Fourier dzakatevedzana: nhoroondo uye pokudzora kwemasvomhu nomazviri kukura nesayenzi

Fourier dzakatevedzana - maonero ichi pamadiro vakasarudzwa Unoshanda kusvika nguva pamusara. Kazhinji mashoko, ichi mhinduro anonzi kuwedzera chinhu pamusoro orthogonal hwaro. Kuwedzera kushanda muna Fourier akatevedzana zvakasiyana chishandiswa chine simba nekugadzirisa matambudziko akasiyana-siyana nokuda ehupfumi hwokushanduka iri kusangana, Kusiyanisa, uyewo iya iri nharo okuti uye convolution.

Munhu asina kujairana dzakakwirira masvomhu, uyewo mabasa French musayendisiti Fourier, Ungangove rwausingazonzwisisi chii "pemisara" uye chii chavanoita. Asi kuchinja uku zvikuru zvakasimba akapinda hwedu. Rinoshandiswa kwete masvomhu chete, asiwo yefizikisi, mishonga, vanachiremba, nyanzvi, seismologists, oceanographers nevamwe. Ngatitsigirewo kunyatsotarisisa nemabasa musayendisiti mukuru French akaita kuwanikwa, mberi nguva yake.

Murume Fourier rokuchinja

Fourier dzakatevedzana mumwe nzira (pamwe ongororo uye nevamwe) of Fourier rokuchinja. Izvi zvinotora nzvimbo nguva dzose munhu anonzwa chero kurira. nzeve dzedu rwoga vatendeuki ruzha wave. Oscillatory kufambira zvokutanga inoramba iri kwakavharwa nenyama svikiro vari kuwedzerwa dzinoti (the chakarondedzera) anotevedzana vhoriyamu tsika kuti zvenzwi zvakakwirira siyana. Next, uropi anomuchinja date ichi mu vanoshamwaridzana ruzha kwatiri. Zvose izvi zviri kuwedzera yedu chido kana kuziva pacharo, asi kuti tinzwisise kwemasayendisiti kuti kutora makore akawanda kudzidza yepamusoro masvomhu.

Verenga zvakawanda pamusoro Fourier rokuchinja

The Fourier rokuchinja anogona kuitwa Analytical, sezviverengo uye dzimwe nzira. Fourier dzakatevedzana dziri nhamba muitiro kuti decomposing chero oscillatory pezvakaitwa - kubva gungwa nemafungu uye mafungu nechiedza rezuva zvinogara (uye vamwe nyeredzi zvinhu) basa. Kushandisa kwemasvomhu nzira idzi, zvinokwanisika kuti nekukatanura basa, kumirira chinhu oscillatory pezvakaitwa munyika dzinoverengeka sinusoidal zvinoriumba kuti ndibve shoma kusvikira kunonyanyisa ndiEzekieri. The Fourier rokuchinja riri basa achirondedzera Pasika uye amplitude pamusoro sinusoids unoenderana imwe maitikiro. Izvi zvinogona kushandiswa pakugadzirisa equations chaizvo kunzwisisa inotsanagura simba pezvakaitwa kuitika pasi zvinoitwa kupisa, chiedza kana yemagetsi simba. Uyewo, Fourier akatevedzana rinoshandiswa kusiyanisa DC zvinoumba mu waveforms yakaoma, zvichiita kuti zvakarurama adudzire experimental zvaanorumbidza mukurapa, kemisitiri uye zvenyeredzi.

nhoroondo mashoko

Kuvambwa baba dzidziso iyi ndiyo yemasvomhu French Zhan Batist Zhozef Fure. zita rake Papera uye kuchinja uku rakashevedzwa. Pakutanga, masayendisiti akashandisa nzira kudzidza uye kutsanangura nzira yaJehovha chafariz conductivity - kupisa kupararira mune zvinhu zvakaoma. Fourier akataura kuti kutanga mugari kuparadzira chafariz chinozunguzirwa inogona yaora kupinda nyore sinusoid, rimwe nerimwe richava kwayo tembiricha zvishoma uye kunonyanyisa, uyewo yaro danho. Nokudaro chikamu chimwe nechimwe yakadaro kuyerwa kubva shoma kune kunonyanyisa ndiEzekieri. The kwemasvomhu basa kuti kurondedzera okumusoro nhongonya ari munyonga, uyewo Pasika mumwe harmonic, yainzi Fourier rokuchinja pamusoro tembiricha kuparadzirwa kutaura. Munyori dzidziso wakaderedzwa kunopindirana Kuparadzirwa basa iri rakaoma tsananguro akarongeka, ari nyore chaizvo kugadzirisa nhamba pakati nenguva dzirikure mashandiro anoita Sine uye cosine, mune mari yekupa kutanga kuparadzira.

Nheyo yokutendeutsa uye maonero vairarama

Vakararama of musayendisiti - vaikudzwa mathematicians chepakuvamba namapfumbamwe remakore - haana kugamuchira dzidziso iyi. chikuru The kuramba raiva kufarirwa Fourier kuti discontinuous basa kutsanangura yakarurama mutsetse kana munyonga chakabvamburanyiwa, rinogona achimiririrwa chitsama sinusoidal mashoko chichiitwa. Somuenzaniso, funga "danho" Heaviside: ukoshi chayo razero kuruboshwe pamusoro pakakoromoka uye mumwe kurudyi. basa iri rinotsanangura kutsamira pane Magetsi pamusoro nguva shanduka nokuda kuvhara cheni. Contemporary dzidziso panguva iyoyo, kumbobvira vasangana ezvinhu akadaro, kana munhu discontinuous mashoko chaizoshatiswa yakarondedzerwa mubatanidzwa kwechiropa, zvakajairika mashandiro, zvakadai exponential, sine wave, Linear kana quadratic.

Chii zvakashatirisa French mathematicians iri dzidziso Fourier?

Pashure pezvose, kana yemasvomhu zvakanaka kuti nharo, Saka, akapedzisa isingaperi trigonometric Fourier series, zvinokwanisika kuwana mufananidzo yakarurama danho rokuti, kunyange kana akaronga matanho zvakafanana. Mukutanga namapfumbamwe remakore, mashoko aya zvaiita penga. Asi pasinei ose chokwadi, mathematicians vakawanda akawedzera yezvinoreva nokudzidza Fani iyi, kutamira mhiri chafariz conduction zvidzidzo. Zvisinei, masayendisiti akawanda akaramba kutamburira mubvunzo: "? Ko Verengai Sine Wave akatevedzana converges kune chaiyo ukoshi discontinuous basa"

Convergence pamusoro Fourier dzino: muenzaniso

Nyaya convergence anomuka nguva dzose unoda summation ane magumo munhevedzano nhamba. funga muenzaniso chaiwo nokuti kunzwisisa chiitiko ichi. Unogona nokusingaperi kusvika rusvingo, kana danho rimwe nerimwe riri hafu yapfuura? Ngatitii muri mamita maviri kubva chinangwa, danho rokutanga pedyo kumativi hafu nzira, rakatevera - chiratidzo a-kumativi matatu, uye pashure wechishanu, iwe richavakunda dzinenge 97 muzana nzira. Zvisinei, hazvinei unofanira sei matanho akawanda kuitwa kana, kuti akaronga chipfuro iwe kusvika vasingaiti kwemasvomhu pfungwa. Achishandisa kusanganiswa Masvomhu, tinogona kuratidza kuti pakupedzisira zvingava pedyo kuti akapiwa kure pamadiro diki. Izvi yakaenzana uchapupu nekuratidza kuti mari imwe hafu, imwe wechina, uye zvichingodaro. E. vanowanzotevedzera kubatana.

Nyaya convergence: Kuuya wechipiri, kana chishandiso Ishe Kelvin

Ataurazve mubvunzo akasimuka mukupera namapfumbamwe remakore, kana Fourier dzino vakaedza kushandisa kufanotaura kusimba pakati ebbs uye Rwako. Panguva iyoyo, Ishe Kelvin yagadzirwa mudziyo chinhu analoginen kombiyuta izvo akabvumira vezvikepe nezvikepe uye mutengesi zvomumvura tarisisa chinhu dzinongoitika kutanga. Izvi mashandiro kunorondedzerwa akagadzirwa maitikiro uye amplitudes petafura kureba nemafungu uye runoenzanirana nguva nenguva, vachinyatsoongorora akayera mu chiteshi rose gore. Mumwe parameter chinhu sinusoidal pakuderedza okuti kumiswa dzakakwirira uye raiva rimwe dzose zvinoriumba. Nokuyerwa zvabuda ndivo mazano kuna kadambari mudziyo Ishe Kelvin, synthesizing munyonga kuti afanotaura Kukwirira mvura ave kushanda zvinotevera gore. Nokukurumidza, makakombama izvi zvakabvumirwa dzose zvengarava zvenyika.

Uye kana izvi huchavhuniwa discontinuous basa?

Panguva iyoyo, zvaiita pachena kuti mudziyo vachifanotaura munhu hyökyaalto, dzakawanda zvinhu nhoroondo anogona ngaaverenge vakawanda maitikiro uye amplitudes, uye saka anopa zvakawanda yakarurama nokufembera. Kunyange zvakadaro, ndakaona kuti muenzaniso uyu haasi kuchengetwa zviitiko apo naTidhari mashoko kuchazoratidzwa synthesized, ine unopinza kusvetuka, kureva, muri discontinuous. Kuri kuti midziyo kupinda umboo kubva patafura nguva pfungwa, izvozvo anoti vashoma Fourier coefficients. Kuwanazve chepakutanga basa nokuda sinusoidal zvinoriumba (maererano akawana coefficients). Kupokana chepakutanga uye reconstructed okuti kuyerwa chero nguva. Kana achidzokorora Masvomhu uye kuenzanisa zvinogona kuonekwa kuti kukosha wokutsauka chikuru haina ipfupi. Zvisinei, vari mupasi munharaunda unoenderana kusvikira atsamwira, uye mumwe pfungwa vanowanzotevedzera razero. Muna 1899, izvi mugumisiro hwakasimbiswa hunonzi Joshua Willard Gibbs pamusoro Yale University.

Convergence pamusoro Fourier akatevedzana uye kukura wemasvomhu rose

Fourier kuongorora hairevi mashoko ine magumo uwandu watsemuka pane imwe penguva. Vakawanda Fourier series, kana chepakutanga basa hunomiririrwa mugumisiro chaiko chaiko kuyerwa, nguva dzose wekuita. Mibvunzo pamusoro convergence kuti muitiro uyu makirasi chaizvo mitambo zvakakonzera mapazi matsva masvomhu, dzakadai dzidziso generalized mabasa. Rinobatanidzwa mazita akadai Schwartz, J .. Mikusiński uye J. Temple. Pasi dzidziso iyi, yakajeka uye chaiwo yokungotaura hwaro kwakadaro okuti hwakagadzwa sezvo Dirac Delta basa (rinotsanangura munzvimbo imwe chete nzvimbo, kusimbiswa ane kamwoyo pedyo pfungwa) uye "danho" Heaviside. Kuburikidza nebasa iri Fourier akatevedzana vakava kushanda kuti nekugadzirisa equations uye matambudziko, izvo zvinosanganisira nzwisisa pfungwa: pfungwa nokuchengeta, pfungwa guru, rinonzi magnetic dipoles, uye kuramba mutoro pamusoro danda.

Fourier nzira

Fourier series, maererano nenhungamiro kukanganiswa, kutanga kupinda nyore pamwe manyowa kunzwisisa siyana. Somuenzaniso, kuchinja kupisa aibuda nokuda ndima yayo kuburikidza siyana zvipingamupinyi kupisa insulating zvinhu kwemashifiti chimiro kana kuchinja pasi pose - nokudengenyeka kwenyika, kuchinja ari muchadenga of zvomuchadenga muviri - zvinoitwa mapuraneti. Kazhinji, equations izvi achirondedzera nyore dzenhoroondo hurongwa zvokutanga akagadzirisa mumwe nomumwe wavelength. Fourier kwakaratidza kuti nyore mhinduro inogona kupfupikiswa sezvo mamwe mabasa akaoma. Mumutauro wemasvomhu, Fourier dzinoti - imwe nzira kuti kuzviisa pasi rwekutaura Verengai harmonic - cosine uye sine wave nemafungu. Saka, kuongorora izvi aizivikanwawo pasi zita "harmonic kuongorora".

Fourier dzakatevedzana - inzira yakanaka kune "zera kombiyuta"

Asati pakasikwa kombiyuta michina Fourier nzira ndiyo chombo vakanakisisa zvombo masayendisiti tichishanda chinozunguzirwa mamiriro edu nyika. Fourier dziri kunzwisisa chimiro unokubvumira kwete chete matambudziko nyore kuti vari amenable kutungamirira nokushandisa mitemo Newton kwakaita makanika, asiwo inokosha equations. Vazhinji Zvakawanikwa Newtonian Sayenzi rechigumi nemapfumbamwe remakore vakava zvinogoneka chete nokuda Fourier nzira.

Fourier dzakatevedzana nhasi

With kukura Fourier rokuchinja makombiyuta zvakakura kuvandudza. nzira iyi zvakasimba midzi zvinenge muminda yose zvesainzi. Somuenzaniso, digital maCD uye mavhidhiyo. Implementation Its wakaitirwa dzidziso kugadzirwa French yemasvomhu okutanga namapfumbamwe remakore zvinogoneka chete kuvonga. Nokudaro, Fourier dziri kunzwisisa chimiro akabvumira kuti budiriro kudzidza muchadenga. Uyezve, zvave sei kudzidzwa wefizikisi ose semiconductor zvinhu uye plasma, microwave acoustics, oceanography, radar, seismology.

Trigonometric Fourier dzino

In masvomhu, mumwe Fourier akatevedzana inzira yokumiririra zvemasanga kunzwisisa mabasa sezvo chitsama nyore. Kazhinji kacho, nhamba mashoko ave ziendanakuenda. The zvikuru nhamba vachiverenga dzemaitirwe, neyakarurama kwokupedzisira mugumisiro inotengwa. Kushandisa vakawanda nyore trigonometric cosine kana sine wave kushanda. Panyaya iyi, iyo Fourier akatevedzana inonzi trigonometric, uye kusarudza mashoko akadai - harmonic manyowa. Nzira iyi inoita basa rinokosha masvomhu. Chokutanga, ivo trigonometric akatevedzana inopa nzira nomufananidzo, uyewo kudzidza mashandiro, ndicho chikuru chairema dzidziso. Uyezve, zvinoita anotibvumira kugadzirisa zvinetso zviri kwemasvomhu wefizikisi. Pakupedzisira, pfungwa iyi kwakaita kuti kukura kwezvinhu kwemasvomhu pachisara nayo chamuita kuti Mapazi zvinokosha kwazvo kwemasvomhu sayenzi (dzidziso integrals, dzidziso nguva nenguva mabasa). Mukuwedzera, pokutangira kukura zvinotevera dzidziso: rinovira, mashandiro ari shanduka chaiyo, chadzo pachisara uye nheyo harmonic kuongorora.