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65 views11 pagesCH 15 Waves
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© © All Rights Reserved
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/ 11
WAVE MOTION
Itisa Alstuvana tak prpagates
Of voawe Constituents from Gn
Ina wave energy and momentum is transmitted by oscillations.
‘A.wave can be classified in two broad categories. Mechanical
Waves and Non Mechanical or Electromagnetic waves. In
‘mechanical waves, energy is transferred due to oscillations of
‘medium particles and in non mechanical waves energy transfer
takes place by.the oscillations of electric and magnetic field
‘Vectors. In this chapter we'll mainly diseus all about mechanical
‘Waves which requires a physical medium for its propagation.
Properties of a Mechanicat Wave
A wave has three basic properties which devide the
characteristic behaviour of wave and these are
© Amplindeofwave Gi) reque
Gil) Velocity of wave
1ey of wave
© Amplitude: Itisthemaximum displacement ofoscitlating
medium particle from their mean position. It depends on the
source of oscillation or the wave generator. Tt may vary with
the distance of medium particle fom the oscillating source.
Gi) Frequency :Itisthe number of oscillations made bythe
tedium particles per unit ime, Italso depends onthe souree
of oscillations, Once a wave is produced by a source it is
‘erred bythe medium, 10 malters what the medium may be,
the frequency of oscillation remains same wherever the wave
ls ene
(Gi) Velocity of wave : It is the speed with which energy
transfers through a medium and it depends tainly on the
physical properties of tie medium, When a waveis ransmited
‘Through a medium, ils frequency and amplitude depends on
the source ofoscilaton which prices the wave ht its speed
is decided by the medium which carries energy.
oe Based or te oscillaboun of wave
Comb tuants 3—
Depending on the way how these medium particle oscillates
and the nature of oscillations, there are several types of
‘mechan cal waves, Mainly mechanical waves are classified in
two broad categories :
@ Transverse waves, (i) Longitudinal waves.
(Transverse Wave: Daring such ypeof wave propagation,
medium particles oscillate in a plane normal tothe direetion of
propagation of the wave energy
KK HAMA
preg
ov place tp Ott with eu move munr
i wth tT
AN
Based en vequrement of mactlum foe te
aisen af wave it 1S of two tye.
Omecnanical wave Non meonanreal wae
— meotiomis vequired —> mectivam nor
lau ty OW.
of ig wove coind
voaae, Woon voave ete oe a
EM waves)
fox ideal mectium whwy |
no wss fs taking
place AMPLITUDE” vemayiny
Same. F
ov lessy mecllum wlirr
lass of “enmnge, takes ploct
AMPLITUDE AOLOdS ov
te digurbaug gous any
from sovrsee- &
bat
4
Morgner
requis reeainy
Same
i) Longitudinal Waves : During such type of wave
propagation, medium particles oscillate along the direction of
_Brrefecigg — co tomprenion
alli, stilt =e
WY
—
wave prepagasen
what disblacenwat
muchum panbe fs teely max Crest
what displace
muchium panbete fs —wely max Trough wave pulse avid wowe brainy
When a disturbance is localised only to a small
part of the space at a time, we say that a wae pulse
{& passing through That part ofthe space. This happens
when the source producing the disturbance (hand in
this case) is active only for a short time. If the source
‘is active for some extended time repeating it motion
several times, We get a wave train or a wave packet
Equation of a Travelling Wave
Suppose, in the example of figure (15.2), the man
starts snapping his hand at f=0 and finishes his job
at f=3%. The vertical displacement y ofthe left end of
the string is a function of time. It is zero for <0, has
nonzero value for O At. Let us represent this function by fi). Take the
“Tet end of the string as the origin and take the X-axis
along the string towards right. The funetion it)
represents the displacement y of the particle at x = 0
sa funetion of time
= 0,0 = AD,
‘The disturbance travels onthe string towards right
the displacement,
reaches the point ©
fend at the time ¢=/0 displacement of the
left end at time ¢—x/v is (tx/v). Hence,
equation is written as
yaft-x/0),
asa
SINE WAVE TRAVELLING ON A STRING
Of Sourea of Ware 1's defn sum
then the ware produad 15 called
Slur wane ov stmple Hormoute wave.
Yarrortet) = Asingot)
(xox, t2t)= A sf (w(tay
y(x-t) = ASfn(w(t x)
wavelengins- The minimum alstaues
ene 14 between two parbee
wolidn oscliare fin
pho
> ak Manly)
mts socf-t] nec 2
This gives
=
or, =A one
= O-2G»
vibrating 7 SMe Phase is obtained by putting n= 1
in the above equation, Thus, the wavelength is
OD
‘The minimum separation between the Fe
“Tu distameg
ve
where n is an tnteg = voave In the
fn whion sourw
Lomple ter ane oseillaror?
displacernunt 4 Particle cur ontafn wos
ony gor octcar
. yorrod = f(y)
tines taken by Hu aisturbourc, to vea th
or xXx D Xp
Tou olistuabance void is od x20 at Hot
will a4 X= 5 a tims (+ Xhy)
ae clistunoouret at x= cub time t
vill or sameas thot of x=o at Hme (eas)
w
Yee tp = Beret )= F(t)
Gennadt equation of travelling wave
Gon? = flex) Gf voane i's
~X ait 9 aed
yore = FORE)
a
x
=— ut —
Suppose this snapsat of ware pulse fs
ar tim t=o 24 wave fs
(xex, fy) jofey -x ols”
le(ried = F(xeut)
# Of & madatmakeal equasen Ts of
fo-ue oy Feay type ana, te
displacimwnr 4 mudium powbele ts
wile. Hun thal madiumabcat equaten
arin Tepresents a fravelliug/ progressive
dae wane oftrousice uot.
(@
4 a = a A
(x ud uy a
(R=
12
1 yt): A Sin] 7 ] 8 The equation of a wave travelling on a string is
Aird: Asin[wlt- 2) Y= (0-10 mm) sink(BL-4 me + B14 Sh
(@) In which direction doos the’Wave travel? (b) Find
the wave speed, the wavelength and the frequency of
the wave. (€) What is the maximum displacement and
Ls the maximum speed of a portion of the string ?
ye Asin(wt ex)
*
==.
when distant bly two Pordicue is A
plore oliff: lw Hun fs 2K
wlin distonc blw hoo ponbeu is x 2 VE Wipe Sis lomls
pho aig 3 a 304
aR. kK angular (me) pastes A= Ox 314
31-gmm|s
Le qoove yo, RENO
A
w= an ke gn [We Dew
T a» [KT
lea) = Asin ( w(t-2)) = Asin(wt - an)
yon = Asin (wr= Kx)
Different forms of stmple Honmonilc wave 5—
out): asin (kK egt-x)) = Asin(k(vt-x)]
K *
yor) = Asin (ant- 22x) = Asin[2n($-3))
Ye Asinlwt- kx) __agefolwt-k) —O
vetoul or Ke | (Up) = = AW Wcot-Kx) »
MY Apart [C= oy O/@ wes
[4s elem gt Qa
Acceleration & particu ap)= dup. - Aw Sin(wot-Kx) aye vies)
art Ox"
a= — oF oy vi (Ea) =0
at
dy: eS Kx) x2 Diffenention arm of aves
ox
: = ~ Aw ws(uot ka?
QS
3 Awcos(vot- k= Vane (2H ) 4ane = dy
on
a Ventas = = Vion (BE) &,
A sin(wot- Kx)
Aus cos(uot~ Kx)
- put sinceot-k) —O
4
= AK cos(cot ka)
ae SAP de
gd Ye
a
MIT SEE 2009 Sheed_of trammeverse ware in stveraud shige
—+v
Tension fn snl T
mass per tr length aM
Mass of Smatt squat Gnp= URdd
Net ove OM. seqroon = atsin(ae)
2 .
(amv? = 3 sin( as) stots 40 4 very
2 small
R
FORA . tx dO Sind 5 ole
x z 2
a hs [Fa
18. A transverse wave deseribed by 17. Two wires of different densities but same area of cross
_ 4 “4 section are soldered together at one end and are
¥= (0-02 m) sin{(10 m”) x + (805 4] stretched to a tension 7. The velocity of a transverse
propagates on a stretched string having a linear mass ¥# in the first wire is double of that in the second
density of 12x 10"‘kgm", Find the tension in the {ye find the ratio of the density of the frst wire to
string. .
Ye Asin (ut + kx) My 2G
= 305! Kem? =
w= 305 7 JE 3(E
ay Ma
Ans 4 My
2 ms 4m
Ge D>
a Sao GS
Fil pe 1:4 2025
21. Two blocks each having a mass of 32 kg are connected
by a wire CD and the system is suspended from the
" 22 Inthe arrangement shown in figure 5-5), hearing
Celing by another wire AB (igure 15:69, The neat <
mass density of the wire AB is 10m" and that of CD has a mass of 455 g How much time will i take for a
in Syms, Pid the sped cfs tetueverse wave plae tranevereedatarbance produced athe for to reach the
produced in AB and in CD. pulley? Toke g = 10m
— sped Te 2q22m0
oR wae fr LO) ah
ve obra, a “tH pee (E:| ‘Zox226
Bly a ony pw Jess
or G
- - vz oomls
4 Tage 1g + Top oe
ts 2 2 002s
100.
3
V_52 19:19 015 (@ e Heouy ape MASS mM On He previow quyhen if fromevere
lengli L. Wars om gennard ar top and toot
; 9 & tromeverse wave Simuttaruously Hun fied He tou wher
" an came cera both woe vuill nur eadr oti and
L take tp each Hu tOp- mane fevgil mat
re ee tof
. dx = | fgar
e- [a ( Ser]
7 Get 2(ax] bgt
a L
afar - fy) Ge
any = Al -aly
alge St
ge tg
50 Huy wall wateh Of a ds STOW?
Un, “pom thas Ioott0e)_ ot hou
aye. [k
Ag Power tramamiitted by sint wave in ie Vea a2
Sreteud Stetng
Ye D shoot ~kx)
EE dig = Aureos (wt- lea)
ar
24 = — AK cos(vot - lex)
ox
rotor tonnes power = -Tsind. Up
i” smout omgk sfa8— 9 tomo
tomp= OY
ax
Pay = we) Aww 0s (vot kor)
_ 2 AT, = AK wos( wot Ica). Aus Cost kx}
» = WMAK Aw cos (wt-Kay
wor = PM uxie wow us (wt- ka)
K
2 ae 2
Ke Pimp = Aw ue COs (uot-kay
LPaug = Pre MUL C cos! (work)?
fag? pee Me | = tisha
£ ea 29. A 200 Hz wave with amplitude 1 mm travels on a long.
string of linear mass density 6 gm" kept under a tension
of 60 N. (a) Find the average power transmitted across a
siiven point on the string. (b) Find the total energy
associated with the wave in a 20 m long portion of the
ame Pou = Cats
“x(IxwW?) x & x
Qs Any 200 x(lxio eG
= 0W73W
) te22 2 ok
ee (fore ae
Tc
& 2 64732 =
me (00
Sead)
To00
Parle of Supenpton and erfonne
o~ |
oo NY
Ye f, (t+ a)
or ok Bilin,
mae Te displacemout of partele or
ar Sr
—_ _ = tue posiben wl both te wars
Re ae
a o On supposing fe give On
— othe
AR ye A(t x) t+ 3)
through a point, the disturbance at the point is given
‘The pulses travel towards each other, overlap and ty the sunt of the disturbances. each’ wave. would
recede from each other. The remarkable thing is that produce in absence of the other wavets).
the shapes of the pulses, as they emerge after the In general, the principle of superposition is valid
overlap, are identical to their original shapes. Each for small disturbances only. If the string is stretched
pulse has passed the overlap region so smoothly as if too far, the individual displacements do not add to give
the other pulse was not at all there. After the the resultant displacement. Such waves are called
encounter, each pulse looks just as it looked before and nonlinear waves. In this course, we shall only be
each pulse travels just as it did before. The waves can talking about linear waves." which obey the
pass through each other freely without being modified superposition principle.
twols torn}
ee -
(@) 7 A vos (es
423 tees @&
15358 ep ond
Intefouncr Of wows waws gring in te fn the Same ddivecten
— Lohwunt wares —- Waves having same frequancy and constanr
frniBor phar
Ye A, sin (uot - kx) 4a2 Arsin(eot- Kx + 6)
At the posiben 5 Suberposib ona
= Wt
Y= Pistn(eot Kx) + Ay Sin (wot-Kx 48)
Y= Aisin ceot-koo + Az sin (wt-ko) cosh + Ay cos(w tex sin€
= (Prt Pa c0s8) Sf (cot kad + (AySin$) cos(eot- kay
eee ee
Rosh Reine
Yr Rofirloot-kx) Cosh + Rvs (eot-Kea) Sfind
Y= RQ sf (cot - ve +4)
Re | ata As +20, A, cos
. 1(P2sing )
a ( Art Ar cosd
Amn pu tude 4 routing war.
FOF Bawa 9 U0s8= 1 for Renin 9 wos Ss -1
Rua = [Arr AZe 2AAL Renin = [P+ Ar 2 Pa
Rare = Ar 4 Re Quin.= [Pi 2]
Comnbuctive Firixfonnnu > Rostrucbive fntofounu 2
who tu amplitude of yolon Hu amplitude vf
rorulting Wave fs max Sudlr soultfng wave fs min. sudlr
type of Infeyouren fs calud ‘gre fnferfonmer fs calle
Srophutive fntogoumer uetiue fndoferrrer
wss= ” woss= 2s
c0$ (2K) c0g8= cogone0A)
G2 0,28/4K,--- Se KBR SR
31. Two waves, travelling in the same direction through the
same region, have equal frequencies, wavelengths and
amplitudes. If the amplitude of each wave is 4 mm and
the phase difference between the waves is 90°, what is
the resultant amplitude? 4 mm ReflecKen oma transmission of wave i
© Reflection ——@) fron rigid end — () from fren end,
@ ‘foorn wigs end a
2 —
www Wane weflect ——~e&>-
cvigid_ end. an He
prom % weflecked wong
cmon Ran Pa
vont inciolanr ware.
1) from fires ond -
When wave. wef leet’s
from free end dwn
thu pax of reflected
voane vemalins same
wet fndiderr wave fhe Ifa wave enters a region where the wave velocity
Trammissions- (i) fom denser to rarer is smaller, the reflected wave is inverted. If enters
: region where the wave velocity is larger, the reflected
OD from varr to denser wave is not inverted. The transmitted wave is never
: . Inwrted aan
= 5, behowen Uke fer end : 2 EME
0 oN 4 q m : : fined eral,
_ _“
medium A ry \ medium@
;
T : T
HW, Me
vi My
f ft
Squation of Incident wave — Y= Aj sin (vot kx)
Suarion of Replectea wave — fp Ar sin (wt +k ,x)
fquati ou of hamamiited wave Yp= At sin(wt - kK)
at foteyfacr toy superposibon prindple 9
displacement: » partic at oviging x20
Gite = Ye .
yay Stn(eot) + Ay sinGot)= Ay Sino)
A Ait Ay = At —O
_e Power dubiverca boy indiouar waves Power cuuivened to veplected +
2 trams mitted warn,
a > patty, = Yortefian + Yae yh sev,
ve(E DAE TRS Ag T+ AE Ty
= vi ve vs
a ab ve = ae a (Ai- Aa) = AKU
vi vi Vi {peees) ve
3 Al-Ay= Meu i)
Ma
fromOs ©
Ye
Aye Bie ) ai Ai e ai(
Vita
Ay= if Va- Vy
Mat
O+@M > aar= eed) @ | Ai=/2y ia
Na.
4 =Vevs)
Vite rounding wane (stabenanry wove)’ ns traveling wave, the strc prod
Stowding wave (stabonany wove) cpt ely aa
; ; «a standing wave its cnfined othe roi where
consiclaring two Tdantsal womes apjmaaaing is produced ne
2. In a travelling wave, the motion of all the
each otun fn ite Samu Shing» particles ae similar in nature. In a sta
diferent partiles move with diferent amplitudes.
3. In a standing wave, 1
Giz A Sli (ot kx) Yas A sfin(ut-aka +6) sivaye romain in ret In travel
particle which always remains
particles at nodes
4, Ina standing wave
By Suberpositien $7 — (Considunig 620) mean postions together
rho instant when all the particles are at the mean
positions together
GF Git th = Asin(wt-kx) +A sin (cot tka
ye A (sintwt-co + dn(wtako]
wave, all the particles between
‘th cations
= A[2 storeot) costen))
the energy of one region To alwaye confined that
Y= 98 SinCoot) 105Ckx) | Equation of Stamoling nanan ;
WOR
Ottun form Y= AA Sin( wt) sfn(k>) Ye aA Sin(wercos (ica)
YF 8A cos(vot) cosCKa) 4 Es = Rsfnca)
= 8A cos (wot) Sfin(xx) for Certain var ox
AA WOS(kx) wsill always lar 2020
iwyerbective time , those
Pink ore Callel nodes.
for nodes 2Aacaskx =o
kx= & 48, Se, 78,
2334
9 Bxsh 9x2
ee + (ab
9 Wye 3K 5 X32
a2 7 ab
9 xr SF 9 x2 5
TOES
9 2x: 1h 9 Xs 7a
a 2 a
Tu Iweoken whint moximur possitk
aiispouerenr *s £24 art called antinoasp
for owtinooles 2ALoskx= £28
Coskas 1 OY coskar -f
x0 kxs
Wye
x
x2 Me
COMA: =A
Kxe 3m
X= 3a
2 Stomaing wawe fn sheteld sing fined loetwean two igi endsy—
Covwicuinivyy mass per wit lengih oh ta shing 6m. amd tension fo the ching fs T
dor rein. frequangys
fe he Az