
db model

thermodynamics aanee solid-steet physicsmadhye, db model hii pieter dbne 1912 madhye vikasit keleli eka paddhat aahe, jee ghanatil vishisht ushnatemadhye (ushnata kshamata) phononchya yogdanacha andaj lavte. heey einstein photoelectron modelchya ulat, eka chaukatitil phonon mhanoon anoo jaali (ushnata) chya compananna manate, jeey ghanala anek vaiyaktik, parasparsanvad naan karananya quantum harmonic ausilatorsarkhe manate. debby model ghan padarthanchya ushnatechya kshamatechya qamee-tapman avalambitvacha yogya andaj lavte, jeey sootra _ 1-debby "tii" 3 kayadyachya pramanat aahe. einstein photoelectron modelpramanech, teey uchch tapmanat dulong-pitt niyam punarprapt karate. gruhitke sopee kelyamule, tyachi achookata madhyavarti tapmanat qamee hote.
vyutpatti.
debby model heey plankchya black boddy kiranotsargachya niyamashi samatulya ghan-sthitee aahe, jithe ekhada vidyutchumbakiy kiranotsargala photon vayoo mhanoon manato. db model anoo compananna eka chaukatitil phonon mhanoon manate (chaukati ghan asate). gananechya bahutek payya eksarkhya asatat kaaran donhi resheey prasaran sambandh asalelya vastumanahin boses vayuchi udaharane aahet.
bajuchya sootrachya ghancha vichar qaraa. chaukatitil lekhatil kanatoon, chaukatitil dhwanik vikshobhanchya pratidhvanit paddhatinmadhye (sadhya fakt eka akshasah sanrekhit kelelya) yaa dware dilelya taranglambi aahet
jethe sootra _ 4 haah eka purnank aahe. phononchi urja aahe
jethe sootra _ 6 haah plank sthirank aahe aanee sootra _ 7 hii phononchi varanvarta aahe. varanvarta tarangalambichya ulat pramanat aahe assa andaj lavtana, aapan
jyaa sutramadhye _ 9 haah ghanatechya aateel aavajacha veg aahe. teen parimananmadhye aapan vaparu
jyaa sutramadhye _ 11 heey phononchya trimiteey gatiche pariman aahe.
varanvarta tarangalambichya ulat pramanat aahe (dhvanicha sthir veg dett) yacha andaj qamee-urja phononsathi changla aahe parantu uchch-urja phononsathi nahee (phononveril lekh pahaa). haah matbhed haah db modelchya maryadanpaiki eka aahe. heey madhyavarti tapmanat chukiche parinam dete, tarr parinam qamee aanee uchch tapman maryadet achook asatat.
aata chaukatitil ekoon urjechi ganana karuya,
jethe sootra 13 heey urja sootra 14 asalelya chaukatitil phononchi sankhya aahe. dusya shabdant, ekoon urja hii tyaa urjesah (eka parimanat) phononchya sankhyene gunalelya urjechya berries itaki aahe. 3 parimananmadhye aamchyakade aaheh
yethe, db model aanee plankcha black boddy kiranotsargacha niyam vegala aahe. boxmadheel vidyutchumbakiy photon kiranotsargachya ulat, phonon urja avasthanchi sankhya maryadit asate kaaran phononchi varanvarta aniyantritpane jaast asoo shakat nahee. tyachi varanvarta tyachya prasarachya madhyamane-ghanchya anoo jaaline-bandhali jaate. khalee aadvya phononche udaharan vicharat ghya.
khalchya aakrutimadhye darshavilyapramane, phononchi kiman taranglambi anoo pruthakkaranapeksha duppat aahe asey gruheet dharane vajvi aahe. ghanamadhye sootra 16 anoo asatat. aapale ghan ghan aahe, jyacha arth aahe qee prati kaath sootra _ 17 anoo aahet. anoo pruthakkaran nantar sootra _ 18 dware dile jaate aanee kiman taranglambi aahe
jastit jaast mode kramank sootra _ 4 (photonsathi anant) banavane
hii sankhya tiheri urja berijchi varchi maryada bandhate
haloohaloo badalatya, changlya vartanukichya karyansathi, berries avibhajyane badalali jaau shakate (jyala thomas-fermi andaje asehi mhanatat)
aataparyant, sootra _ 24 chaa ullekh jhalela nahee, urja sootra _ 25 phonon asalelya phononchi sankhya boses-einsteinchya aakadevariche palan karate. tyanche vitaran prasiddha boses-einstein sankhyiki sutradvare dile jaate.
kaaran phononchya teen sambhavya dhruvikaran avastha asatat (eka anudairdhya aanee done aadvya jyaa andaje tyachya dhruvikaranavar parinam karat naheet).
urja) vareel sutrala 3 neey gunakar karane aavashyak aahe,
pratyakshat eka 'prabhavi dhwanik veg' sootra _ 28, i vaparato.eae. debby tapman sootra _ 29 (khalee pahaa) heey sootra _ 30 chya pramanat aahe, adhik tantotant sootra _ 31, jethe konee rekhansh aanee anuprastha dhwani-lahari veganmadhye farak karato (yogdan anukrame 1/3 aanee 2/3). debby tapman kinwa prabhavi dhwanik veg heey sphatikachya kadakpanache mojmap aahe.
urja avibhajya utpannat badal karane
photonsathi yaa samakalananche moolyankan jyaa sahajatene kele jaate, teey prakashachi varanvarta, kiman ardh-shastriyadrushtya, unbound asalyamule aahe. vareel aakrutivarun spasht hote qee, heey phononchya babatit kharey nahee. yaa tiheri samakalitcha andaj lavnyasathi, debine golakar nirdeshank vaparale.
aanee eka golakachya aathvya bhagane ghan andaje kele
jethe sootra _ 35 hii yaa golakachi trijya aahe, jee ghanatil aanee golachya aathvya bhagat kananchi sankhya jatan karoon aadhalate. ghanache aakarman sootra 16 ekak-peshi aakarman aahe,
tyamule aamhala milate
yogya samakalanasathi golavar samakalan badalalyane pratimanat ashuddhatecha aanakhi eka srot yeto.
urja avibhajya banate
ektrikaran chull badaloon sootra _ 40 qaraa,
swaroop sope karnyasathi
yaa abhivyaktiche, debby tapman sootra _ 29 paribhashit qaraa.
jethe sootra _ 44 heey bajuchya sootra _ 2 chya ghanpetiche praman aahe.
anek sandarbhanmadhye debby tapmanache varnan kahee sthirank aanee bhautik-aadharit chalansathi keval laghulipi mhanoon kele jaate. tathapi, khalee darshavilyapramane, sootra _ 46 heey andaje kiman taranglambi modachya phonon urjechya barobariche aahe aanee mhanoon aapan debby tapmanacha arth assa kadhu shakato qee jyaa tapmanavar sarvoch-varanvarta mode (aanee mhanunach pratyek mode) uttejit aahe.
pudhe chaloo thevun, nantar aaplyakade vishisht antargat urja aaheh
jethe sootra _ 48 heey (tisare) deby karya aahe.
sootra _ 49 chya sandarbhat farak kelyas aaplyala parimanahin ushnata kshamata milateh
hii sutre db modelver serve tapmanat upchar karatat. adhik prathamik sutre khalee dilee gelyas qamee aanee uchch tapmanachya maryadet lakshanrahit vartan milate. aadhich namood kelyapramane, heey vartan madhyavarti vartanachya ulat achook aahe. qamee aanee uchch urjevareel achukateche anukrame aavashyak kaaran mhanje db model (i) qamee frequenciver achook 'dispersion relation' sootra _ 51 dete aanee (ii) prati frequency madhyantaratil compananchya sankhyeshi sambandhit sthitee sootra _ 52 chya achook ghanateshi sambandhit aahe.
debiche vyutpann.
debine tyache samikaran kahise vegale aanee adhik sope kele. satatya yantrikicha vapar karoon, tyala aadhalale qee vishisht mulyapeksha qamee varanvarta asalelya companshil avasthanchi sankhya hii lakshane nasaleli aahe.
jyaa sutramadhye _ 54 heey pariman aahe aanee sootra _ 55 haah eka ghatak aahe jyachi ganana tyanni lavchikta gunank aanee ghanatevarun kelee. heey sootra tii tapmanavar (einsteinne tyachya modelmadhye aadhich vaparlele) harmonic ausilatorchya apekshit urjesah ekatra kelyane urja milel
jarr compan varanvarta anantparyant chaloo rahili. haah form 1 vartan sootra deto jeey qamee tapmanat yogya aahe. parantu dbla samajale qee aine anoonsathi sootra _ 58 companshil avasthanpeksha jaast asoo shakat nahee. tyanni asey gruheet dharale qee anoo ghanamadhye, companshil avasthanchya varanvarteche varnapat vareel niyamache palan karat raheel, jastit jaast varanvarta sootra _ 59 paryant nivadalele jenekarun ekoon avasthanchi sankhya 
debila maheet hote qee hii dharana kharokharach yogya nahee (uchch varanvarta gruheet dharalyapeksha adhik jawalun antar asate), parantu tee uchch tapmanavar (dulong-pitt kayada) yogya vartanachi hummy dete. tyanantar urja dilee jaate
sootra _ 63 sathee sootra _ 29 badaloon,
jethe sootra _ 65 heey nantar third-aurder db functionche naav dilele karya aahe.
aanakhi eka vyutpatti.
pratham aapan compan varanvarta vitaran midvato; khaleel vyutpatti appendix vi pasoon aadharit aahe. tri-aayami isotropic lavchik ghanpadarth aanee aayatakruti samantar aakarachya anoonsah bajuchya lambichya sutrasah _ 66 chaa vichar qaraa. lavchik lahari lahari samikaranache palan karel aanee samtal lahari asateel; tarang vahak sootra _ 67 chaa vichar qaraa aanee sootra _ 68 paribhashit qaraa. lakshat ghya qee aamchyakade aahe
lahari samikaranache upay aahet
aanee sootra _ 71 varr seema aut sootra _ 70 saha, aamchyakade aahe
jethe sootra _ 72 heey sakaratmak purnank aahet. () laa () madhye badalane aanee dispersion relation sootra _ 73 dekhil vaparane, aamchyakade aahe
vareel samikaran, nishchit varanvarta sutrasathi _ 75, "mode space" madhye lambvartulachya aathvya bhagache varnan karate (aathve kaaran sootra _ 72 sakaratmak aahet). sootra _ 75 peksha qamee varanvarta asalelya paddhatinchi sankhya mhanje lambvartulachya aat avibhajya bindunchi sankhya, jee sootra _ 78 (i) chya maryadet aahe.eae. khoop mothya samantar paipasathi) lambvartulachya aakarmanacha andaj lavala jaau shakato. mhanoon, shreni sootra _ 80 madheel varanvartesaha _ 79 sutranchi sankhya aahe
jethe sootra _ 81 heey samantar paip kelele aakarman aahe. lakshat ghya qee rekhansh dishene laharincha veg anuprastha dishepeksha vegala aahe aanee laharinche dhruvikaran eka prakare rekhansh dishene aanee done prakare anuprastha dishene kele jaau shakate; asha prakare aapan sootra _ 82 paribhashit karato.
yaa vyutpattinantar, aapan compan sootra _ 83 chya varanvartesathi eka uchch maryada paribhashit karato; ghanamadhye n anoo asalyane, 3n quantum harmonic ausilator (pratyek x-, y-, z-dishene 3) aahet jeey frequency sootra _ 84 chya shrenivar dolan karatat. mhanoon aapan sootra _ 85 vaparoon nirdharit karoo shakato.
sootra _ 86 paribhashit karoon, jethe "k" haah boltzmann sthirank aahe aanee "h" haah plank sthirank aahe, aanee () gunile () badaloon, aaplyala milate.
hii vyakhya adhik manak aahe. frequency sootra _ 75 varr dolan karananya serve ausilatorsathi aapan urjeche yogdan shodhu shakato. quantum harmonic ausilatormadhye urjeche sootra _ 88 asoo shakate jethe sootra _ 89 aanee maxwell-boltzmann sankhyiki vaparoon, urja sootra _ 90 asalelya kananchi sankhya aahe
frequency sootra _ 75 asalelya ausilatorsathi urja yogdan nantar aahe
heey sootra _ 93 lakshat gheuun (kaaran tethe frequency sootra _ 75 saha dolan karnare sootra _ 94 mode aahet), aamchyakade aahe
varoon, aaplyala 1/a sathee eka abhivyakti miloo shakate; tyala () madhye badaloon, aaplyala aahe
ν utpannachya sandarbhat ektrikaran
qamee tapman maryada.
jarr _ 99 sootra asel tarr db ghanache tapman qamee asalyache mhatale jaate, jyamule
yaa nishchit avibhajyache achook moolyankan kele jaau shakateh
qamee-tapman maryadet, varr namood kelelya debby modelchya maryada lagoo hott naheet aanee teey (dhvanyatmak) ushnata kshamata, tapman, lavchik gunank aanee prati anooche praman (nantarche praman debby tapmanat samavisht kelele aahe) yanchyatil yogya sambandh dete.
uchch tapman maryada.
jarr sootra _ 102 asel tarr db ghanache tapman jaast asalyache mhatale jaate. sootra _ 104 asel tarr sootra _ 103 vaparoon
kuthe?
haah dulong-pitt kayada aahe aanee jaree too susangatata vicharat ghet nasla taree baryapaiki achook aahe, jyamule ushnatechi kshamata aanakhi vadhate. ghan padarthachi ekoon ushnatechi kshamata, jarr too vahak kinwa ardhavahak asel tarr tyat electronche naan-nagany yogdan dekhil asoo shakate.
debby viruddha einstein.
debby aanee einstein model prayogik detashi jawalun julatat, parantu debby model qamee tapmanat barobar aahe tarr einstein model barobar nahee. modelmadhil farak pahanyasathi, doghannahi naisargikaritya ekach akshanchya sanchavar rekhatale jaeel, parantu heey lagech shakya nahee kaaran einstein model aanee debby model donhi ushnata kshamatesathi "karyatmak swaroop" pradan karatat. model mhanoon, tyanna tyanchya vastavik jagachya samkakshanshi sambandhit karnyasathi scalechi aavashyakta asate. ekhadyala asey disu shakate qee einstein modelche praman _ 107 yaa sutradvare dile aaheh
debby modelche praman sootra _ 29, debby tapman aahe. donhi samanyatah prayogik mahitishi models basavoon aadhaltat. (dhwani aanee sphatik parimananchya vegavarun saiddhantikdrushtya debby tapman mojale jaau shakate.) kaaran done paddhati vegvegalya dishene samasyekade jaatat aanee vegvegalya bhoomiti, einstein aanee debby scale samaan aahet, mhanje
yacha arth tyanna akshanchya ekach sanchavar rachanyat kahee arth nahee. teey ekach goshtiche done model aahet, parantu vegvegalya pramananche aahet. jarr konee ashee vyakhya kelee
mug konee taree mhanoo shakel
aanee, yaa doghancha sambandh jodanyasathi, aapan gunottar shodhale pahije
einstein ghan heey-frequency quantum harmonic ausilator, sootra 114 neey banalele aahe. tee frequency, jarr kharokhar astitvat asel tarr tee ghanatil aavajachya vegashi sambandhit asel. jarr ekhadyane dhvanichya prasarachi kalpana anooncha ekmekanna "marnyacha" kram mhanoon kelee, tarr heey spasht hote qee dolan varanvarta hii anoo jaali, sootra _ 115 dware tikanari kiman tarangalambishi sambandhit asane aavashyak aahe, jethe
jeey karate 
sootra _ 117
aanee mhanoon magitalele praman aahe
gunottar vaparoon, donhi models ekach aalekhavar rekhatalelya asoo shakatat. heey gunottar heey trimiteey golakachya eka ashtakachya ghanatechya aanee tyat asalelya ghanchya ghanatechya gunottaratil ghanmool aahe, joou vareel urja avibhajya andaje kartana dbdware vaparla jaanara fakt sudharana ghatak aahe. vaikalpikaritya, done tapmananche gunottar heey einsteinchya ekach varanvarteche gunottar mhanoon pahile jaau shakate jyavar serve ausilator dolan karatat aanee debychi kamal varanvarta. einsteinchi ekal varanvarta nantar db modelsathi upalabdh asalelya varanvartanchi sarasari mhanoon pahili jaau shakate.
tapman sarani kharab qaraa.
jaree db model purnapane barobar nasale taree, itar yogdan (jase qee atyant gatishil vahak electron) nagany asalelya insulating, sfatikeey ghan padarthanchya qamee tapmanachya ushnatechya kshamatesathi teey eka changla andaj dete. dhatunsathi, ushnatemadhye electronche yogdan sootra _ 49 chya pramanat asate, jeey qamee tapmanat jaalichya companansathichya db sootra _ 1 parinamanvar varchasva gajavate. yaa prakaranat, debby model keval vishisht ushnatemadhye jaaliche "yogdan" andaje asalyache mhatale jaau shakate. khaleel taktyat anek shuddha ghatak aanee nilmanisathichya debby tapmananchi yadee aaheh
debby modelche prayogik mahitishi julane heey debby tapmanala tapmanavar avalambun rahoo deuun anekada ghatanatmakdrushtya sudharle jaate; udaharanarth, tapman nirpeksh shunyapasoon sumare 100 keey paryant jaat asatana panyachya barfache moolya sumare 222 keey teey 300 keey paryant vadhate.
itar ardh-kananparyant vistar.
itar bosonic ardh-kanansathi, udaa.jee., phononaivaji pheromagnetmadhil magnonsathi (parimanit firki lahari) ekhadi vyakti samaan parinam midvu shakate. yaa prakaranat qamee varanvartevar gatee aanee urjeche vegvegale prasaran sambandh asatat, udaa.jee., magnonchya babatit sootra _ 121, phononsathi sootra _ 122 aivaji (sootra _ 123 saha). ekachi avasthanchi ghanata dekhil vegvegali asate (udaa.jee., sootra _ 124). parinami, ferromagnetmadhye ekhadyala ushnata kshamatet magnon yogdan milate, sootra _ 125, jeey purese qamee tapmanat, sootra _ 126, phonon yogdanavar varchasva thevate. dhatunmadhye, yaulat, ushnata kshamatet mukhya qamee-tapman yogdan, sootra _ 127, electronmadhun yete. heey fermionic aahe aanee somerfeldchya mukt electron modelkade parat jaatana vegvegalya paddhatinni ganana kelee jaate.
dravpadarthanparyant vistar.
dirghkalapasun assa vichar keela jaat hota qee phonon siddhant dravanchi ushnatechi kshamata spasht karoo shakat nahee, kaaran dravpadarth keval anudairdhya tikavoon thevatat, parantu aadvya phononla tikavat naheet, jeey ghanpadarthanmadhye ushnatechya kshamatechya 2/3 bhagasathi jababdar asatat. tathapi, neutrons aanee ksh-kiranansah brilloin vikhuranyache prayog, yakov franklechya antardnyanachi pushti karatat, asey darshavile aahe qee trikoni phonon dravpadarthanmadhye astitvat aahet, jaree frankle frequency navachya threasholdchya varchya frequenciparyant maryadit aahet. bahutek urja yaa uchch-varanvarta paddhatinmadhye asalyane, db modelcha eka sadha badal sadhya dravpadarthachya prayogik ushnata kshamatecha changla andaj denyasathi puresa aahe.
deby frequency.
db frequency (chinh sootra _ 128 kinwa sootra _ 129) haah db modelmadhil eka mapdand aahe. heey vastumanachya harmonic sakhalichya laharinsathi cutt-auph koneey varanvartecha sandarbh dete, jyacha vapar sphatik jaalimadhye aayanchya halchaliche varnan karnyasathi aanee adhik visheshtah, asha sphatikanmadhil ushnata kshamatecha uchch tapmanasathi sthir asalyacha yogya andaj lavnyasathi keela jaato (dulong-pitt kayada). pieter debby yanni 1912 madhye hii sandnya pahilyanda mandali.
yaa sampoorn lekhat niyatkalik seema atty gruheet dharalya jaatat.
vyakhya.
failav sambandh asey gruheet dharale
sphatikatil aavajacha veg _ 131 yaa sutrasah; aanee tarang vahak k, deby frequencyche moolya khalilpramane aaheh
eka aayami ekparmanvik sakhalisathi debby frequency samaan aahe
sootra 133 saha, jevha pranali tyachya urjechya jaminichya sthiteet asate tevha sakhalitil done shejari anunmadhil antar (yaa prakaranat yacha arth assa aahe qee konatehi anoo ekmekanchya sandarbhat firat naheet); sootra 16 sakhalitil anoonchi ekoon sankhya; aanee sootra 2 pranalicha aakar (khand) (sakhalichi lambi); aanee sootra 136 hii resheey sankhya ghanata aahe. jethe khaleel sambandh aaheh sootra _ 137.
dvimiteey ekparmanvik chauras jaalisathi debby frequency samaan aahe
jethe sootra 133 aanee sootra 140 purvisarkhech aahet; sootra 141 heey prushthbhagacha aakar (kshetra) aahe; aanee sootra _ 142 heey prushthbhagachi sankhya ghanata aahe.
trimiteey ekparmanvik adim ghan sphatikasathi, db frequency samaan aahe
jethe sootra 133 aanee sootra 140 purvisarkhech aahet; sootra _ 146 pranalicha aakar; aanee sootra _ 147 khand sankhya ghanata.
sphatikatil dhvanicha veg anoonche vastuman, tyanchya parasparsanvadachi takad, pranalivaril daab aanee/kinwa firki lahariche dhruvikaran (rekhansh kinwa aadvya) yavar (itarambarobarach) avalambun asoo shakato, parantu pudhil goshtinmadhye aapan pratham konatyahi dhruvikaranasathi dhvanicha veg samaan asalyache gruheet dharoo (heey gruheet matra durgami parinam dett nahee).
vastumanachya eka-aayami sakhalisathi gruheet vikhurane sambandh sahajpane chukiche siddha jhaale aahe, parantu debichya modelmadhye heey samasyapradhan siddha jhaale nahee.
debichya tapmanashi sambandh.
debby tapman sootra _ 148, debby modelmadhil aanakhi eka mapdand, sambandh sootra _ 149 dware debby frequencieshy sambandhit aahe jethe sootra _ 150 haah qamee jhalela plank sthirank aahe aanee sootra _ 151 haah boltzmann sthirank aahe.
debiche vyutpann.
trimiteey sphatik.
debichya ushma kshamatechya vyutpattimadhye too pranalichya serve sambhavya paddhatincha saransh deto. mhanjeh vegvegalya disha aanee dhruvikaranansah. tyane gruheet dharale qee pratyek dhruvikaranachya paddhatinchi ekoon sankhya sootra _ 58 (sootra _ 16 saha pranalitil vastumanachya pramanat) kinwa ganiteey bhashet aahe.
jethe donhi bajunche sootra _ 155 heey teen dhruvikarananmule aahe, mhanoon eka vishisht dhruvikaranasathi berries serve paddhatinvar chalate. debine heey gruheet dharale kaaran tyala shastriy yantrikivarun mahit hote qee vastumananchya sakhalitil prati dhruvikaran paddhatinchi sankhya nehmi sakhalitil vastumanachya pramanachya barobarichi asavi.
davi bajoo aata db frequenciver kashee avalambun aahe heey darshavinyasathi spasht kelee jaanar aahe (yethe fakt cutt-auph frequency mhanoon sadar kele aahe, mhanjeh db frequencipeksha jaast frequency astitvat nahee), jenekarun tyasathi eka abhivyakti miloo shakel.
sarvapratham, sootra _ 2 khoop mothe aahe asey gruheet dharoon (sootra _ 2, sootra _ 2 saha teenpaiki konatyahi dishene pranalicha aakar) konatyahi dishene sarvat lahan tarang sadish andaje asey manale jaau shakateh sootra _ 159, sootra _ 160 saha. niyatkalik seema paristhitimule lahan tarang sadish astitvat asoo shakat naheet. tyamule berries 4 hoil
jethe sootra _ 162; sootra _ 163 haah pranalicha aakar aahe; aanee avibhajya serve sambhavya paddhatinvar (berries mhanoon) aahe, jeey eka maryadit kshetra (cutt-auph frequencine vedhalele) manale jaate.
tribhuj samakalan sootra _ 164 chya nirpeksh mulyachya serve sambhavya mulyanvar ekal samakalan mhanoon punarlikhit kele jaau shakate (pahah golakar nirdeshankansathi jacobian). parinam assa
sootra 166 saha, debby frequencieshy sambandhit wave vectorche paripoorn moolya, mhanoon sootra 167.
aaplyala dispersion sambandh sootra _ 168 mahit asalyane, heey serve sambhavya sootra _ 169 varr avibhajya mhanoon lihile jaau shakate.
avibhajya sodavalyanantar teey punha sootra _ 171 samatulya kele jaate
nishkarshah
3 dii spacemadhye eka aayami sakhali.
anoonchya eka aayami sakhalisathi samaan vyutpatti kelee jaau shakate. modchi sankhya aparivartit rahate, kaaran ajunahi teen dhruvikaran aahet. tyamule
urvarit vyutpatti mageel samaan aahe, mhanoon punha davya hatachi bajoo punha lihili jaate;
shevatchya tappyat donane gunakar karane mhanje pahilya avibhajyamadhil samakalan summ aahe aanee samakalanchya seema moodabaddal samamiteey aahet, mhanoon avibhajya 2 chya ghatakane scaling kelyanantar 0 teey sootra 176 paryant punarlikhit kele jaau shakate. badal formula _ 177 chya badalyat lagoo karane, aata aaplya seema 0 teey formula _ 178 aahet, jeey aaplyala aapale sarvat yogya avibhajya dete. 
aamhi chaloo thevato;
nishkarshah
dvimiteey sphatik.
dvi-aayami sphatikasathi samaan vyutpatti kelee jaau shakate. punha, modchi sankhya aparivartit rahate, kaaran ajunahi teen dhruvikaran aahet. vyutpatti mageel done samaan aahe. aapan tyach samikaranane suruvat karato,
aanee mug davya hatachi bajoo punha lihili jaate aanee sootra _ 58 chya barobarine kelee jaate.
jethe sootra _ 184 haah pranalicha aakar aahe.
nishkarsh
dhruvikaranamule farak padoo shakato.
parichayat namood kelyapramaneh sarvasadharanpane, anudairdhya laharincha lahari veg aadvya laharinpeksha vegala asato. spashtatesathi teey aadhi samaan manale jaat hote, parantu aata aapan teey anuman sodoon deto.
prasaran sambandh sootra _ 186 banato, jethe sootra _ 187, jeey teen dhruvikarananshi sambandhit aahe. cutt-auph frequency (db frequency) tathapi sootra 188 varr avalambun nahee. aanee aapan ekoon modchi sankhya sootra 189 mhanoon lihu shakato, jeey punha sootra 171 chya barobariche aahe. yethe modavareel berries sootra 188 varr avalambun aahe (jaree spashtapane namood keleli nahee).
eka pariman.
punha ekada modavareel berries punha lihili jaate
parinam assa
asha prakare db frequency aadhalate
kinwa done aadvya dhruvikarananche samaan asane asey gruheet dharoon (samaan tappyacha veg aanee varanvarta asane)
heey sambandh purvi aadhalalelya (jevha dhruvikaranane farak padala nahee) samatulya aahet heey sootra _ 196 seet karoon tapasale jaau shakate.
done pariman.
dvimiteey sphatik shodhanyasathi samaan vyutpatti kelee jaau shakate (vyutpatti mageel vyutpattishi samaan aahe)
kinwa done aadvya dhruvikaran samaan aahet asey gruheet dharoon (jaree done parimanansathi serve dhruvikaran bhinn asalyas teey adhik tarkik asel):
punha, heey sambandh heey purvi aadhalalelya sambandhanchya samatulya aahet heey _ 196 heey sootra seet karoon tapasata yete.
teen pariman.
tri-aayami sphatik shodhanyasathi samaan vyutpatti kelee jaau shakate (vyutpatti mageel vyutpattishi samaan aahe)
kinwa done aadvya dhruvikaran samaan aahet asey gruheet dharoon (jaree teen parimanansathi serve dhruvikaran samaan asel tevha teey adhik tarkik asel):
punha, heey sambandh heey purvi aadhalalelya sambandhanchya samatulya aahet heey _ 196 heey sootra seet karoon tapasata yete.
vastavik prasaran sambandhasah vyutpatti.
hii samasya adhik guntaguntichi banavoon tee adhik antardrushtine kelee jaau shakate. prasaran sambandh sootra _ 203 vaparnyaivaji, yogya prasaran sambandh aata gruheet dharale jaanar aahe. shastriy yantrikivarun asey dnyat aahe qee, ekmekanshi susanvad sadhananya vastumanachya samaan antarachya sakhalisathi prasaran sambandh khalilpramane aaheh
sootra _ 204
haah sambandh rachalyanantar, heey spasht hote qee cutt-auph tarangalambicha debicha andaj shevati yogya hota. kaaran sootra 205 peksha mothya pratyek tarangsankhyesathi (mhanjeh sootra 136 heey sootra 207 peksha lahan aahe) sootra 208 peksha lahan asaleli tarangsankhya samaan koneey varanvartesaha aadhalu shakate. yacha arth mothya tarangsankhyesah modasathi parinami bhautik prakatikaran lahan tarangsankhyesah vegale karata yet nahee. tyamule, prasaran sambandhancha abhyas pahilya brilloin zoan i paryant maryadit asoo shakato.eae. formula_209.this sathee heey shakya aahe kaaran pranalimadhye animated chitrat darshavilyapramane discretised points asatat. dispersion sambandhala _ 210 yaa sutrane vibhajit karane aanee _ 210 yaa sutrasathi _ 208 heey sootra ghalane, aaplyala _ 213 yaa sutrasah laharicha veg aadhalato
sootra _ 214
mool prasaran sambandhat fakt sootra _ 215 samavisht karoon aaplyala aadhalate
sootra _ 216
heey parinam ekatra karoon punha ekada toch parinam disun yeto
sootra _ 217
tathapi, dviaadhari sakhalisathi (aanee adhik jatil sakhalisathi) sambandhit cutt-auph varanvarta (aanee taranglambi) phaar achook nasate, kaaran cutt-auph taranglambi duppat mothee asate aanee prasaran sambandhat done shakha asatat (dviaadhari sakhalisathi). adhik aayami pranalinsathi cutt-auph frequenciecha andaj dbne achookpane vartavala hota qee nahee heey dekhil yavarun nishchit nahee.
paryayi vyutpatti.
eka aayami sakhalisathi haah parinam aliyasingavaril siddhant vaparoon dekhil punarutpadit keela jaau shakato. niqvist-shannon sampling pramey khaleel vyutpattimadhye vaparla jaato; mukhya farak assa aahe qee pudhil vyutpattimadhye vivechana velevar nahee, tarr avkashat aahe. jarr aapan shevatchya parichchhedatun yogya prasaran sambandh vaparla, tarr cutt-auph frequencyche moolya purviche (donada) moolya qaa aahe heey dusarya antardrashtipurna margane spasht hoil. tyamule punha,
sootra _ 218
asey gruheet dharale jaate.
heey vyutpann mageel vyutpattichya purnapane samatulya aahe, mhanjeh parinam punarprapt karnyasathi samaan gruhitke kelee jaatat. too qamee-adhik pramanat achook nahee, too fakt eka vegala drushtikon aahe.
cutt-auph frequency kuthe asavi heey tharvanyasathi, tarangalambicha cutt-auph kuthe asava heey pratham tharvane upayukt tharate. dispersion sambandhavarun aaplyala mahit aahe qee sootra _ 219 sathee pratyek modchi punaravrutti kelee jaate, mhanoon cutt-auph taranglambi sootra _ 220 varr asel. yavarun aanee niyatkalik seema atinvarun tumhi lagech pahoo shakata qee prati dhruvikaran modchi ekoon sankhya sootra _ 16 asel. mageel parichhedachya jee. aay. eff. madhye pahilyapramane yache kaaran asey aahe qee sootra _ 222 peksha qamee taranglambi asalelya pratyek vevhchi jaaga sootra _ 223 peksha jaast taranglambi asalelya waveney ghetali jaau shakate jenekarun samaan bhautik parinam parat miloo shakel.
tathapi, cutt-auph _ 224 yaa sutravar qaa asava hyachya tarkat mageel parichhedacha (yogya) prasaran sambandh dekhil aavashyak nahee. kaaran, darshavilyapramane, sootra _ 223 peksha jaast taranglambi asalelya laharich dusyachyyaa samaan bhautik parinam deu shakatat. tyamule yogya prasaran sambandh naan vaparta (kinwa debipramanech shastriy yantrikiche dnyan) phononchya cutt-auph tarangalambicha yogya andaj lavnyacha haah aanakhi eka marg aahe. tathapi, debyne gruheet dharalelya chukichya prasaran sambandhancha vapar karoon, qamee taranglambi asalelya laharinchi varanvarta jaast asel, parantu vastumanachi sapeksh halchal samaan asel, tyamule heey naveen paddhati prastut karat nahee.
yacha parinam punha sootra _ 226 madhye hoto, jeey prastut karate
sootra _ 217
yethehi konata dispersion sambandh vaparla gela aahe yane kahee farak padat nahee (yogya kinwa vaparleli eka dbi), teech cutt-auph frequency aadhalel.
durdaivane, dvi-kinwa tri-aayami sphatikasathi teech paddhat (titaki sahaj) vaparali jaau shakali nahee, kaaran karnareshechya laharinchi lambi mothee asel, jyacha andaj lavane dekhil katheen aahe.