wave theory (physics 2 )
DESCRIPTION
AIUB Physics 2 courses notesTRANSCRIPT
; .
v-Reh+ion br.{ws..
" !,{ave
por{icle, vcloci'ty ond kiale velocily i
Ne knor.r.l ,
-The displqccmont .
: t usin*
\le(e, v is *he w..vQ-
Thp- pon.l iele rreloci ly.
gi. dv
0+-4 - 0vo. loci ly.
dt
A I e* t ,1ff 1o <'n ;- t"t-"rJ
t - o^o co5 ;- et - x-) . ; ! 'v
Nor^r,rne
2rro-vA
vnonimu.vn va\utQ-
2x avA
a-^: -:jl].- X L0OVq
A
, cos f (" t -E
of pa,tficle velocity
velocity
0t5In2r-x- ("t -r")1
- . . )
e.-E (# -)2n\
coe -;- (vt -9
Ima"rc
N oul,dyd:r-
dra;tOco5 +
- 2nq
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-v:3Ld1-
+ ' . : -v _E_dr-
df Ener.ql o[ o. Pr6 BE!,5Live-
We- kno6,
' .dv-dt
UJarJE :
)(x,t) = y,n6in(kx-uJt)
Porc{ic\e
P".nficl e ,oP
vc\oci[y , ut
o.cct\ grcq+i on
L.:y'',, cor (xx - ^9
- u!1.t ln(xx-ot)
' ^ - /
volurnc- = L. pt'
Lop 1{"{co*(xx--)
r- p . \,{l-. v--' coJ (vx - u' t)
2 2 1;*l"cos- (kx - art)
-f- = -^Vol ,
'PL= -p;t
il [.la- consi&ru on\/
rno.Xnt*^dc -
F = p"fvvol .
No,,r,,
kinetic
Wonk
dY i :
srelCEY perc unri
. k '8.---'---:-
dore perL unil \ olurn e $o c. smc\\ diePlocerunt
# = {t='ov
# = J. d. t t= Jl /-- i 'd/
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7^-J1r dv
/*" L+J'"Pult- -,2*
L ,o e"+\\*t V . \dt \-
e7n"l\v
I,,J
Vol
Vol .
Esl---grelur-\ol .
/ \ {x-rdt){ -v - siri". 2dP
PF
,L PCvol .
o:v^ !\, '\,'
KEVol '
[sn^oa)
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a
* N,{{nr.nn\lo\ Eq',rotion o{ Stotion o.cy !,jova. i
-the tdso\ eguohon,
y . 2aco: ! .;," {
+ # : 2o,co3 + $ sin &jY! -Fo
: 2o.co: ) euu +L T
= 4^ou n^. 2r:( nn. 2T 'lL
.1 A4
Av.
" F - -+*
oo. {L oi" \L. n;u oo
- - 2o, $f .un 2!x sin TL -
O*,t 2:,vtY -- zoCos :i: 5 | n -;-
' d" zrr l t , : - 2rr . 2 ' r ,. tt- = _ 2o.srn =i "'r
-A- -1-
:. - 2^. ff- sin 2r-;!l sin 2!f
" #- -2".q{ s io} ! -oou\
' da--
^"
fn1vl
Anoin.
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,L l - . t | ,-+F Jtc\ttorv\Kv l^JOVe- D
Hem-,-The
) l
TI*
vt 'L
Usi no.U
1"
po+,{i.re. c- dit.e,c+i6n wovo eguo{ion \i ,
= qsinff= (vt-x) --O
"ugd,'ln. -r- dir@c+ion t^rcv{- oquation
= ou,^f- ( tr+rt) I
C
supercposition poinciple wL f'
Yr + 5z-
o. i" f (vt-r)
^ [ . r^ ( t* ! t -q1""\^
2n'ttO
,/ 5rn -;-
^ o 2n'ttA 5rn l{- ftot , ?-aeos
15 u
* o.sinf ('t+")
qx.-\ + "1" (3+L
- "X,A/ ' \
"ou 2f [_: <\^ (o+u)+sl"(o-u)
- 2;,1o"o.b]
. - - r - . , | l
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+ ber.iv" *'o fu'cml,l^ So posi.fiorur of
1\o sfon din6 t,rqvol f,r<rned by '{h"
{o rc-"d bv *ho sqpeopoui*,on o{ t}'ryz (x,t) -- Asin (Vl + Lall).
t..
{he node. o.d an{ino&n' in
.1, r l lStlPQtLPost na., ot Tlir wa!{-\
r,oavc yl Q,t) 'Asin (t<r - t,o9 o"d
. Q, l ,a,3. ' - ' - '
92' , (r t> .)2 G,t) -
Y.:in (kx-L^rt)
Y- sin (xx+,r)
NodcD
Res\^\{o,nt wo.ve, y(x,1) = y, (x,t) + seft,t)
. yra ein (**-^9 * t-sin (xx+urt)
= Y,a fain (xr-'^rt) + sinQx+ot)]
- 2l- sin (t<x) coa ( u: t)
= pv. ;i. k!cos*,t
Equa'lio r'' O decrcibao o. standing trcrve -
Ncw atnpli Ldq, , Y.' = 21^sinkX
f . lodcAl\ , ' - " .tm
' ' . : inkx = o
. ' . \<x : mr ,
Poei{ion o+ d,o,rL
= 2" sink{
ohUna- , m
rnX= --------------Q^A
^ mx,,\-2F
= rn f-4-)\ - . /
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On$ oode^ :
-The om p\i.|,^& ol -lhe alcnding ,^rwe o.l eqn O ha.
R yno{mutm vql\ e o+ 2y, , ruhich occr^ru, {orc .,n1,^€^ o I kX
{f-+ S'{e I sf n kx | = 1 -Tho se vo,l,^{r" oK42
: (z- +){
-_ (rn+ _b) x_
I
Po.i ton of qn{ incdu' >x: ( rn++)T /k
# Mo.x. mwm ond minirnuvn Js g\ocoment tirne perriod of 4adon^tcy r^ra"al
vr-TgnEE--t ' - - t '2-
27a/.
(-*{) 4.
t&^l aq,"^-fion,
y . eoui' ?t oot 3f
+r=Asinf f ! -
4{ si.1 4l = t +..c^t we r,.rill see +ja, morimu- displccm-tA
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.'. mqximum dttpf,ce.er,f -li_" prrcio d. .rs
3r-+ =+qYz'
l ll\01,l ,
I - . 2n. , I{o'. srn 3it- = O , W0 # minimurn diaplacement '
. 2av L
^ 'o ' " i ' 2 '7 ' 3q' - - - ' nn
-\ 2n t.
' + - n/l - n-f__4;_2_
mioi muem displocemotrt I'i^o Per'io d i 3 -
-T-z
* Enetrgl of sla{i6nq.c7 kJ(ve i
o:- t r -dY-dY'
nT*t
f . 'v .ErE"t-d
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N;*,
w--
- ' . Ro{c
Jnu at
J e* . i . , ry, 2ocoo ?]3- co. $ + ' df
, l ,^uo. i , . ' fL oor S! 'dt I v.-- z^o".4* '24)
r,* Jr- sin (ni" " '*e) at
P*v* I { sin AnJt .o*) -4
t lT " r rxvt .61-t P,tvx
Jo .,n -i
"{ ruorck dono = +F J- "1" ry- ' 6t
p,uun lT o,dt= _oa
Jo
50,
t^t o,vo .
*ntY:o
la l '15 not *tcvvr-te nrud \n
- r \ .co,5e ot s+qtlonoi9/
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. l , .\hc sfRft.hed stKin* trcoquency e.quqton '5
r lTn = -;;- \l 7-,,/.\ I '
O lo" ol ltn$h ?
n o + [ur]"tr, T o"d / ot* co"olo't*f
$ [o"r of -fmsion i
n c{ {i fwh.m, I *'d ,r^ 6rz1 com'lant]
O 1", of mq.3i o[ u",it lengl4r i
, l
n o {^ L*\- , -l- o",d I o'* co*{"n11
En r,oi nr ,
donr i tT is / 'k
t :1? Lo\^r of .vi brrq*ion 6t 5t s infihrd stont o
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. l -L i , , . , "-+r Yqloe tty Ot C,. *no,yr\prse r^tovc o\qnq q s"|trqlchod s*11i.'^ '
o " f '
S\ncq *h" posi+ion o{ -}ho.
s\,tu..X ftr.^ on Ncc o+ G c\rcc\e of rcadio'e tc.' 'rt
.r^ri\\ mo!e- .il^r.t the ci'tcr'\ort po.lh o't d -|l*n'{"n'-
otgcttionccr o, ee"t *rc, pata\ froco a\o^3 c0 ' 1\ /^ i'
Loi, consldrrc q ernall e\ crnrn* :
o{ i-l"o- sfnitsL MN isU
'ro"4w5,^---P
-leo" 6/t-l . r r , F l r . - €qisPbee d Lenq+h Ot thr5
- \ )
Tsrno.t\ elsment Mn i: t1. p 2a
6\--:- g
+|u- rnqs5
o+ MN.
5o , Tt\t-
pert unit leng*h 'lhnn //Er- \c {''' moes
L -- -{/"
- l'$t:"l-,("
-T.+.\ ver+i cql comPomnt
2T3'nO/2-
cql"ipd"\ Qoce- o. l in6 on. MN i6
t{ow ,
Qs i :
o+ {onsion -T- alon6
{+ 0 'tu sflqll +l^r,
2rs'rn9"_ :
, ainQ ,a 6
2Ta.ro-r, S-"-F-J-qL
YJ [ : , 'e '
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' 14 hlare Volocity sr., Phase Vetocity i
-The relo citl of o- wave ohieh maitrlait* covrstanl
phare of ol\ sr,tccexi.le, poei{i6r,6 d*,cin1 pnopoqotion i s
knoun u,5 t{avr velocily drL pfosc olocity.
Pl-nsr \e\ocrty, vp = *jF Gnoup ve\ocity j
-fie ve\ocilf of o. g1o(p o{ wave',' trAich
rnqintq\''^ constat pdi{iorn duninr '0* prc"po 6a{ion \s
knouln o.e qrtoup l elocit y ,
-Thn nxoug vclocity ,, ut =
*ts
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-rr Dr+lar&ncq ba*ular-n t^tcrvo vqJqci{t ond 6.o"p vcloci 17 I
B bJqut veloql[, iu dqfln{.d {.n. b"{h, fi* sin0lq wore s a.cl
Suporcrmposed wave^. $,.t *hc B*rp velocity i, ds{i r, J
on\y $ *he-
Jhe qtoup
touerc Sugurncy
o{ -lho hiavp-
s,^pttt'-psed wavQa
valocity t '14 lh. vels1i1, o{
, br.l +L pho..e rrelocitT i:
r^,i*h higher {ag"teney.
tht .waue h'r+1,r
.lL ve.[o citoI
r.^:,t - klur' -^4jk'*
-)-l-tt Reblon -bolwnon qr.or^p velocitT a'nd pl'nse ve\ocity i
Yl
t tL
Noc0 -
q€rn
06rn
k,4
xrD
(",t
(^ut
:
' l t2-
.a:in (r^r,1 - x'n) + asin(ol - x2t)
o I sin (r,r,f - k,rO + ain (t'r.t- xr+)]
a, '2 ' itno,)t - krx- + 1bL: l3x-
Co5
-- 2q sio (t''*t")t:lF'j!d! coa
-L^kD-- 4.a aln z (q't '- t(x), ... ( t a^t
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Notr.l ,
G.<o"p f . lvelocity Chont'nt- tqeTor.
l r | \
col (daurt - 12-akt
co. *or. I-++-r -")z \4K - . /
, /Cos 12- Ax ( \ t t -9
oAur dtt
dk
noJolo. be*l'teon qnoup ve'locilT und
4R
t+*
pln ":c ve\oci ty .
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+ 3+ l.,r o lrcoe\\inn t,t,qval or.e. y, -- od,n (r.*_*9 , .!r,- asin(x.-."l tO)
us'nn -fhl- supenpcei{ion pr<inciple {i",d -fin" t r^,..\d"& unue," rh.4
**o 1rcava1\1"'6 .o,7* a,rc taue[\1n6.... 5o,n\ s dinoc*io", a"rJ hain6
o p!'osc dri"lnnlt uo B ^hen
.41 "
Heto,
Y r = osin (r< r- ^+)
'i "
-' asin (k't - ^1 + B)
- , Y. y,+yz
" c { s,n (xt-co+) + sin(t<,,r,--+ +p)J
' l<r--s'r{ tkt - c'"* +o - W-6e - k< yat -p= O . 2 Sin "l-:-:::::----- CO5 ----- 2--
: 2osin (3#) "" . (1)
: 2o. srn (r.- - ,r+ + P/) ao;$
Foo, g -o, Resu.l.fon{ A^pltlude,
A , zaw'(PQ: ?o. (4*)
f .o, A, T\ ,
A = 2o, "r'
(Tt)
= t. B{ a [A_)
[lo, 0 . T"_
' A . zacosQel)= {Z
- to.)E", O=r
A = 2.,.C,or (f,/2->
: o (&-)
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