wave theory (physics 2 )

; .

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

http://nissongo.blogspot.com

-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/


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)


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.


,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


+ 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-)\ - . /


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


.'. 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


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/


. 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


. 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 '


' 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


-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


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 .


+ 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 (&-)


wave theory (physics 2 )

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o i kx

t r dy

i f vtr

t usin

o f sncq

stotion o

rotion o

o fucml