is a simple
harmonic of frequency 2 π. The positive number A is called the amplitude of
ω
the oscillatory motion described by equation (1.3). The number α is called
thephaseofthose oscillations. BothAandα dependontheinitial conditions
x(0)=x and v(0)=v . In terms of x and v , the amplitude and the phase
0 0 0 0
are expressed by A=(x20+v02ω−2)1/2, α=−arctan(vω0 x0).
Equation (1.2) describes the motion of the material point without friction
and free of any external force. When other forces are involved, we have to
modifyequation(1.2)accordingly.Weareledinthismannertodealwiththe
so-called forced oscillations, as opposed to the free oscillations described by
equation (1.2). The new equation will have the form mx¨(t)=−kx(t)+F(t),
with F(t) designating the external force acting on the point at the moment t.
Denoting f(t)=m−1F(t), the equation of forced oscillations becomes
x¨(t)+ω2x(t)=f(t), (1.4)
in which f(t) is the density of the external force, or the force on a unit of
mass of the moving point.
Inmathematicalterms,equation(1.4)forforcedoscillationsofthematerial
point is an inhomogeneous equation of the second order. Using the variation