39133855 Curs Fizica Engleza

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Chapter 1

Mechanics. Oscillations.

1.1

Quantities, Principles and Fundamental Laws in

Classical Mechanics1.1.1. Notions and Basic Quantities in Classical Mechanics

Mechanics studies the simplest form of motion of the matter, the mechanical motion. The mechanical motion is the motion that causes the change in the position of the bodies, ones relative to the others or of their parts in space and in time. Classical Newtonian mechanics studies the motion of the bodies that have muchslower speeds than the speed of the light in vacuum, c=3 10 m/s. Under classicalmechanics, there is a series of fundamental laws (principles): the law of inertia or of the conservation of the impulse, the law of motion of a material point or of variation of the impulse, the law of reciprocal actions, the law of superpo

sition of the forces, the law of gravitation. There are other laws besides the fundamental ones, like those concerning material: the law of elasticity, the lawof friction etc.8


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The principle represents a statement suggested by an observation which meets thecondition that all the consequences that result from its acceptance do not cont

radict the observation. The pattern from which the classical mechanics developsis based on the causality principle and states that, under given initial conditions, a physical process always develops in a particular way. To the same causesthere correspond the same effects. The law represents a mathematical relationship which relates different physical quantities amongst themselves. The main physical quantities presented in this chapter are: speed, acceleration, force, impulse, moment of force, angular momentum, kinetic energy, potential energy, mechanical work. Kinematics is that part of mechanics that establishes the mathematicalequations describing the motion of the bodies, disregarding the cause of the motion. Dynamics is concerned with the study of the causes that produce the motionand establishes the mathematical equations that describe the motion. In mechanics there is introduced the notion of material point. By material point we understand a body of which dimensions can be neglected when its motion is studied. Themotion is studied with respect to a referential system, arbitrarily chosen, as there is no absolutely fixed system of reference relating to which all motions ca

n be studied. Hence, the motion and the rest are relative. The reference systemcan be inertial or non-inertial. The inertial reference system is the system that has a rectilinear uniform motion or is at relative rest. The non-inertial reference system has an accelerated motion. Under classical mechanics and special relativity, the inertial reference systems are considered.


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To the reference system there is rigidly attached a reference frame. The motionof a moving body is univocally determined if, each moment, its coordinates are known in relation with the reference system chosen. In mechanics there are especially used: 1) Cartesian coordinate system 2) spherical coordinate system In theCartesian coordinate system the position of a point P is given by the Cartesiancoordinates x, y and z (Fig. 1.1). The vector r , that connects the origin withpoint P, is called position vector or radius vector.

r

r r r r r = x i + y j + zk

.

(1.1)

z

r zr k

zP r j

r r

r Az k

r A

r O ir x

r y

y

Or Ax i

r Ay j

y

x

x

Fig. 1.1 In general, any arbitrary vector A can be written:

Fig. 1.2

r r r r A = Ax i + Ay j + Az k ,

(1.2)

where Ax , A y and A z are called the components of vector A


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r

(Fig. 1.2).

r According to the components, the length (magnitude) of vector A is:


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r r 2 A = A A = Ax2 + Ay + Az2

(1.3)

r r r i j and k are unit vectors of the coordinate axes. One knows they satisfytherelations:

r r r i = j = k = 1; r r r r r r i i = j j = k k = 1; r r r r r r i j = i k = j = 0O x

z

r

P y

In the spherical coordinates system the position of a point P is given by the spherical coordinates r, , (Fig. 1.3).

Fig. 1.3

Let Oxyz be a Cartesian coordinates system against which the motion o

a material point P is studied. The motion can be determined i

the law o

variation withrespect to time o

coordinates x, y, z o

the point and the initial conditions are known:

x = x(t),vector varies with time:

y = y(t),

z = z(t)

(1.4)

The relations (1.4) are equivalent to the relations indicating how a position

r r r = r (t),

r r r r r (t) = x(t) i + y(t) j + z(t) k .

(1.5)

The relations (1.4) or (1.5) represent the law o

motion o

the material point.

a) SpeedConsider a material point that moves along a trajectory. At the moment t1 the material point is in position P1 and at the moment t 2 is in P 2 . We denote s thelength o

the curve P 1P 2 . The average speed is de

ined by the ratio between the space crossed by the moving body and the time needed

or crossing the space:


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s v m t .

(1.6)

The average speed gives us a rather vague idea about the motion o

the moving body. In order to increase precision, there needs to be introduced the notion o

speed at a particular time, also called instantaneous speed. In point P 1 , the position vector is r ( t1 ) whereas in point P 2 , it is r ( t 2 ). Vector r in Fig. 1.4 represents the di

erence between them/ the displacement. r is the curveP 1P 2 . The ratio r / t is a collinear vector with curve P 1P 2 and de

ines velocity. Velocity is the rate o

displacement. The magnitude o

velocity is speed.I

t tends towards 0, then P 2 tends towards P 1 and curve P 1P 2 tends towards the tangent in

r

r

r

r

P 1 . According to the de

inition o

the derivative, we have:

r r dr r . = lim dt t 0 tBy de

inition, instantaneous velocity vector is:

r r dr v dtr

(1.7)

d r having the orientation o

the tangent line at the curve. Hence, in each point o

trajectory, velocity v has the orientation o

the tangent line at the trajectoryand its direction coincides with the direction o

material point.P2 r r P1 r r (t1 )r r / t

r r (t2 )

P2 P1

r dr / dt

O

Fig. 1.4


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Like any vector, velocity can be written as

ollows:

r r r r v = v xi + v y j + v z k ,

(1.8)

where v x , v y , and v z are the velocity components. From the relations (1.1)and (1.7) it results:

r v=

r r r d (xi + yj + zk ) dt

=

dx r dy r dz r i+ j+ k dt dt dt

(1.9)By comparing the previous two relations, it results:

vx =

dx dy , vy = dt dt

, vz =

dz dt

(1.10)

The length (magnitude) o

velocity is:

v=

v2 + v2 + v2 . x y z

(1.11)

b) AccelerationA change in velocity is called an acceleration. Acceleration is de

ined as the rate o

change o

velocity o

an object with respect to time. Objects are only accelerated i

a

orce is applied to them. The instantaneous acceleration vector is de

ined by the relation:

r r r d v d 2r = a dt dt 2In terms o

acceleration components, we can write:

(1.12)

r r r r a = axi + a y j + az k .

(1.13)

From the relations (1.8) and (1.12) it results:


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r d v r d v y r d vz r j+ k . a= xi + dt dt dtBy comparing the previous two relations, we get:

(1.14)

d vy d 2 y d vx d 2x d vz d 2z = 2 , ay = = 2 , az = = 2 ax = dt dt dt dt dt dt ollowing equation:

(1.15)

We can compute the length (magnitude) o

the acceleration vector using the

r 2 2 a = a x + a y + a z2 .

(1.16)

Let deno e he angen uni -vec or. Considering v = v , we can wri e he rela io

n (1.12) as beside:r

r

r

As is a uni ary vec or, we have:

r

r r d r d (v ) d v r = a= +v . d d d

(1.17)

= 1.By deriva ion wi h ime, he rela ion becomes:

r r

(1.18)

r r r d r r d r d + = 0 = 0. (1.19) d d d r r This rela ion demons ra es he vrs and d / d are perpendicular. Hence, r d / d has he orien a ion of he normal

line a he rajec ory: r r d d r (1.20) = n, d d r n being he normal uni -vec or. Consequen ly, rela ion (1.17) becomes:

d r r dv r a = + v n. d d

reciprocally perpendicular:

(1.21)

This rela ion demons ra es ha he accelera ion vec or has wo componen s,


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- a angen ial componen a he rajec ory, a , de ermined by he speed varia ion wi h respec o ime:

a =

dv , d

(1.22)

- a normal componen a he rajec ory, an , de ermined by he speed varia ion wi h respec o direc ion

an = v

d d

.

P(1.23)r

Considering ha poin s P and P (Fig. 1.5) are very close o each o her, we can wri e :d = d ds d 1 ds v = = , R d R d R

r r r n d

P

r

=

d

O

Fig. 1.5 where R is he curve r dius of he r jec ory in he proximi y of poin

P. Hence, i resul s h :

d r v 2 r r n = v n = n d RBy repl cing (1.24) in (1.21), we ob in:

(1.24)

r d v r v2 r = + n. d RFrom his rel ion i resul s:

(1.25)

= nd


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dv d

(1.26)


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v2 n = . RP r icul r si u ions

(1.27)

1. In line r mo

ion, R nd i

resul

s n = 0. If

he line r mo

ion is lso uniform, v = c nd, ccordingly, is lso null. 2. In circul r uniform mo ion, v =c a = 0, a = an .

c) Mechanical WorkThe physical quan i ies can be classified as follows: 1) s a e quan i ies/parame ers (s a e func ions) depend only on he curren s a e of he sys em, no on he way in which he sys em go o ha s a e; for example: kine ic energy, po en

ial energy, he in ernal energy of a hermodynamic sys em e c.; 2) process quan

i ies/process func ions which depend on he ype of he process, on he pa h followed by he process/ are physical quan i ies ha describe he ransi ion of asys em from an equilibrium s a e o ano her equilibrium s a e. As an example, me

chanical work and hea

are process quan

i

ies because

hey describe quan

i

a

ively he ransi ion be ween equilibrium s a es of hermodynamic sys ems; for ins ance. Le X deno e any s a e quan i y or process quan i y. The infini esimal varia ion of a s a e quan i y is deno ed by dX because i is an exac differen ial.For a process quan i y, deno a ion X is use . Since quantity X epen s on the path followe , X is not an exact ifferential. Consequently, let L an Q enote the elementary work an the quantity of elementary heat respectively.


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r Consi er that upon a material point there is applie a force F that etermines the isplacement of a bo y along a trajectory. Generally, - force canvary with respect to time, F = F(t); - force irection may not coinci e with th

e irection of the material point isplacement.

r The elementary work

one by force F , that applies to the material point r when it isplaces its application point along the istance r , is efine by thefollowing relation

r The total work one by force F when it isplaces its application point from Ato B,is:B

L = F r

r

r

(1.28)

LAB

=

A

r r F r .

(1.29) ) Kinetic EnergyKinetic energy is that part of the mechanical energy etermine by the motion ofa material point, generally of the bo y.

Ec or T enotes kinetic energy.Force is equal to the impulse erivative with respect to time:

r r p r r , p = mv F= tWe calculate elementary work as follows:

(1.30)r r 2 r r r r p r L = t r = p t = (m v ) v = m v . 2

(1.31)

Kinetic energy is efine by the relation:


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mv 2 T Ec . 2by the motion of a material point, generally of the bo y. From the previous tworelations we obtain:

(1.32)

Kinetic energy (energy of motion) is that part of the mechanical energy etermine

L = T .L AB = T = T (B ) T ( A) =A B

(1.33)

In he c se of he mo ion be ween A nd B, by in egr ing rel ion (1.33), we ge

:

m v2 m v2 B A 2 2 ,

L AB = T

(1.34)

Relation (1.34) is the mathematical expression o

the theorem o

the kinetic energy variation. The theorem o

the kinetic energy variation shows that the work done between two positions A and B is equal to the kinetic energy variation o

the material point between the two positions.

e) Conservative ForcesThe conservative

orces are the

orces

or which the work done between two points A and B (Fig. 1.6) does not depend on the path taken by the application pointo

the

orce, they are path-independent. In a more general sense, a conservative

orce is any

orce which may be expressed as a gradient o

a scalar potential.B B r r r r F dr = F dr . A ( C2 )

(C1)

A(C2)

B

Fig. 1.6 (1.35)

A ( C1 )


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I

the working point is displaced along the closed graph A B A, the work done vanishes.

LAB A = 0 .

(1.36)

Hence, the work done by a conservative

orce along a closed graph is zero:(C )

r r F dr = 0 .

(1.37)

By using the Stokes-Ampre theorem, A dr = rotA dS , we obtain(C ) S

rr

r

r

curl F = 0 . are: gravitational

orce, electric

orce, elastic

orce.

r

(1.38)

The curl o

a conservative

orce vanishes. Examples o

conservative

orces The non-conservative

orces are the

orces

or which the work done between two pointsdepends on the path. For example:

riction and magnetism.

) Potential EnergyPotential energy is that part o

mechanical energy that depends on the system con

iguration, which is on the position o

the particles in a

orce

ield. Potentialenergy depends on the system coordinates. We can speak only o

the variation o

the potential energy. For instance, this variation can be turned into kinetic energy or vice versa. As a result, the choice o

the point where U = 0, called thezero-point o

the

potential energy or re

erence point, is arbitrary. In some cases, the choice o

acertain zero-point o

the potential energy is more convenient. The choice o

anyother zero-point o

the potential energy gives the same result.


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The potential energy can be uniquely de

ined only

or the conservative

orces. Let us consider material point in a conservative

orce

ield. I

the curl o

a vector

ield is null, then this vector

ield results

rom a scalar

ield by applying the

r gradient (rot A = 0

r A = grad ) since rot grad = 0,

or any . r With the conservative

orces we obtain rot F = 0 .r F = gr dU ,

By defini ion, he sc l r func ion from which F resul s is he po en i l energy:(1.39)

nd we h ve F = gr d U = x i + y j + z k . We compute elementary work:

r

U r

U r

U r

r r r r r U r U r U r L = F r = k x i + y j + z k = j+ i + z z = dU

(

)

an we obtain

L = dU

(1.40)

If he pplic ion poin of he force is disclosed from A o B, in egr ing herel ion (1.40) we ge :B

L A B = dU = U ( A) U (B ) .A

(1.41)

Le poin A be rbi r rily design e he zero poin of he po en i l energy, h is we h ve U(A) = 0, Lpoin ref .B = U (B ) nd we ge


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U (B ) = L

B poin ref

=

poin ref . B

r r F dr

(1.42)

The po en i l energy of m eri l poin , in ny posi ion B from conserv iveforce field, is equ l o he work done by he forces of he field in order o disclose

he m

eri l poin

from

he considered posi

ion

o

he reference poin

.

Ex mples

1) El s

ic Po

en

i l Energy We

ke

he equilibrium posi

ion s reference poin

:U(0) = 0.

U ( x ) = F dx = k x dx = k x , 2 x x2

0

0

nd

U (x ) = k x 22) Elec ric l Po en i l Energy

2

(1.43)

According o Coulombs l w, he in er c ion force be ween wo elec ric ch rges Q nd q is:

r Qq r F= r 4 r3W tak th r f r nc oint to infinity : U(r = ) = 0.

(1.44a)

r r Qq 1 r r Qq r dr = U (r ) = F dr = 4 r3 4 r r

1 r

Qq r 2 dr = 4 r . (1.44b)


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Consid ring that th l ctric ot ntial at distanc r from charg Q is giv n byth r lation

V (r ) =on g ts:

Qq , 4 r

(1.44c)

U (r ) =

Qq = qV (r ) . 4 r

(1.44d)

3) Gravitational Pot ntial En rgy According to th univ rsal attraction law, th

int

raction forc

b

tw

n two mass

s m1 and m2 (Fig. 1.7) is:m1

r F

r F m 2

Fig. 1.7

r mm r F = K 13 2 r, r

(1.45a)

wh

r

K is th

univ

rsal gravitational constant. L

t b

th

r

f

r

nc

oint: U(r=

) = 0.r r U (r ) = F dr = K m1 m2 12 dr = K m1 m2 1 , r r r r(1.45b)

U (r ) =

K m1 m2 r

(1.45c)


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For th bodi s in th gravitational fi ld from Earth, it is conv ni nt to hav z ro oint of th ot ntial n rgy at th Earth surfac (Fig. 1.8): U(r = R) = 0.R r U (h ) = F dr = K m M 1 dr = K m M 1 1 2 R R+h R+h R+h r Rr

(

)

(1.45d)

U (h ) = K m M

h . R R+h

(1.45e)

For low heights we have: h


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1.1.2 Fundamental Principles of Classical Dynamics1) The Law of Inertia A material point tends to stay in a rectilinear uniform motion or at relative rest unless acted upon by an (external) force. 2) The Fundamental Law of Dynamics In classical mechanics, mass is constant: m = const.

r r r r dp d (m v ) dv r F= = =m = ma , dt dt dt

r r F = ma .

(1.47a)

Statement. If a force F acts upon a body, it applies an acceleration directly

r

proportional to and having the same orientation with F and inversely proportional with its mass. r r r If F = 0 a = 0 v = cons .

r

By using he componen s of force F , we ob ain:

r

r r r r r r d 2r F = m 2 ; F = Fx i + Fy j + Fz k d d 2 r d 2x r d 2 y r d 2z r= i + 2 j+ 2 k d d 2 d 2 d


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r r r d 2x r d 2 y r d 2z r Fx i + Fy j + Fz k = m( 2 i + 2 j + 2 k ). d d d r r r Iden ifying he coefficien s of i , j , and k i resul s d 2x Fx = m 2 , d d2y Fy = m 2 , d d 2z Fz = m 2 . d

(1.47b)

(1.48)

(1.49)

(1.50)

The rela ions (1.48), (1.49) and (1.50) are equa ions of mo ion. They demons ra

e ha any mo ion can be decomposed in hree linear mo ions by hree perpendicular direc ions. The solu ions o hese equa ions represen he law ofmo ion of he ma erial poin :

x = x(

),y = y ( ),

z = z ( ).

(1.51)

By removing ime from rela ions (1.51) we ob ain he equa ion of he rajec ory.3) The Law of Reciprocal Ac ions If one body ac s upon ano her body wi h a forc

e, he second one reac s wi h an equal force in magni ude bu opposi e in direc

ion, calledreac ion.


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4) The Law of he Superposi ion of Forces If more forces ac upon a ma erial poin , each force ac s independen ly of he o hers.

1.1.3. Laws of Conserva ion1) The Law of Conserva ion of Momen um

The ma hema ical expression is

r r dp r r F = ; F = 0 p = cons . d

(1.52)

S a emen . An applied force is equal o he ra e of change of momen um.

2) The Law of Conserva ion of Angular Momen um

The angular momen um of a par icle wi h respec o a fixed poin , called

r r r r r L = r p = r mv . r r For a circular mo ion, r p and we have:

origin, is defined by he rela ion:

(1.53a)

L = m v r sin 90o = m v r. (1.53b) r r The momen of force ( orque) M , wi h rpec o he same fixed poin isdefined by he rela ion:

r r r M =r F.

(1.54)


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posi ion vec or r , if he origin of he coordina e axes is chosen in he cen erof he field. In his si ua ion we have:

r

r r r r r F = f (r ) r , M = r f (r ) r = 0.

(1.56b)

Examples of cen ral force fields: gravi a ional field, elec ric field crea ed byan elec ric charge, e c.

3) The Law of Conserva ion of Mechanical Energy

From he rela ions (1.33) and (1.40) one ob ains:

L = T;

L = U; T + U = 0;T + U = const. (T + U) = 0;(1.57)

The statement of the law of mechanical energy conservation: on con ition conservative forces act upon a material point, the sum of the kinetic energy an potential energy is constant with respect to time.

The systems upon which only conservative forces act are calle conservative systems.

The systems upon which non-conservative forces can act as well, are calle issipative systems. In this situation, the mechanical energy ecreases in time, turning into other forms of energy, like thermal energy.Views upon the Conservation Laws

The conservation laws are a result of the fact that Eucli ian space is homogeneous an isotropic an time is uniform. The free space is homogeneous, in other wor s, it oes not iffer from one point to another. If a figure is isplace , without rotation, from one place to another,


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no change in its size or its geometrical properties occurs. Furthermore, the physical properties (inertia, internal forces) of a bo y o not change if that bo yis isplace to another point in space. Therefore, the geometrical an physicalproperties are invariant with the space isplacement of the object. The space i

s isotropic, that means that all the irections are equivalent. The geometricalan

physical properties of an object

o not change if we rotate it in space. That is to say the geometrical an physical properties are invariable with rotation. Time is uniform. In other wor s, the laws of motion of any given system applyin epen ently from the origin of time. For instance, Coulombs law or the gravitational law is the same at any time. The space homogeneity etermines the conservation of momentum. The space isotropy lea s to the conservation of the angular momentum as the time uniformity lea s to the energy conservation.


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1.2. Oscillations1.2.1. Classification

A system is in stable equilibrium on con ition it has minimum potential energy.If the system is isplace from the stable equilibrium position, there applies aforce that ten

s to bring back the system into its initial position. For short isplacement from the equilibrium position, this force is elastic: F = kx.The oscillatory motion is the motion of a bo y, of a system, from one si e to the other of the equilibrium position. A more general efinition is the following:The oscillatory motions are those motions that repeat perio ically or quasiperi

o ically in time.

A harmonic oscillator consists in a system which, when isplace from its equilibrium position, experiences a restoring force F proportional to the isplacementx, F = kx where k is a positive constant, the spring constant. When there is a

single force F acting on the system, the system is calle asimple harmonic oscillator. The system un ergoes simple harmonic motion: sinusoi

al oscillations about the equilibrium point, A constant amplitu

e ischaracterizing the motion an there is also a constant frequency (which oes not epen on the amplitu e). Amplitu e is maximal isplacement from the equilibrium. Frequency represents the number of cycles the system performs per unit time,an is = 1T

.

I the prese ce of a frictio al force (dampi g) proportio al to the velocity, the harmo ic oscillator is described as damped. I this case, the freque cy of the


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oscillatio s is smaller tha i the o -damped case. oscillatio s is ot co sta

t, a d decreases with time.

The amplitude of the

If a

exter

al time-depe

de

t force is acti

g o

the oscillator, the harmo

ic oscillator is described as drive . Examples of oscillatio s: pe dula (with small a gles of displaceme t), masses co ected to spri gs a d acoustical systems, electrical harmo ic oscillators (a LC circuit, a RLC circuit). I periodic oscillatio , the values of all physical qua tities characteristic of the oscillatio process repeat at equal time i tervals. The mi imum time i terval after which thisvalue repeats is called period, T. Period represe ts the duratio of a completeoscillatio or period is the umber of cycles as a result of time (time/cycle) or the time it takes the system to complete a oscillatio cycle. Period is alsothe i verse of freque cy. The (oscillatio ) freque cy, v, is defi ed by the relatio v = 1/T a d represe ts the umber of the complete oscillatio s do e withi the time u it. I quasi-periodic oscillatio s, o ly a part of the physical qua t

ities get values that repeat at equal time i

tervals. Accordi

g to the physicalcharacter of the oscillatio , there are: a) mecha ical oscillatio s, whe the ki etic e ergy is tur ed i to pote tial e ergy a d vice versa, like the oscillatio s of a pe dulum, vibratio s of a stri g, of a membra e etc.; b) electromag etic

oscillatio s, whe electric e ergy tur s i to mag etic e ergy a d vice versa, like the oscillatio s from a oscillatio circuit c) electromecha ical oscillatio s, whe electric e ergy tur s i to mecha ical e ergy or vice versa.


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1.2.2. No -Damped/Simple Harmo ic Oscillatio s

This type of motio is ge erated i a elastic force field, i the abse ce of frictio . The object (system) is displaced from its equilibrium positio a d leftfree. Let us co sider a mass m, u dergoi g such type of motio alo g axis Ox, a

d the origi

of the coordi

ate system which ide

tifies with the equilibrium positio . Displaceme t of mass m about the equilibrium positio at a give poi t isx . The simple harmo ic oscillator has o drivi g force, a d o frictio (dampi

g), so the oscillator experie ces the elastic force:

F = k x,

(1.58)

where k is the spri g co sta t, he ce the motio equatio of the material poi t.Accordi g to the fu dame tal law of dy amics, o e has:

r d 2x r d 2 y r d 2z r r F = m a = m( 2 i + 2 j + 2 k ) dt dt dtAs y = 0, z = 0, o e gets:

(1.59)

d 2x F =m 2 . dtCombi i g the relatio s (1.58) a d (1.60) o e obtai s:

(1.60a)

d 2x d 2x k k x = 2 ; 2 + x = 0. dt m dtBy th d notation:

(1.60b)


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02 =one obtains:

k m

(1.61)

d 2x + 02 x = 0 . dt 2

(1.62)

E uation (1.62) is the differential e uation of the non-damped/simple harmonic oscillator. It is a 2nd order homogeneous linear differential e uation. The so called characteristic e uation is:

r 2 + 02 = 0 r1, 2 = i 0 .The solution to the e uation (1.62) is:

(1.63a)

x = B1e i t + B2 e i t .0 0

(1.63b)

Using Eulers formulas e

i x

= cos x i sin x , one gets:(1.63c)

x = ( B1 + B2 ) cos 0 t + i ( B1 B2 ) sin 0 t .We denote:

C1 = B1 + B2 C2 = i ( B1 B2 )and on has:

x = C 1 cos 0 t + C 2 sin 0 tand substitute:

(1.63d)

C1 = A sin ;it results:

C2 = A cos ,(1.63e)

x = A(sin cos 0 t + cos sin 0 t )and

x = A sin(0 t + ) .

(1.63f)


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This relation represents the la of the non-damped/simple harmonic oscillatory motion. Alternatively, ith the formula sin = cos( /2), th r lation (1.63f) can b writt n as follows:

x = A cos( 0 t + 1 ) 1 =

2

,

(1.64)

wh r x is th dis lac m nt, A is th am litud that quals th maximum dis lac m nt.

0 is the angular fre uency of the solution, as it is determined only

by its characteristics, (measured in radians per second).

0 t + is the phase, hile

i itial phase (at t = 0).The period T0 is the mi imum time duri g which the material poi t stays i the same positio , has the same speed, i terms of value a d directio (therefore, the duratio of a complete oscillatio ). We have x(t) = x( t + T0 ), Asi ( 0 t + )

= Asin[ 0 (t + T0 ) + ], 0 (t+ T0 ) + = 0 t + + 2 , T0 = 2/ 0 .Using relation (1.61) one obtains:

T0 =

2

0

= 2

m . k

(1.65)

Th fr qu ncy (m asur d in h rtz), is giv n by:

0 =

1 . T0

(1.66)

We determi e the speed a d acceleratio of the material poi t u dergoi g harmo ic oscillatio s (oscillatory motio ) as follows:

v = vx =

dx = 0 A cos(0 t + ) , dt

(1.67)


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a = a x = 02 A sin(0 t + ) = 02 x .the displacement.

(1.68)

In simple oscillations, acceleration is proportional and inversely directed to We determine kinetic, potential and total energy of the oscillator at any t timeas follo s:

mv2 1 T= = m02 A2 cos 2 (0 t + ) , 2 2 k x2 1 U= = m02 A2 sin 2 (0 t + ) , 2 2

(1.69) (1.70) (1.71)

1 1 E = T + U = m02 A2 = k A2 . 2 2

The total energy is constant relative to time. Conse uently, the simple harmonicoscillator is a conservative system.

The Complex Numbers and the Harmonic Oscillation

The displacement of the harmonic oscillatory motion is represented by the complex number:

x = Aei ( t + ) , x = A[cos(0 t + ) + i sin(0 t + )] .0

(1.72)

Under these conditions, the follo ing convention is made: the displacement is given by the imaginary part of this complex.


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The Vectorial Representation of the Harmonic Oscillation

Any harmonic oscillation can be graphically

r represented, by a rotating vector A , called phasor(Fig. 1.9). The maximum displacement is given by the phasor s length and the phasor rotates in

y

trigonometric (counterclock ise) direction at an angular frec uency

O

t B

r A(t ) r A(0)x

0 . The phasor travels ithFig. 1.9

velocity 2 A / T, which is th maximum s d

of th oscillator. So, th angl it mak s with th x axis giv s th has angl .At t = 0, th has denotes the angle made by the phasor with axis Ox, while atany t time, we have

0 t + . The projection B o

the terminal point o

the

phasor on axis Ox (as well as the projection on Oy) describes a harmonic oscillation. The vector method (phasor) is mainly used in the study o

the two parallel/orthogonal harmonic oscillations o

equal

requency.

1.2.3. Particular Case o

Simple/non-Damped Harmonic Oscillation. Simple Pendulum (Bob Pendulum).

Simple (mathematical) pendulum consists o

a weight m attached to a massless inelastic wire (rigid rod). Because o

an initial push, it will swing back and

orth under the in

luence o

gravity over its central point. We assume that the bobis a point mass and motion occurs in a 2 dimensional plane (Fig. 1.10). The harmonic oscillations o

the pendulum occur

or small angular displacements about the equilibrium point, namely < 4 (sm ll ngle pproxim ion). The mo ion is c used

by he ngen componen of G , G :

r


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G = m g sin m g , he bob c uses decre se in ngle. We pply he fund men l l w of dyn mics:

(1.74 )

wi

h minus bec use

he gr vi

ion l force on

l

G B Gn

G = m = m

dv . d

(1.74b)

A

Fig. 1.10 We c lcul

e line r cceler

ion

(AB = s):

ds d dv d 2 =l a = =l 2 , s = l v = d d d d d 2 G = ml 2 . d

From he wo expressions of G one ge s:

(1.74c) (1.74d)

d 2 d 2 g m g = ml 2 2 + = 0 . d d lLe

us deno

e:

(1.74e)

g d 2 2 = 0 2 + 02 = 0 . l d

d mped h rmonic oscill or, (1.62), o he new equ ion is:

(1.74f)

We ge n equ ion iden ic l wi h he differen i l equ ion of he simple/non

being repl ced by x. Therefore he solu ion(1.74g)

= 0 sin(0 t + )and the period is given by:


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T0 =

2

0

= 2

l . g

(1.75)

A llications

According to M. Schul r (1923), a ndulum whos riod xactly quals th orbital riod of a hy oth tical sat llit orbiting just abov th surfac of th arth (about 84 minut s) will t nd to r main ointing at th c nt r of th arth wh

n its su

ort is sudd

nly dis

lac

d. This r

r

s

nts th

basic

rinci

l

of Schul r tuning that has to b includ d in th d sign of any in rtial guidanc syst m that will b o rat d n ar th arth, such as in shi s and aircraft. B caus of th valu s of th gravitational acc l ration g in th quation (1.75) th ndulum fr qu ncy is diff r nt at diff r nt lac s on arth. If w consid r an accurat ndulum clock in Glasgow (g = 9.815 63 m/s2) and tak it to Cairo (g = 9.793 17 m/s2), w must short n th ndulum by 0.23%. Doubl ndulum: in horology, a doubl ndulum r r s nts a syst m of two sim l ndulums on a common mounting which mov in anti has . in math matics (dynamical syst ms), a doubl ndulum is a ndulum that has anoth r ndulum attach d to its nd. This syst m is a sim l hysical syst m that xhibits rich dynamic b havior. A s t of cou l d ordinary diff r ntial quations d scrib th motion of a doubl ndulum. Also, for a valu of n rgy gr at r than a c rtain on its motion is chaotic. A ro

rty of th

doubl

ndulum is that it und

rgo

s chaotic motion, and shows a s nsitiv d nd nc on initial conditions. Th imag from Fig. 1.11 shows


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32

th amount of la s d tim initial conditions *.

b for th ndulum "fli s ov r", as a function of

Fig. 1.11*

1.2.4. Dam d Harmonic Oscillations

Th harmonic oscillator is known as dam d in th r s nc of a frictional forc (dam ing) ro ortional to th v locity. Th fr qu ncy of th oscillations is sm

all r than in th non dam d cas , and th am litud of th oscillations is notconstant and d cr as s with tim . Dam d oscillations ar thos oscillations with r duc d am litud du to th dissi ation of n rgy, und r th action of friction forc s. Th friction forc acting u on th oscillator d nds u on th giv nmotional conditions. L t us consid r th oscillator moving through fluid. In this cas , at r lativ ly low s d, th body is subj ct to a frictional forc ro ortional to v locity, as in th Fig. 1.12:


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34

and e denote

m

= 2 an

02 =k , here is calle amping factor an 0 is called the m

(undamped) natural fre uency (angular fre uency of the oscillator in the absenceof friction) of the system. Both parameters represent angular fre uencies and h

ave for units of measure radians per second. Using these denotations, the e uation becomes:

d 2x dx + 2 + 02 x = 0 . 2 dt dthomogeneous linear differential e uation ith constant factors. To the e uation(1.78) e attach the e uation:

(1.78)

E uation (1.78) is the differential e uation of oscillations. It is a 2nd order

r 2 + 2 r + 02 = 0 .The solutions to the e uation (1.79) are:

(1.79)

r1, 2 = 2 02 .We distinguish t o situations. a) If the medium resistance is high, the factor

(1.80)

ets a hi h value, so that

2 02 > 0 . In this case, solutions r1, 2 are real and the solution to the differentiale uation (1.78) becomes:

x = A1e (

2 2 0 ) t

+ A2 e(

+2 2 0 ) t

(1.81)x

According to relation (1.81), displacement decreases exponentially ith time. The motion is no longer periodical. The body, no longer in

t

e uilibrium, returns asymptotically to it, ithout exceeding it.

Fig.1.13


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35

In this case, the motion is called aperiodic (Fig. 1.13). The system is said tobeover-damped. An overdamped door-closer ill take longer to close the door than a

critically damped door closer. When

2 02 = 0 , is real an the system is critically ampe . An example of

critical amping is the oor-closer of many hinge oors present in public buil ings. b) If the me ium resistance is low,

2 02 < 0 . In this case, r1, 2 are complex:(1.82)

r1, 2 = i 02 2 .We enote:

= 02 2 .The solution to (1.78) becomes:

(1.83)

x = A1e ( +i ) t + A2 e ( i ) tand one obtains:

(1.84)

x = e t ( A1e i t + A2 e i t ) .Using Eulers formula, one gets:

(1.85)

x = e t [( A1 + A2 ) cos t + i ( A1 A2 ) sin t ] .By substitution:

(1.86)

A1 + A2 = A0 sin , i ( A1 A2 ) = A0 cos .The displacement becomes:

(1.87)

x = A0 e t (sin cos t + cos sin t )and one has:

(1.88)

x = A0 e t sin( t + ) .

(1.89)


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In this case, the system is under-damped. In this case, the system oscillates atthe damped fre uency, hich is a function of the natural fre uency and the damp

ing factor. Relation (1.89) represents the la of the damped oscillatory motion.The amplitude is:

A(t ) = A0 e t .(1.90)

The amplitu e is not constant, but ecreases exponentially (Fig. 1.14). With asthe damped fre uency the period T can be defined as the minimum time interval bet een t o successive moves of the oscillator through the e uilibrium point, in the same direction:x

2 = 2 T= 0 2

2t

(1.91)

Fig. 1.14 and th riod of th oscillations is bigg r than in th non dam d cas . Th dam d oscillations ar also d t rmin d by th following quantiti s: i)Th dam ing ratio (d cr m nt of dam ing),

defined

y the relation:(1.92)

A0 e t A(t ) = ln = ln = ln e T = T ( t +T ) A(t + T ) A0 ean one gets:

= T .The stronger the amping is, the higher

(1.93)

gets.

ii) Equili

rium time of motion, , is he ime rel ive o which he mpli ude de

cre ses e imes.


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A( ) A( + ) = , e

A0 e

( t + )

A0 e ( t ) = e = e 1 = 1 e

=l w reg rd o s below:

1

.

(1.94)The smaller is, he s ronger he d mping ge s. We c n develop he mo ion

x = A0 e / sin( t + ) .ii) The uality (performance) factor, Q is defined by the relation:

(1.95)

Q = 0 .mo ion we h ve:

(1.96)

This rel ion is v lid for n oscill or which is no s rongly d mped. In periodic

0 Q = 0 =In periodic mo ion:

0 1.

(1.97)

< 0 Q =

0 > 1.

(1.98)

The higher Q is, the more weakly the motion gets ampe . Accor ing to relation (1.89), oscillations stop after an infinite time. In reality, when amplitu e becomes comparable to interatomic istance, we cannot speak of an oscillation of thewhole bo y with such an amplitu e. Thus, actually, the oscillations stop after

a efinite time. In the ampe oscillations, the energy that the oscillator nee s in or er to be isplace from the equilibrium point gra ually turns into thermal energy ue to friction. The ampe oscillator is a issipative system.


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38

Analogy between the mechanical case an the electrical case, series RLC circuit:- force F ( N ) voltage U (V ) ; - spee v (m/s) current intensity I ( A) ; -

isplacement x ( m) electric charge q (C ) ; - amper constant ( k / s ) resistance R () ; mass m ( k ) inductance L ( H ) ; sprin constant k ( N / m) inverse of capacitance (elastance)

1 C

(1 / F ) ;

1 2

k m

1 2

1 LC

.

Analogy

et een the mechanical case and the electrical case, arallel RLC circuit: force F (N ) voltage U (V ) ; s eed v (m/s) du dt ;

dis lacement x (m) voltage U (V ) ; dam er constant ( k / s ) conductance mass m (k ) capacitance C (F ) ; sprin constant k ( N / m) susceptance 1 2 k

m 1 2 1 LC . 1 L (1 / H ) ; 1 R ( ) ;1


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1.2.5 Driven Harmonic Oscillations. Resonance.

In order to prevent the decrease of amplitude, due to ener y dissipation, external ener y needs to be communicated to the oscillator. This can be achieved if aperiodic external force acts upon the oscillator. In driven oscillations, the followin

forces act upon the oscillator: elastic force Fe , resistance force Fr ,and a periodic force, Fext = F0 sin( t + ) , here

is the angular fre uency of the

external force. The oscillations are called driven, but they are also damped. Inthe first moments, after the application of the external force, the oscillation

s are not stationary, in other ords, they do not have a constant amplitude andpulsation. Mean hile, the amplitude and pulsation vary according to a complicated la , the oscillations being under transient conditions. In time, the steady-state conditions are reached. We determine the e uation of motion and the la of motion:

Fe + Fr + Fext = ma ,d 2x dx m 2 + + k x = F0 sin( t + ) , dt dt F d 2 x dx k + + x = 0 sin( t + ) dtm dt m mand e denote

(1.99) (1.100) (1.101)

m

= 2 an 02 =

k ,

here is calle

amping factor an

0 is called the m(undam ed) natural frequency (angular frequency of the oscillator in the a

senceof


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friction) of the system. Both arameters re resent angular frequencies and havefor units of measure radians er second. We o

tain:

F d 2x dx + 2 + 02 x = 0 sin( t + ) . dt 2 dt m

(1.102)

This is the differential e uation of the driven harmonic oscillations. It is anunhomogeneous 2nd order linear differential e uation, ith constant factors. The

solution to this type of e uation is the sum of the homogeneous e uation solution and a particular solution, like the second members, x = xom + x part . The solution to the homogeneous e uation is: xom = A0 e The particular solution is: x =

A sin( t + ) . In time, due to the damping, xom 0 and, as a result, the solutionto e uation (1.102) ill be:

t

sin( 1 t + ) .

x = x part = x = A sin( t + ) .

(1.103)

Since the moment e can have such a solution (1.103), the oscillator is under steady-state conditions. Under these conditions, the oscillations have a fre uency

e ual to the fre uency of the external force. Considering (1.103) a solution tothe e uation (1.102), e can determine the constants A and .

dx = A cos( t + ) , dtd 2x = 2 A sin( t + ) . 2 dtIntroducing (1.103), (1.104) and (1.105) in (1.102) it follo s:

(1.104) (1.105)

(02 2 ) A sin( t + ) + 2 A cos( t + ) =

F0 sin( t + ) . m

(1.106)


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By developing sinus and cosinus and e uating the coefficients of sin t and

cos t it follo s:A(02 2 ) cos 2 A sin = F0 , m(1.107) (1.108)

A(02 2 ) sin + 2 A cos = 0 .From relation (1.108) one gets:

tg =

2 . 2 02

(1.109)

From the multi lication of (1.107)

y cos and (1.108) by sin and sum of the ne relations, one obtains:

A=We develop cos :

F0 cos . m(02 2 )

(1.110)

cos =

1 1 + tg 2

=

( 2 02 ) ( ) + 4 2 2 2 0 2 2

.

(1.111)

Com

ining (1.110) and (1.111) one o

tains:

A=

F0 m (02 2 ) 2 + 4 2 2

.

(1.112)

According to relations (1.109) and (1.111), the am litude, A and the initial hase,

of the driven oscillations depend on pulsation of the external force.r . The phenomenon of developing

The angular fre

uency value/modulus of the external force

ith maximum amplitudeis called angular resonant fre uency


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a driven oscillation of maximum amplitude hen

= r , is called resonance. The

angular resonant fre uency is obtained from the maximum condition:

dA = 0 4 ( 2 02 + 2 2 ) = 0 and e get

(1.113)

r = 02 2 2 .relative to resonance:

(1.114)

Embe ing (1.114) in (1.112) we etermine the value of the amplitu e

Ar =

F0 2 m 2 0 2

.

(1.115)

The amplitu e with regar to resonance is as higher as the amping factor/coefficient,

is lower. If the me

ium resistance is null, = 0, r = andA

Ar . This situation does not actuallyoccur, as the medium resistance al ays interferes. The curves, as a result of the gra h re resentation of the am litude relative to the angular frequency of the

external force, are called resonance curves. (Fig. 1.15). An exam le of resonance in Fig. 1.16.

=0

2 < 1

r r1

2

0

1

Fig. 1.15


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Fig. 1.16 The Tacoma Narro s Bridge (sho n t isting) in Washington colla sed s ectacularly, under moderate ind, in art

ecause of resonance*. In hysics, resonance is the tendency of a system to a

sor

more energy hen the frequency of its oscillations matches the system's natural frequency of vi

ration (its resonantfrequency) than it does at other frequencies. Exam les of resonance: the acoust

ic resonances of musical instruments, the tidal resonance, or

ital resonance asexem lified

y some moons of the solar system's gas giants, the resonance of the

asilar mem

rane in the

iological transduction of auditory in ut, and resonance in electronic circuits. 1) Acoustic resonances. Strings under tension (lutes,har s, guitars, ianos) have resonant frequencies hich are directly connected to the mass, length, and tension of the string. The avelength that corres onds to the first resonance on the string is equal to t ice the length of the string.For higher resonances corres ond avelengths that are integer divisions of the fundamental avelength. 2) Tidal resonance. Tidal resonance (oceanogra hy) occurs hen the time it takes for a large ave to travel from the mouth of the

ay tothe o osite end, then


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44

reflect and travel

ack to the mouth of the

ay, equals the time from one high tide to the next. 3) Or

ital resonance a ears in the case of some moons of the solar system's gas giants. When t o

odies have eriods of revolution that are asim le integer ratio of each other a mean motion or

ital resonance a ears. De ending on the conditions, the or

it can

e either sta

ilized or desta

ilized. When the t

o

odies move in a synchronized fashion and they never closely a

roachsta

ilization occurs. There are four gas giants in our solar system's: Ju iter,Saturn, Uranus, and Ne tune (Fig. 1.17). Uranus and Ne tune are a se arate su

class of giant lanets, 'ice giants', or 'Uranian lanets',

ecause they are mostly com osed of ice, rock and gas, unlike the "traditional" gas giants Ju iter orSaturn. They share the same qualities of the lack of the solid surface; their differences stem from the fact that their ro ortion of hydrogen and helium is lo

er,

ecause they are situated at greater distance from the Sun.

Fig. 1.17 From to : Ne tune, Uranus, Saturn, and Ju iter (sizes not to scale)*.


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4) Resonance of the

asilar mem

rane in the

iological transduction of auditoryin ut. The mem

rane is ta ered and it is stiffer at one end than at the other. Because of this, there is a sound in ut (frequency) that makes to vi

rate a articular location of the mem

rane more than other locations due to the hysical ro erty of resonance. Georg von Bksy (No

el Prize) sho ed in ex eriments that high frequencies lead to maximum vi

rations at the

asal end of the cochlear coil (narro mem

rane), and lo frequencies determine maximum vi

rations at the a ical end of the cochlear coil ( ide mem

rane). 5) Resonance in electronic circuits. Inan electrical circuit, resonance a ears at a articular frequency hen the inductive reactance equals the ca acitive reactance. This determines electrical energy to oscillate

et een the magnetic field of the inductor and the electric field of the ca acitor. An analogy is re resented

y the mechanical endulum. At resonance, the series im edance of the t o elements is at a minimum and the arallel im edance is a maximum. Resonance has a lications in tuning and filtering,

ecause resonance a ears at a articular frequency for given values of inductance

and ca acitance. Resonance can roduce un anted sustained and transient oscillations in electrical circuits that may cause noise, signal distortion, and damage

to circuit elements.*


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1.2.6. Mechanical Oscillations

a) T o Parallel Harmonic Oscillatory Motions of Equal Frequency

Let us consider a material oint undergoing t o harmonic oscillatory motions ofequal frequency along axis Ox:

x1 = A1 sin( t + 1 ) , x2 = A2 sin( t + 2 ) .The displacement of the resulted motion is given by:

(1.116)

(1.117)

x = x1 + x2 = A1 sin t cos 1 + A1 cos t sin 1 + + A2 sin t cos 2 + A2 cos t si= = ( A1 cos 1 + A2 cos 2 ) sin t + + ( A1 sin 1 + A2 sin 2 ) cos tWe denote: . (1.118)

B1 = A1 cos 1 + A2 cos 2 , B2 = A1 sin 1 + A2 sin 2 .One gets:

(1.119)

x = B1 sin t + B2 cos t .By substitution:

(1.120)

B1 = A cos ; B2 = A sin and the displacement becomes:

(1.121)

x = A sin( t + ) .

(1.122)


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The resulted motion is a harmonic oscillatory motion as ell. Let us determine amplitude A and the initial phase of the resulted motion:

A cos = A1 cos 1 + A2 cos 2 , A sin = A1 sin 1 + A2 sin 2 .By dividing these relations, it follo s that:

(1.123)

(1.124)

tg =

A1 sin 1 + A2 sin 2 . A1 cos 1 + A2 cos 2

(1.125)

S uaring and summing relations (1.123) and (1.124), e get:

A2 = A12 + A22 + 2 A1 A2 cos( 2 1 ) .

(1.126)

The vector method leads to the same result. The vector representation (Fig. 1.18) is useful mainly in the composition study of more oscillations, of similar fre uency and direction. In this case, the polygon rule applies: a vector polygon is dra n; the vector that closes up the polygon outline represents the resulted oscillation (Fig. 1.19).y

r A2 r A1

r A

y

r A

r A3

O

x

O

r A1

r A2x

Fig. 1.18

Fig. 1.19


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48

Particular Cases

1) If

2 1 = 2 k , from relation (1.126) it follo sA = A1 + A2(1.127)

The resulted am litude equals the sum of the com onent am litudes. The t o oscillations are considered in hase (Fig. 1.20).x x

t

t

Fig. 1.20 2) If

Fig. 1.21

2 1 = 2 (k + 1) , from relation (1.126) one gets:A = A1 A2 .(1.128)

The oscillations, in this case, are in hase o osition (Fig. 1.21). 3) If

2 1 = 2 (k + 1) / 2 , from relation (1.126) it results:A2 = A12 + A22 .(1.129)

The t

o oscillations are in a square relationshi

.

) T o 0rthogonal Harmonic oscillatory Motions of Same Frequency

Let us consider that a material oint simultaneously undergoes t o harmonic oscillations, one along Ox, the other along Oy:

x = A sin t ,

(1.130) (1.131)

y = B sin( t + ) .

In order to obtain the trajectory e uation, e leave out the time from the relations.


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49

As a result, relation (1.130) becomes:

x x2 2 sin t = cos t = 1 sin t = 1 2 A ACom

ining (1.131) and (1.132) one o

tains:

(1.132)

y = B sin t cos + B cos t sin ,

x2 x y = B cos + B 1 2 sin , A A x x2 y B cos = B 1 2 sin . A ABy s uaring this relation, one gets: (1.133)

y2 x2 2 x y + 2 cos = sin 2 . 2 B A ABmotion (Fig. 1.22).y D1 D2 x

(1.134)

The trajectory is an ellipse. The consecutive motion is an elliptical periodical

Fig. 1.22Particular Cases

1) If

2 1 = 2 k , from relation (1.134) it follo s:


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y2 x2 2 x y x y + 2 =0 = . B2 A AB A Brepresen s he firs diagonal from he ampli ude rec angular. 2) If

(1.135)

The consecu

ive mo

ion is a harmonic oscilla

ory mo

ion along line D1

ha

2 1 = 2 (k + 1) , from relation (1.134) one gets:x y = . A B(1.136)

The consecutive motion is a harmonic oscillatory motion along line D2 . 3) If

2 1 = 2 (k + 1) / 2 , from relation (1.134) it results:y 2 x2 + = 1. B 2 A2(1.137)

The motion is

eriodical on an elli

se around an axis. In

articular, if A = B ,the elli se

ecomes a circle:

x 2 + y 2 = A2 .harmonic oscillations, of similar

(1.138)

Reci rocally, any circular eriodical motion can

e decom osed in t o cross

a d A , havi g the phase differe ce

2 (k + 1) / 2 .If the frequencies are different, the consecutive trajectory has a more com

licated form. If the frequency ratio is a rational num

er,

1 / 2 = 1 / 2 ,

1 , 2 N what we get are closed trajectories, called Lissajous figures.


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Chapter 2Waves2.1 Defi itio . Classificatio . Physical Descriptio of a Wave. Wave Equatio . Propagatio of Waves. E ergy of Waves. Wave I te sity.2.1.1 Defi itio . Classificatio .A wave represe

ts a disturba

ce that propagates through space, ofte

tra

sferri

g e ergy. A mecha ical wave exists i a medium but waves of a electromag etic radiatio , a d probably gravitatio al radiatio ca travel through vacuum. Wavestravel a d tra sfer e ergy from o e poi t to a other, without a y of the particles of the medium bei g displaced perma e tly (there is o associated mass tra sport) but there are oscillatio s arou d fixed positio s. Waves are characterizedby crests (highs, maximums) a d troughs (lows, mi imums), either perpe dicular (i the case of tra sverse waves) or parallel (i the case of lo gitudi al waves)

to wave motio . A wave ca be described by the wave fu ctio as:

r = f ( x, y , z , t ) = ( r , t ) .

(2.1)r r field H , electric field intensit E ) or a scalar quantit (electrical potentialdifference, pressure).

The wave function can be a vector (a displacement in mechanics, magnetic


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The wave front is the locus (a line or surface in an electromagnetic wave) of the points that have the same phase. We denote:

( x, , z , t ) = S = const.

(2.2)

We have plane waves and spherical waves. Characteristic of a plane wave: has a constant-frequenc and the wave fronts (surfaces of constant phase) are infiniteparallel planes normal to the phase velocit vector. The waves that are approximatel plane waves in a localized region of space are also called plane waves. For example, a localized source such as an antenna produces a field that is approximatel a plane wave in its far-field region (the region be ond approximatel 10

wavelengths from the antenna).

Fig. 1.23 Plane waves* A spherical wave is the wave that has spherical wave front. For example, an antenna produces a field that is approximatel a spherical wa

ve in region less than the far-field region (the near-field region). Spherical waves are the wave function given b :

= ( x, , x, t ) = ( r , t ) ,where r =

(2.3)

x 2 + 2 + z 2 and:

1 = F (r vt ) r

(2.4)


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Also, e have surface ave hich is a ave that is guided along the interface

et een t o different media for a mechanical ave. Exam les: the aves at the surface of ater and air, ocean surface aves, or ri les in the sand on the interface of ater or ind. Another exam le is internal aves, aves that are transmitted along the interface of t o ater masses of different densities. Surface aves

are encountered in seismology. There is an analogy

et

een surface

aves and

ater aves. Surface aves travel over the Earth's surface. Their velocity is smaller than the velocity of

ody aves. Their lo frequency determines them to

e more likely than

ody aves to stimulate resonance in

uildings. They are therefore the most destructive ty e of seismic ave. There are t o ty es of surface aves: Rayleigh aves and Love aves. Plane aves are descri

ed

y ( x, y, t ) :

r r = F (r k vt ) , r r r r = xi + y j .

(2.5)

here

(2.6)

Circular aves are descri

ed

y ( r , t ) ith r =

x 2 + y 2 and:(2.7)

=

1 F (r vt ) r

Fig. 1.24 Plane aves and circular aves*


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Exam les of Waves*: 1. Ocean surface aves. They re resents ertur

ations that

ro agate through ater (surfing and tsunami). Ocean surface aves are surface aves hich a ear at the surface of an ocean. They are aves guided along the interface

et een ater and air. The ind transfers a art of its energy into the

ater, the ater gains energy from the ind

ecause of the friction

et een the

ind and the

ater. The surface

articles have an elli

tical motion,

hich is a com

ination of longitudinal (

ack and forth) and transverse (u and do n) ave motions.

Fig. 1.25 Breaking aves at La Jolla* Individual "freak aves" (also "rogue aves", "monster aves" and "king aves") can occur in the ocean, often as high as 30 metres. Such aves are different

y tides and tsunamis. There are three main ty es of aves that are identified

y surfers: lunging aves ("dum ers"), s illing aves and surging aves. Their ro erties make them more or less suita

le forsurfing and resent different dangers. 2. Electromagnetic radiation: radio ave

s, micro aves, infrared rays, visi

le light, ultraviolet rays, x rays, and gammarays. In this case, ro agation is ossi

le


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ithout a medium, through vacuum. These electromagnetic aves travel at 299 792458 m/s in a vacuum. In the case of micro aves the avelengths is longer than those of infrared light,

ut relatively short for radio aves. Micro aves have avelengths a roximately in the range of 30 cm (frequency = 1 GHz) to 1 mm (300 GHz).

Fig. 1.26 Micro ave image of 3C353 galaxy at 8.4 GHz (36 mm)*. Infrared (IR) radiation is electromagnetic radiation of a avelength longer than that of visi

lelight,

ut shorter than that of micro ave radiation. The name means "

elo red"(from the Latin infra, "

elo "), red

eing the color of visi

le light of longest avelength. Infrared radiation s ans three orders of magnitude and has avelengths

et een a roximately 750 nm and 1 mm.The visi

le s ectrum (o tical s ectrum) is the region of the electromagnetic

s ectrum that is visi

le to the human eye. Electromagnetic radiation in this range of avelengths is called visi

le light or sim ly light. There are no fixed

ounds to the visi

le s ectrum,

ecause a ty ical human eye can res ond to avelen

gths from 400 to 700 nm, although

avelengths from 380 to 780 nm can

e

erceived. A light ada ted eye has its maximum sensitivity at around 555 nm, in the green


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region of the o tical s ectrum. The s ectrum does not contain all the colors that the human eyes and

rain can distinguish, for exam le

ro n and ink are a

sent.

Fig. 1.27 Ultraviolet (UV) radiation is electromagnetic radiation of a avelength shorter than that of the visi

le region,

ut longer than that of soft X rays.There is near UV (380200 nm avelength), far or vacuum UV (20010 nm FUV or VUV), and extreme UV (131 nm EUV or XUV). X rays have a avelength in the range of 10 to

0.1 nanometers, corres onding to frequencies in the range 30 to 3000 PHz. X rays are rimarily used for diagnostic medical imaging and crystallogra hy. Because

X rays are a form of ionizing radiation they can

e dangerous. Gamma rays ( ) areelectroma netic radiation produced by radioactive decay or other nuclear or sub

atomic processes such as electron positron annihilation. They are be in at an ener y of 10 keV, a frequency of 2.42 EHz, or a wavelen th of 124 pm, althou h electroma netic radiation from around 10 keV to several hundred keV is also referred to as hard X rays. Gamma rays and X rays of the same ener y do not present physical difference. Gamma rays are distin uished from X rays by their ori in. Gamm

a rays are a form of ionizin

radiation. 3. Sound is a mechanical wave that propa ates throu h air, liquid or solids. Its frequency can be detected by the auditory system. Similar are seismic waves in earthquakes, S, P and L kinds. 4. Gravitational waves, which are fluctuations in the ravitational field predicted by General Relativity. These waves are nonlinear.


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2.1.1.1 Transverse and Lon itudinal Waves

Other classification of the waves includes transverse waves and lon itudinal waves. This classification is made in function of the position of the vibrations and the direction of the propa ation of the wave. Transverse waves are the waves with vibrations perpendicular to the direction of the propa

ation of the wave. Some examples: waves on a strin , seismic waves and electroma netic waves. Lon itudinal waves are the waves with vibrations parallel to the direction of the propa

ation of the wave. Some examples include most sound waves, ripples in water, and certain types of waves from earthquakes, where the particle motion is in the direction of travel. In seismolo y transverse waves are called S (for secondary) waves as they arrive later than the P (primary) waves from an earthquake. The absence of transverse waves travelin throu h the earths core demonstrates that it is liquid. Also, transverse waves are connected to the curl operator and are overned by a vector wave equation. The lon itudinal waves are connected to the div operator and are overned by a scalar wave equation. A lon itudinal wave represents

compressions movin throu h a plane. In this case the ener y from the wave is t

ransmitted as mechanical ener

y. An example is a slinky which was pushed forwardand backwards, compressin and extendin it as the motion of the wave was transmitted. Opposite, li ht is composed of transverse waves (electric

r r component E and ma netic component H ).

Examples of combination of transverse and lon itudinal waves are the Ripples onthe surface of a pond. Therefore, the points on the surface follow elliptical paths*.


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Fi . 1.28* For an object that bobs up and down on a ripple in a pond, the trajectory is elliptical because ripples are not simple transverse sinusoidal waves.

2.1.1.2 Physical Description of a Wave

There are a number of standard variables that can be used for describin

the waves. These standard variables are: amplitude, frequency, wavelen th and period. The amplitude of a wave represents the ma nitude of the maximum disturbance in the medium durin one wave cycle. The amplitude is measured in units dependin onthe type of wave. For examples, waves on a strin have an amplitude expressed in

meters, sound waves as pressure (pascals) and electroma netic waves have the amplitude expressed in units of the electric field (volts/meter). If the amplitude

is constant we have the case of the continuous wave or the amplitude can vary with time and (or) position. The envelope of the wave is the form of the variation of amplitude.


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x t x ( x, t ) = A sin (t ) = A sin 2 ( ) = A sin( t k x) , (2.11.b) T v here:

k=

2

r is the ave number and is the abso ute va ue of the ave vector k r r 2 r r k =kn = n = n. v r ave vector k e have:

here

=

2 = vT v

(2.11.c)(2.11.d)

In the case of the ro agation of the ave in s ace in the direction of ther r ( x, t ) = A sin( t k r ) = Im{Aei ( t k r ) } ,

(2.11.e)

= t k ris the hase of the ave.

r

(2.11.f)

The grou ve ocity of a ave is the ve ocity ith hich the enve o e of the ave ro agate through s ace. The grou ve ocity is defined by the equation:

vg =

. k

(2.12.a)

About the grou ve ocity e can say that is the ve ocity at hich energy or information is conveyed a ong a ave. There is a direct connection bet een the grou

ve ocity and the dis ersion. Dis ersion is the henomenon that causes the se aration of a ave into com onents of varying frequency ( ave ength). If is direct y ro ortiona to k, then the grou ve ocity is exact y equa to the hase ve ocity. In the ro agation of signa s through o tica fibers and in the design of short u se asers the "grou ve ocity dis ersion" is aying an im ortant ro e.


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2.1.3 Pro agation of Waves 2.1.3.1 Pro agation through So ids1) Longitudina Waves

In so ids, there are t o kinds of sound aves, ressure aves and shear aves. Anon zero stiffness both for vo umetric and shear deformations is resent in a s

o id. In a so id there are conditions for generating sound

aves

ith differentve ocities de endent on the deformation mode. We have t o cases: a) the ro agation of ongitudina aves ( ressure aves) through a so id rod hen the ength of the so id rod is finite (a bounded so id rod) or ith thickness much sma er than the ave ength, and b) the ro agation of transverse aves (shear aves) hen the ength of the so id rod is infinite (bound ess) or ith atera dimensions

much arger than the ave ength. a) We consider a so id rod and the ength of the so id rod is finite (a bounded so id rod). We denote the cross section of the

rod ith S , the inear density ith

and

the modulus of elasticity o

Young's modulus with E . A vib

ation in a

od is awave. Tension is a eaction fo ce applied by a st etched od (o a simila object) on the objects which st etch it. The di ection of the fo ce of tension is pa allel to the od, towa ds the od. The magnitude of the fo ce of tension has aninc easing with the amount of st etching. In the case of a small st etching, the

fo ce is given by Hooke's law. The Hooke's law is given by:


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( x, t ) =

1 F ( x, t ) = E ( x, t ) = E , S x

(2.13)

wh r th xt n ion ( train) i lin arly proportional to it t n il tr , con tant factor, th modulu of la ticity E . Al o, on ha :

by a(2.14)

l l0 1 l F ( x, t ) = E = E S l0 l0and

=

l l0 . = l0 x(2.15)

Al o, w can writ for th infinit imal di plac m nt dx with dm = r ultant forc :

S dx the

dx S . x From th cond law of dynamic F = m a on g t :dF = 2 dF = dm 2 = dx S x tand

(2.16)

(2.17)

2 dx S S dx 2 = t xand it esults:

(2.18)

2 . 2 = t xFo the od one has:

(2.19)

( x, t ) = E ( x, t ) = Eand

x

(2.20)


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The sound velocity is highe than in the fi st case.


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2) T ansve se Waves

In the case of the t ansve se waves the esultant fo ce that is acting upon the

dm is pe pendicula to the di ection of p opagation and we have the connection tothe late al shea given by:

dF =

dx S . x

(2.27)

She r ppe rs he cu ing of he shee iron, he s ress of he rive . Suchdeform ions re homogeneous. The second l w of dyn mics is:

2 dm 2 = dx S

x nd one ge s:

(2.28)

2 2 = . t xF om the Hooke's law it esults:

(2.29)

= G = G

, x

(2.30)

where = F / S is he she r s ress, G is he she r modulus or modulus of rigidi y

nd

is the shear strain. The shear modulus is a quantity used for measurin the

stren th of materials. The shear modulus describes the material s response to shearin strains. All of them arise in the eneralized Hooke s law. The shear modulus influences the value of the speed of sound and also controls it. One ets:

2 2 2 =G 2 t x

(2.31)


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and

2 2 = 0. x 2 G t 2By comparing to th wav quation on obtain :

(2.32)

vt =

G

.

(2.33)

We have the connection between the modulus of elasticity E and the shea

modulusG :

G=and it follows:

E 2(1 + ) E < vl . 2(1 + )

(2.34)

vt =

G

=

(2.35)

In the case when the length of the st ing (solid od) is infinite (boundless) o with late al dimensions much la ge than the wavelength one gets:

v l' 2(1 ) = > 1. vl 1 2

(2.36)

In i mology tran v r wav ar call d S (for condary) wav a th y arriv lat r than th P (primary) wav from an arthquak . Th ab nc of tran v r wav

trav ling through th arth s core sho s that it is iquid. Sound i trave s o er in denser materia s, and faster in "s ringier" ones. Sound i trave faster in a uminium than uranium, and faster in hydrogen than nitrogen because of the o er density of the first materia of each set. At the same time, sound i trave faster in a uminium than hydrogen, as the interna bonds in


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a uminium are much stronger. Genera y so ids i have a higher s eed of soundthan iquids or gases.

2.1.3.2 Pro agation in a F uidIn f uids there are on y ongitudina aves. The on y non zero stiffness is to vo umetric deformation (a f uid does not sustain shear forces). Hence the s

eed of sound in a f uid is given by:

v= here

adiab ,

(2.37)

adiab is the adiabati bulk modulus. The bulk modulus K of a fluid o solid adiab = V

p , Vis the inve se of the omp essibility: (2.38)

whe e p is p essu e and V is volume. The bulk modulus thus measu es the esponsein p essu e due to a hange in elative volume, essentially measu ing the subst

an e's esistan e to unifo m omp ession. The speed of sound in wate is of inte est fo mapping the o ean floo . In saltwate , sound t avels at about 1500 m s-1and in f eshwate 1435 ms-1. These speeds va y due to p essu e, depth, tempe atu e, salinity and othe facto s.

2.1.3.3 P opagation in Ideal Gases and in Ai In gases the e a e only longitudinal waves. Newton conside ed that the p opagation of sounds in gases is an isothe mal one. He conside ed the speed of


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sound befo e most of the development of the modynamics. He inco ectly used isothe mal calculations instead of adiabatic. The facto of esult, which was othe wise co ect. The Boyle-Ma iotte s a is:

does not appear in his

pV = const. ,

ln p + ln V = ln(const.) = 0 , (and

dp p ) isot = dV V

(2.39)

isot = V (given by:

dp ) isot = p , dV

(2.40.a)

the isothe mal bulk modulus is equal to the p essu e of gas. The speed of soundis

v=

p

.

(2.40.b)

The expe iments do not onfi m this equation. Lapla e made the onne tion between theo y and expe iments. He onside ed that the p opagation of sounds in gasesis an adiabati p o ess. The Poisson law is:

pV = const.

=and

Cp CV

=

cp cV

(2.41)

ln p + ln V = ln(const.) = 0 , (dp p ) adiab = , dV V


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adiab = p = isotand we have the speed of sound in ideal gases and ai given by:

(2.42)

v=

p . m p , = V RT

(2.43)

U ing th id al ga law Clap yron M nd l v:

pV =

m

RT =

(2.44)

the speed of sound is identi al to:

v=

p RT . =

(2.45)

In th quation abov w hav R (287.05 J kg-1K-1 fo ai ) is the gas constant fo ai : the unive sal gas constant R, which units of Jmol-1K-1, is divided by the mola mass of ai , as is common p actice in ae odynamics, mass,

i th molar

is the adiabatic index and T is the absolute temperature in kelvins.For an ideal as, the speed of sound depends on temperature only, not on the

pressure. We can consider that the air is almost an ideal as. The temperature of the air varies with altitude, this cause the followin variations in the speed

of sound usin the standard atmosphere actual conditions may vary. Given normal atmospheric conditions, the temperature, and thus speed of sound, varies with

altitude*.


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Altitude

Temperature m s-1 kmh-1 mphSea level

15 C (59 F)

340 1225

761

11,000 m20,000 m (C uising altitude of comme cial jets, -57 C (-70 F) 295 1062 andfi st supe sonic flight) 660

29,000 m (Flight of X-43A)

-48 C (-53 F) 301 1083673

In ai a ange of diffe ent methods exist fo the measu ement of sound, like thesingle-shot timing methods and Kundt's tube.


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2.1.4 Ene gy of Waves. Wave Intensity.P opagation of waves means p opagation of ene gy. The pa ticles of the medium oscillate and we have a p opagation of the state of motion in the medium. Fo thelongitudinal waves that p opagate in a solid od with the c oss section S , when

the length of the solid od is finite (a bounded solid od) o with thickness much smalle than the wavelength, if the fo ce that is acting is F and the displacement is ( x) , the oss se tion has the speed:

v' =

. t

(2.46)

Also, upon the oss se tion is a ting the elasti fo e Fel = F . The me hani al powe is:

P=W . = ( F ) v' = F t t

(2.47)

If the elasti wave (sound) is sinusoidal:

= A sin( t k x) ,one has:

(2.48)

v =

= A cos( t k x) t

(2.49)

and from the Hooke s a it fo o s:

=and

F = E S = E S Ak co ( t k x) . x

(2.50)

F = S E = S ETh m chanical pow r i :

(2.51)


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P=

W = E S k A2 cos 2 ( t k x) . t

(2.52)

With the equations:

v=and

E

,

v2 = E(2.53)

k=the powe be omes:

v

,

(2.54)

W 2 2 2 = v S A os 2 ( t k x) P= t vand

(2.55)

P=

W = v S 2 A2 cos 2 ( t k x) . t2

(2.56)

From (2.56) it fo o s that even the o er is time de endent it has a ays ositive va ues because of cos ( t k x) that has ositive va ues. The o er de ends on

t k x , so it corres onds to a ave of energy.W 1 T W P =( dt = ) med = t T 0 tT 1 1 2 2 = v S A cos 2 ( t k x)dt = v S 2 A2 . T 2 0

We have the time medium va ue of the o er:

(2.57)

For so id rod the roduct v S is the vo ume here the aves have ro agated erunit time. We define the quantity:

=


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1 2 A2 , 2

(2.58)


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that is the energy density of aves or ave energy density and one has:

= 2 2 2 A2a d it follows:

(2.59)

P = vS w .or regio of space per u it volume.

(2.60)

E ergy de sity of waves is the amou t of e ergy of waves stored i a give system From (2.60) it results that the e ergy de sity of waves is proportio al to the

square of amplitude, the square of freque cy a d the li ear de sity. The sou de ergy de sity or sou d de sity describes the sou d field at a give poi t as asou d e ergy value. The sou d e ergy de sity describes the time medium value of

the sou

d e

ergy per volume u

it; it gives i

formatio

about the sou

d e

ergy which is at a defi ed place of room. The sou d e ergy de sity is give i J/m3. For sou ds i air we have w = 2,5 105

J / m3 .

The wave intensity, I , is defined as the wave powe P pe unit a ea:

I=

1 1 P = v 2 A2 . S 2

(2.61)

Because the maximum of s eed is v max = A one gets:

I=

1 2 v v 'max . 2

(2.62)

F om (2.62) it esults that the wave intensity is p opo tional to the wave ene g

y density and the wave velo

ity. Fo

sounds in ai

we haveI = 2,5 10 5 J / m 3 340 m / s = 8,5 10 3 W / m 2 .


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Some Appli ations of the Waves*: The te ms of wave powe efe s to the ene gy ofo ean su fa e waves and the aptu e of that ene gy to do useful wo k this in

ludes ele t i ity gene ation, desalination, and the pumping of wate (into ese voi s). Wave powe an be onside ed a fo m of enewable ene gy and it is diffe ent than the diu nal flux of tidal powe and the steady gy e of o ean u ents.La ge waves a e mo e powe ful and the wave powe is dete mined by wave height, wave speed, wavelength, and wate density. Wave size is dete mined by wind speedand fet h and by the depth and topog aphy of the seafloo . Fo a given wind speed the e is a p a ti al limit ove whi h time o distan e will not p odu e la ge

waves. This limit is alled a "fully developed sea." Wave motion is highest atthe su fa e and in eases exponentially with depth. Wave ene gy is also p esentas p essu e waves in deepe wate . Fo a set of waves the potential ene gy is p opo tional to wave height squa ed times wave pe iod (the time between wave ests). Be ause longe pe iod waves have elatively longe wavelengths they move faste . The potential ene gy is equal to the kineti ene gy. Wave powe is exp essed in kilowatts pe mete .


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Fig. 1.31 Inte fe en e of two i ula waves* Examples of inte fe en e: inte fe en e of sounds, inte fe en e of su fa e waves on the wate , inte fe en e of g avitational waves, and inte fe en e of light. Examples of inte fe en es of light:light f om any sou e an be used to obtain inte fe en e patte ns, Thomas oung's double slit expe iment, and Newton s rings that can be roduced ith sun ight. About the

hite ight,

e notice that it is not so suitab e for

roducing c ear interference atterns, because is a mix of co ours, that each has different s acing of the interference fringes. An exam e of ight c ose to monochromatic is the sodium ight. This is more suitab e for obtaining interference atterns. The aser ight exhibits the same ro erty, and is a most erfect y monochromatic. In

the case of the interference of t o aves, the resu ting aveform de ends on the frequency (or ave ength) am itude and re ative hase of the t o aves. If e

have t o aves of the same am itude A and ave ength the resu tant aveform i have am itude bet een 0 and 2 A . The first case corres onds to t o aves th

at are in hase, and the second case describes t o aves that are out


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of hase. In the figure be o are resented t o aves in hase and t o aves 180out of hase.

Fig. 1.32 T o aves in hase and t o aves 180 out of hase* If these t o aves are in hase, and have the am itudes A1 and A2 ,

39133855 Curs Fizica Engleza

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