Waves: Class 11 Physics Chapter 14 Hinglish mein

14.1 Introduction

Socho, ek tense IPL chase ke dauraan cricket stadium khachakhach bhara hai. Ek end par koi haath upar karke khada hota hai aur phir baith jaata hai, agla insaan ek pal baad wahi karta hai, phir uske baad wala, aur khade hote fans ki ek lehar poore stands mein ghoom jaati hai. Koi bhi apni seat nahi chhodta. Har spectator sirf uthta aur baithta hai, phir bhi kuch to clearly travel kar raha hai: motion ka ek pattern (Fig. 14.1).

Nature mein yahi trick har jagah hai. Talaab mein tairti ek patti, girte pathar se bani ripples ke guzarne par upar-neeche hilti hai, lekin kinaare tak nahi pahunchti. Mandir ki ghanti apne aas-paas ki hawa ko kampa deti hai, aur awaaz aangan paar karke tum tak pahunchti hai, phir bhi ghanti se tumhare kaan tak hawa ka koi jhonka nahi aata. Is tarah ka travelling disturbance, jismein ek pattern medium se hokar aage badhta hai jabki medium khud sirf fixed positions ke aas-paas hilta hai, wave (तरंग) kehlaata hai.

Waves energy aur information le jaati hain, matter nahi. Guwahati mein rehne wale cousin ko phone call ke baare mein socho. Tumhari awaaz sound waves banati hai; phone ka microphone unhe electrical signal mein badalta hai; phone us signal ko radio waves ke roop mein tower tak bhejta hai; phir woh optical fibre mein light pulses ki tarah travel kar sakta hai, aur doosre end par poori chain ulti chalti hai. Har step message ko ek tarah ki wave se doosri tarah ki wave ko saunp deta hai.

Waves ki teen badi families hain. Mechanical waves (ripples, strings par waves, sound, earthquake ke jhatke) ko material medium chahiye; yeh isliye travel karti hain kyunki medium ke paas-paas ke hisse ek doosre ko push aur pull karte hain, isliye inhe Newton ke laws aur medium ki elastic properties control karti hain. Electromagnetic waves (light, radio waves, X-rays) ko koi medium nahi chahiye: taaron ki roshni lagbhag khaali space ko sadiyon tak paar karke hum tak pahunchti hai. Vacuum mein yeh sab same speed se travel karti hain, jo 1983 se metre ki definition ke dwara exactly fix hai:

Teesri family, matter waves, quantum physics ka hissa hai: electrons, protons aur poore atoms bhi wave jaisa behaviour dikhate hain. Yeh abstract lagta hai, lekin electron microscopes viruses aur materials ke atomic structure ki image banane ke liye electrons ki waves par hi depend karte hain. Electromagnetic waves se tumhari mulaqat Class 12 mein hogi; yeh chapter mechanical waves ke baare mein hai.

Ek disturbance medium mein aage kaise badhta hai? Morning assembly ki line mein kandhe se kandha milakar khade students ko socho. Pehle student ko ek halka dhakka do, aur har student agle ko takkar maarkar waapas apni jagah par jhool jaata hai: dhakka line mein aage daudta hai jabki har student apni jagah par rehta hai. Asli medium springs se jude chhote masses ki chain jaisa behave karta hai. Solid mein yeh springs crystal ke paas-paas ke atoms ke beech ke bonds hote hain. Ek atom ko displace karo, to uske bonds use waapas kheenchte hain aur saath hi uske neighbours ko bhi kheenchte hain.

Hawa mein bonds nahi hote, phir bhi usmein springiness hoti hai. Hawa ki ek patli slab ko dabao, to uski density thodi badhti hai, isliye uska pressure bhi badhta hai. Yeh extra pressure aage wali slab ko push karta hai, jo phir dab jaati hai, jabki pehli slab relax hoti hai aur thoda aage nikalkar ek patli, low-pressure state mein chali jaati hai. Is tarah pressure differences springs ka kaam karte hain, aur dabe hue regions, compressions, aur khinche hue regions, rarefactions, ka ek pattern hawa mein aage badhta hai. Energy medium ke paas-paas ke hisson ke beech in elastic links se aage pahunchti hai. Isliye elastic media mein waves Chapter 13 ke oscillators ki kareebi rishtedaar hain: medium ka har hissa oscillate karta hai, aur neighbours ke beech ka coupling oscillation ko travel karwata hai.

14.2 Transverse aur longitudinal waves

Mechanical waves ko is basis par baanta jaata hai ki medium ke hisse wave ki travel direction ke comparison mein kaise move karte hain. Transverse wave (अनुप्रस्थ तरंग) mein har hissa travel ki direction ke right angles par move karta hai. Longitudinal wave (अनुदैर्ध्य तरंग) mein har hissa travel ki direction ke along aage-peeche move karta hai. Ek slinky spring dono dikhati hai (Fig. 14.2).

Gate se bandhi ek rassi transverse wave ko saaf dikhati hai. Apne end ko ek baar jhatke se upar aur waapas neeche karo, to ek akela kubad, ek pulse, gate ki taraf daud jaata hai. Apna haath pendulum bob ki tarah steadily upar-neeche karo, to tum ek continuous harmonic (sinusoidal) wave bhejte ho: barabar doori par crests aur troughs ki ek train. Tum kuch bhi karo, rassi mein bandhi gaanth sirf upar-neeche hoti hai; woh kabhi gate ki taraf nahi badhti.

Ek wave ko do bilkul alag tareekon se socha ja sakta hai, aur tumhe dono chahiye. Ek instant par li gayi photo poori rassi par phaila poora shape pakad leti hai. Ek akeli gaanth ka slow-motion video dikhata hai ki samay ke saath ek jagah kaise upar-neeche hoti hai. Section 14.3 dono pictures ko ek formula mein pack karta hai.

Sound roz ki longitudinal wave hai. Hawa ki ek lambi tube ke end par loudspeaker rakho (Fig. 14.3). Jab speaker cone aage uchhalta hai to woh paas ki hawa ko compression mein thoons deta hai; jab woh peeche hatta hai to ek rarefaction chhod deta hai. Ek steady tone tube mein compressions aur rarefactions ki ek regular train bhejti hai, aur unke guzarne par hawa ki har patli layer tube ke along aage-peeche move karti hai.

Ceiling fan ke neeche khade ho, to tumhe hawa lagti hai: hawa room mein ek jagah se doosri jagah le jaayi ja rahi hai. Fan ki bhinbhinahat bhi tum tak pahunchti hai, lekin wave ki tarah, bina kisi hawa ke safar kiye. Isi tarah nadi ka current paani ko neeche ki taraf le jaata hai, jabki shaant tank ko paar karti ripple koi paani nahi le jaati. Aisi waves, jinka pattern medium ke ek region se agle region tak aage badhta hai, progressive (travelling) waves (प्रगामी तरंग) hain, transverse ya longitudinal.

Ek taani hui string transverse waves carry kar sakti hai jabki uske aas-paas ki hawa nahi, aisa kyun? Medium ko patli layers ki tarah socho. Jab transverse wave guzarti hai, to har layer apne neighbours ke relative sideways ghaseeti jaati hai, jo ek shear hai (Chapter 8). Solid shear ke against waapas kheenchta hai, isliye woh displace hui layers ko waapas line mein push kar sakta hai. Liquid ya gas apni layers ko bas phisalne deta hai, isliye fluid ke andar transverse wave ke liye koi restoring force nahi hota aur woh turant khatam ho jaati hai. Dabana alag baat hai: solids, liquids aur gases sab compress hone par waapas push karte hain. Isiliye longitudinal waves har tarah ke medium se guzarti hain, isiliye copper rod dono tarah ki waves carry kar sakti hai, aur isiliye hawa mein sound purely longitudinal hai. Dono tarah ki waves alag-alag elastic constants par depend karti hain, isliye same solid mein bhi aam taur par unki speeds alag hoti hain.

Jheel par uchhalte cork ko dhyaan se dekho, to dikhega ki woh har crest par thoda aage aur har trough mein thoda peeche bhi khisakta hai, aur ek chhota loop banata hai (Fig. 14.2 ka teesra step). Isliye surface water waves transverse aur longitudinal motion ko milati hain, aur gehraai ke saath loops chhote hote jaate hain. Surface ko waapas level kaun kheenchta hai, yeh wave ke size par depend karta hai. Lagbhag do centimetre se chhoti tiny ripples ke liye surface tension jeetta hai; yeh capillary waves hain. Khule samudra ki lambi waves ke liye, lagbhag ek metre se lekar saikdon metre tak, Earth ki gravity yeh kaam karti hai; yeh gravity waves hain.

14.3 Progressive wave mein displacement relation

Ek formula ko do kaam karne hain. Fixed time par padhne par use poori string ka shape dena chahiye; fixed jagah par padhne par use string ke us hisse ki motion deni chahiye. Isliye use is par depend karna chahiye ki tum kahan dekhte ho, $x$, aur kab dekhte ho, $t$. Maan lo string $x$-axis ke along padi hai aur $y(x,t)$ uska sideways displacement hai. Sabse simple travelling wave, $+x$ ki taraf jaati ek harmonic wave, hai

Eq. (14.2) ko teen tarah padhne se dikhta hai ki yeh sach mein aisi hi wave hai.

• Ek crest ko follow karo. Crest woh point hai jahan phase $kx-\omega t+\varphi$ ki ek khaas value hoti hai. Jaise-jaise $t$ badhta hai, us value ko banaye rakhne ke liye $x$ ko badhna padega, isliye crest $+x$ ki taraf move karta hai.
• Ghadi rok do. Kisi bhi fixed time $t_0$ par formula $a\sin (kx+\text{constant})$ hai: snapshot ek sine curve hai.
• Ek jagah ko dekho. Kisi bhi fixed $x_0$ par yeh $a\sin (\text{constant}-\omega t)$ hai: string ka woh hissa simple harmonic motion karta hai.
• $\omega t$ ke aage ka sign badal do, to crest ko ulti taraf move karna padega, aur $-x$ ki taraf jaati wave milti hai:

Table 14.1 mein saare symbols ek jagah hain, aur Fig. 14.4 Eq. (14.2) ko kaam karte hue dikhata hai: ek crest (orange marker) steadily right ki taraf move karta hai, jabki $x=0$ par bead apni jagah par hi sirf neeche, upar aur phir neeche move karta hai.

14.3.1 Amplitude aur phase

Sine ke aage ka constant $a$ amplitude (आयाम) hai: medium ka koi bhi hissa apni rest position se zyada se zyada kitni door jaata hai. Kyunki $\sin$ ka size kabhi 1 se zyada nahi hota, $y$ hamesha $+a$ aur $-a$ ke beech rehta hai. Hum sign convention aisa chunte hain ki $a$ khud positive ho, jabki $y$ axis ke upar positive aur neeche negative ho. Snapshot ke sabse upar wale points crests hain aur sabse neeche wale troughs.

String ke har hisse ko ghadi ki sui ki tarah ek cycle mein ghoomta hua socho. Angle $kx-\omega t+\varphi$ batata hai ki time $t$ par $x$ wala hissa apni cycle mein kitna aage hai; ise phase (कला) kehte hain aur ise radians mein naapte hain. Ek baar $a$ pata ho, to displacement nikaalne ke liye bas phase chahiye. Jab ghadi zero dikhaati hai tab origin par iski value $\varphi$ hai, yaani initial phase angle. $x=0$ kahan rakhein, ya ghadi kab shuru karein, yeh badalne se $\varphi$ badal jaata hai; inhe sahi chunne se $\varphi =0$ ho jaata hai, aur ab se hum aam taur par yahi karenge.

14.3.2 Wavelength aur angular wave number

Do points jo same instant par apni cycle ke same stage par hon, in phase hote hain: jaise do paas-paas ke crests, ya do paas-paas ke troughs. In phase points ke beech ki sabse chhoti doori wavelength $\lambda$ (तरंगदैर्ध्य) hai. Ise Eq. (14.2) se nikaalne ke liye $\varphi =0$ rakho aur $t=0$ par snapshot dekho:

$x$ ke along $2\pi \mathrm{/}k$ aage badhne se angle mein $2\pi$ jud jaata hai aur sine waapas same value par aa jaata hai; $4\pi \mathrm{/}k$, $6\pi \mathrm{/}k$ waghairah se bhi yahi hota hai. In repeat distances mein sabse chhoti wavelength hai:

Hum $k$ ko angular wave number kehte hain; kuch books ise propagation constant kehti hain. Iski SI unit radian per metre hai; kyunki radian ek ratio hai, ise aksar sirf ${\mathrm{m}}^{-1}$ likha jaata hai. $k$ ko per metre phase change ki tarah padho: ek metre mein jitni wavelengths fit hoti hain, har ek ke liye $2\pi$.

14.3.3 Period, angular frequency aur frequency

Snapshot ki jagah ab ek hi jagah ko dekhte raho, jaise Fig. 14.4 ke step 4 ke history graph mein. $\varphi =0$ ke saath, $x=0$ wala hissa $y(0,t)=a\sin (-\omega t)=-a\sin \omega t$ ki tarah move karta hai. Iska period $T$ ek poori cycle ka time hai, jiske baad motion bilkul repeat hoti hai: har $t$ ke liye $\sin \omega t$ aur $\sin (\omega t+\omega T)$ barabar hone chahiye. Aisa karne wala sabse chhota $T$, $\omega T$ ko $2\pi$ ke barabar banata hai:

$\omega$ wave ki angular frequency hai, $\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{s}}^{-1}$ mein. Frequency $\nu$ (Greek 'nu') per second oscillations ginti hai, aur ise hertz (Hz) mein naapte hain:

Upar ki koi bhi baat wave ke transverse hone par depend nahi karti thi. Longitudinal wave mein har hisse ka displacement travel ki direction ke along hota hai, aur hum aam taur par ise $y$ ki jagah $s$ kehte hain:

Baaki saare symbols ke matlab wahi rehte hain. Sound ke liye $s(x,t)$ batata hai ki jo hawa ki layer normally $x$ par hoti hai, woh kitni aage ya peeche khisak gayi hai.

14.4 Travelling wave ki speed

Pattern kitni tezi se move karta hai? Wave ka koi bhi feature chuno, sabse aasaan ek crest, aur dekho ki woh ek chhote time $\mathrm{\Delta }t$ mein kitni door jaata hai. Fig. 14.4 mein poori curve rigidly right ki taraf khisakti hai, isliye fixed phase wala har point same speed $v$ se move karta hai; agar alag phases alag speeds se move karte to shape bacha nahi rehta.

Eq. (14.12) ko timing ke baare mein ek statement ki tarah padho. Jab tak medium ka koi ek hissa ek cycle poori karta hai, tab tak jo crest uske upar tha woh wahan pahunch jaata hai jahan agla crest tha: ek wavelength. Yeh har travelling wave ke liye sach hai, chahe woh kisi bhi tarah ki ho. Lekin equation yeh nahi batati ki speed kitni hai. Mechanical waves ke liye yeh medium decide karta hai. Do properties muqabla karti hain: medium deform hone ka kitni sakhti se virodh karta hai (string mein tension, ya koi elastic modulus: Young's, bulk ya shear) aur uski inertia kitni hai (string ki mass per metre, ya density). Ek baar medium ne $v$ fix kar diya, to frequency chunne se wavelength fix ho jaati hai. Medium shear aur compression ka alag-alag stiffness se virodh karta hai, isiliye usmein transverse aur longitudinal waves ki speed aam taur par alag hoti hai.

14.4.1 Taani hui string par transverse wave ki speed

String par wave ki speed kis par depend kar sakti hai? Mudi hui string ko seedha karne wala khinchaav tension $T$ hai, isliye zyada taani hui string waves ko tez le jaani chahiye. Jise move karana hai woh string ka apna mass hai, jise linear mass density $\mu =m\mathrm{/}L$, yaani uski mass per metre, se naapte hain, isliye bhaari string dheemi honi chahiye. (Is subsection mein $T$ ka matlab tension hai; chapter mein baaki jagah yeh period hai, aur context se hamesha saaf hota hai ki kaunsa.) Newton ke second law se poori calculation is book ke level se aage hai, lekin $T$ aur $\mu$ ke dimensions lagbhag baat tay kar dete hain.

Eq. (14.14) ke baare mein do baatein underline karne layak hain (Fig. 14.6). Pehli, ismein sirf string aur uska tension hai; tension bahar se set hota hai, sitar ki khootiyon ya latakte weight se, akele material se nahi. Doosri, ismein na wavelength hai na frequency, isliye ek given string par har frequency ki waves same speed se travel karti hain. Frequency woh chunta hai jo string ko hilaata hai, aur phir string wavelength ko fit hone par majboor karti hai:

Har medium itna sahyogi nahi hota. Jahan speed khud frequency ke saath badalti hai, jaise glass mein light ke liye ya gehre paani par waves ke liye, us medium ko dispersive kehte hain; tab ek pulse, jo kai frequencies ka mel hai, chalte-chalte phail jaata hai.

14.4.2 Longitudinal wave ki speed (sound ki speed)

Longitudinal wave mein medium ki layers baari-baari se dabti aur khinchti hain, isliye jo elastic property matter karti hai woh yeh hai ki medium volume ke change ka kitni zor se virodh karta hai. Yeh Chapter 8 ka bulk modulus hai:

Minus sign $B$ ko positive banata hai, kyunki pressure badhne se volume ghatta hai. $B$ ke dimensions pressure wale hain aur ise pascals mein naapte hain. Ab inertia density $\rho$ hai, $[\mathrm{M}{\mathrm{L}}^{-3}]$.

Ek lambi patli bar ke sides thoda phoolne ya sikudne ke liye free hote hain, isliye bar effectively ek simple longitudinal stress aur strain mein hoti hai, aur relevant modulus Young's modulus $Y$ hai. Wahi dimensional argument aur exact theory dete hain

Table 14.2 mein typical speeds di gayi hain. Rod wali values seedhe Eq. (14.20) se handbook ki $Y$ aur $\rho$ values lekar aati hain; usi metal ke bade block mein, jahan material side mein phool nahi sakta, longitudinal speed thodi zyada hoti hai.

Liquids aur solids gases se kahin zyada dense hain, to unmein sound tez kyun hai? Kyunki unhe compress karna aur bhi mushkil hai. Paani aur hawa ko compare karo. Paani ka bulk modulus, lagbhag $2.2\times {10}^9\mathrm{P}\mathrm{a}$, hawa ke effective bulk modulus ka lagbhag $15000$ guna hai, jabki paani sirf lagbhag $780$ guna zyada dense hai. Isliye ratio $B\mathrm{/}\rho$ lagbhag $20$ guna bada hai, aur speed lagbhag $\sqrt{20}\approx 4.5$ guna zyada: lagbhag $1480\mathrm{m}\mathrm{/}\mathrm{s}$ versus $331\mathrm{m}\mathrm{/}\mathrm{s}$.

Gas ke liye hum bulk modulus predict kar sakte hain. Newton ne pehli koshish ki, aur woh jis tarah galat hui, usse kaafi kuch seekhne ko milta hai.

Ise $0\mathrm{°}\mathrm{C}$ aur normal pressure, $P=1.013\times {10}^5\mathrm{P}\mathrm{a}$, par sookhi hawa par try karo. Hawa ke molar mass, $M=2.897\times {10}^{-2}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{o}\mathrm{l}$, ke saath ideal gas law $\rho =PM\mathrm{/}RT=1.29\mathrm{k}\mathrm{g}\mathrm{/}{\mathrm{m}}^3$ deta hai, aur Newton ka formula predict karta hai

Dhyaan se kiye gaye measurements $331\mathrm{m}\mathrm{/}\mathrm{s}$ dete hain. Newton ki value lagbhag $15\mathrm{\%}$ kam hai, jo experimental error se kahin zyada hai. Galti assumption mein thi, algebra mein nahi (Fig. 14.7). Sound wave ke compressions aur rarefactions ek second mein saikdon baar badalte hain, aur involved regions centimetres se metres tak ke hote hain. Itne kam time mein heat na to dabe hue region se bahar nikal sakti hai, na rarefied region mein andar ja sakti hai. Isliye compressions gas ko thoda garam karte hain aur rarefactions use thanda: changes adiabatic hain, isothermal nahi. Adiabatically dabayi gayi gas zyada zor se virodh karti hai, kyunki uska badhta temperature pressure ke rise mein aur jod deta hai.

Hawa zyaadatar diatomic hai, $\gamma =1.40$ ke saath, isliye Laplace ka formula Newton ke estimate ko $\sqrt{1.40}=1.18$ se multiply karta hai: $280\mathrm{m}\mathrm{/}\mathrm{s}\times 1.18\approx 331\mathrm{m}\mathrm{/}\mathrm{s}$, jo experiment se bahut achhi tarah match karta hai.

14.5 Waves ka principle of superposition

Ek lambi slinky ke opposite ends par khade do dost ek hi pal mein ek-ek pulse bhejte hain. Beech mein pulses milte hain. Tum shayad takkar expect karo, jismein dono pulses waapas uchhal jaayein ya ek doosre ko bigaad dein, lekin woh bas ek doosre ke aar-paar nikal jaate hain aur aage badhte rehte hain, har ek apne original shape aur speed ke saath. Jab tak woh overlap karte hain, slinky ek naya shape leti hai, aur woh shape predict karna aasaan hai: har point par dono pulses ke alag-alag displacements ko jod do (Fig. 14.8). Yeh rule principle of superposition hai.

Sum mein signs matter karte hain. Agar ek crest-shaped pulse ulti taraf se aate ek jaise trough-shaped pulse se milta hai, to ek pal aata hai jab woh bilkul overlap karte hain aur string poori seedhi hoti hai. Energy gaayab nahi hui: us instant par string ka har hissa tezi se move kar raha hai, isliye saari energy kinetic hai. Ek pal baad pulses phir dikhte hain aur door chale jaate hain. Yeh principle interference ki chaabi hai: overlap karti waves ka combined effect.

Ab do sources ko same frequency ki harmonic waves same string par bhejne do, jismein ek doosre se cycle ke kuch hisse baad shuru hota hai. Tab dono waves ka same $\omega$, same $k$ aur isliye same wavelength aur speed hai, aur hum unke amplitudes $a$ barabar lete hain. Farq sirf ek constant phase difference $\varphi$ ka hai. String kya karti hai?

Do special cases khaas hain (Fig. 14.9). Jab waves in phase hon, $\varphi =0$ (ya $2\pi$ ka koi poora multiple), to crest crest se milta hai aur

sabse bada possible amplitude, $2a$: constructive interference. Jab woh bilkul out of phase hon, $\varphi =\pi$, to har jagah crest trough se milta hai aur

saare points par aur har time: destructive interference. In do extremes ke beech amplitude $2a$ se lekar zero tak har value leta hai; jaise $\varphi =2\pi \mathrm{/}3$ se $2a\cos {60}^{\circ }=a$ milta hai.

14.6 Waves ka reflection

Asli media kahin na kahin khatam hote hain. Ghaati ke paar ek cliff ki taraf chillao, to cliff tumhari awaaz echo ki tarah waapas bhejti hai: rigid boundary se reflect hua sound. Ek halki string ko bhaari rassi se baandho aur string ko hilao: har wave ka kuch hissa rassi mein aage chala jaata hai aur kuch string ke along waapas aata hai. Aam taur par do media ke beech ki boundary wave ka kuch hissa reflect karti hai aur baaki transmit karti hai. Jab wave boundary se kisi angle par takraati hai, to transmitted (refracted) wave Snell's law ke hisaab se mudti hai aur reflected wave laws of reflection follow karti hai.

Ek string ko deewar se mazbooti se baandho aur uske along ek upward pulse bhejo (Fig. 14.10, upar). Agar deewar uski koi energy absorb nahi karti, to pulse apne shape ke saath waapas aata hai lekin ulta ho kar. Hum kehte hain ki uska $\pi$ (yaani ${180}^{\circ }$) ka phase change hua hai. Pehle forces ke baare mein socho. Pulse pahunchne par string ka aakhri hissa upar uthta hai, jo deewar ko upar kheenchta hai; Newton ke third law se deewar string ko neeche kheenchti hai, aur yahi neeche ka khinchaav woh ulta pulse hai jo waapas daudta hai. Superposition se bhi yahi jawab milta hai: bandha hua end kabhi move nahi kar sakta, isliye jo bhi waapas aaye use wahan har instant par aane wale displacement ko cancel karna chahiye, jo sirf ek ulti copy hi kar sakti hai.

Iska ulta case woh end hai jo poori tarah free ho. Textbooks ise ek halki ring ki tarah model karti hain jo chikni vertical rod par bina friction ke slide karti hai (Fig. 14.10, neeche). Ring ko waapas neeche koi nahi kheenchta, isliye pulse use apni height ke lagbhag do guna tak upar phenk deta hai, aur reflection seedha waapas aata hai: koi phase change nahi. Sound ke liye flute ya organ pipe ka open end hawa ke displacement ke liye free end jaisa behave karta hai.

Ise likhne ke liye, ek harmonic wave $y_i=a\sin (\omega t-kx)$ ($+x$ ke along ek wave) ko left se $x=0$ par ek boundary tak aane do. Reflected wave $-x$ ke along travel karti hai aur, do tarah ke ends ke liye,

$x=0$ par rigid case har time $y_i+y_r=a\sin \omega t-a\sin \omega t=0$ deta hai, bilkul fixed end ki condition. Free case $2a\sin \omega t$ deta hai: end do guna amplitude se oscillate karta hai. (Books inhe alag sign conventions ke saath likhti hain; physics same hai: rigid end par sum zero hota hai aur free end par do guna.)

14.6.1 Standing waves aur normal modes

Guitar string par, ya flute ke andar ki hawa mein, do boundaries hoti hain, aur ek wave unke beech kai baar aage-peeche reflect hoti hai. Second ke ek chhote hisse mein string par opposite directions mein jaati do wave trains hoti hain, aur unka sum ek steady pattern mein tik jaata hai jo kahin jaata hua nahi lagta: ek standing wave ya stationary wave (अप्रगामी तरंग).

Eq. (14.37) mein position aur time alag ho gaye hain: ek factor sirf $x$ par depend karta hai, doosra sirf $t$ par, aur moving combination $kx-\omega t$ gaayab ho gaya hai. Ise ek oscillation $\cos \omega t$ ki tarah padho, jo string ke har hisse ke liye same hai, aur jise ek amplitude $\mathrm{\mid }2a\sin kx\mathrm{\mid }$ se multiply kiya gaya hai jo har hisse ke liye fixed hai lekin string ke along badalta hai. Kuch points, jahan $\sin kx=0$, kabhi move hi nahi karte: yeh nodes (निस्पंद) hain. Unke theek beech mein swing sabse bada hota hai: antinodes (प्रस्पंद). Do paas-paas ke nodes ke beech ke saare hisse ek saath uthte aur girte hain, jabki string ka agla section ulti taraf move karta hai, kyunki node par $\sin kx$ ka sign badal jaata hai. Crests aage nahi badhte, isiliye 'standing'. Aur kyunki node kabhi move nahi karta, uske aar-paar koi work nahi hota: energy har loop ke andar aage-peeche chhalakti hai lekin string ke along aage nahi jaati.

Nodes wahan hain jahan $\sin kx=0$, yaani $kx=n\pi$. $k=2\pi \mathrm{/}\lambda$ ke saath:

Antinodes wahan hain jahan $\mathrm{\mid }\sin kx\mathrm{\mid }=1$, yaani $kx=(n+\frac{1}{2})\pi$:

Paas-paas ke nodes ek doosre se $\lambda \mathrm{/}2$ door hote hain, paas-paas ke antinodes $\lambda \mathrm{/}2$ door hote hain, aur har antinode do nodes ke theek beech mein hota hai, har ek se $\lambda \mathrm{/}4$ door.

Ab asli fayda dekho. Ek kabhi na khatam hone wali string par tum jo chaho us frequency ki waves bhej sakte ho. Lekin $x=0$ aur $x=L$ par bandhi string kahin zyada choosy hoti hai, kyunki dono ends nodes hone chahiye. $x=0$ wala end Eq. (14.37) mein pehle se hi node hai. $x=L$ ko bhi node banane ke liye Eq. (14.38) kehti hai ki length mein half-wavelengths ki poori sankhya fit honi chahiye:

Isliye allowed wavelengths aur frequencies ($\nu =v\mathrm{/}\lambda$, aur $n=1,2,3,\dots$ ke saath) hain

Yeh natural frequencies string ke normal modes hain. Sabse neechi, ${\nu }_1=v\mathrm{/}2L$, fundamental mode ya first harmonic hai; $n=2$ second harmonic deta hai, $n=3$ third, aur aage bhi isi tarah, har ek fundamental ka whole-number multiple (Fig. 14.12). Yahan $v=\sqrt{T\mathrm{/}\mu }$ string se fix hota hai, isiliye string ko kasne (zyada $T$) ya ungli se chhota karne (kam $L$) se uski har frequency badh jaati hai.

Sitar ki string chhedo, to tumhe ek pure mode nahi milta. Ek saath kai milte hain, fundamental aur harmonics ki ek family, har ek ki apni strength ke saath. Kaunse harmonics strong honge, yeh is par depend karta hai ki string ko kahan chheda ya bow kiya gaya: jis harmonic ka us point par node hai, woh mushkil se excite hota hai. Players jaan-boojh kar iska use karte hain, aur harmonics ka yeh mel hi kaafi had tak woh cheez hai jo same note par veena, violin aur sitar ki awaaz ko alag banati hai.

Air columns ke bhi normal modes hote hain. Test tube ya patli bottle ke munh par phoonko, to tumhe andar ka air column sunai deta hai, jo neeche closed aur upar open hai. Closed end par hawa glass se chipki rehti hai, isliye wahan displacement hamesha zero hota hai (displacement node); lekin woh layer upar ki hawa se kisi bhi aur jagah se zyada dabti aur khinchti hai, isliye pressure change wahan sabse zyada hota hai (pressure antinode). Open end par ulta hota hai: hawa free move karti hai aur pressure room ke pressure ke kareeb rehta hai, yaani displacement antinode aur pressure node. Closed end $x=0$ par lene se ek node pehle se hi wahan hai, aur phir Eq. (14.39) $x=L$ par ek antinode maangti hai:

isliye allowed wavelengths aur normal-mode frequencies, $n=0,1,2,\dots$ ke liye, hain

Fundamental ($n=0$) $v\mathrm{/}4L$ hai: tube mein ek quarter-wavelength fit hoti hai. Upar ke modes $3v\mathrm{/}4L$, $5v\mathrm{/}4L$ waghairah hain, sirf odd harmonics, jinhe third, fifth, … harmonics kehte hain. Dono ends par open pipe ke dono ends par displacement antinodes hote hain; unke beech half-wavelengths ki poori sankhya fit hoti hai, isliye usmein har harmonic hota hai, ${\nu }_n=nv\mathrm{/}2L$, bilkul string ki tarah (Fig. 14.13).