Imagine a Source (S) of sound waves that emits waves having frequency f and therefore

Transcription

1 heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing relaie o () will also obsere ha he sound wae has requency. I on he oher hand () is moing relaie o (), () will deec a dieren requency ha is shied o eiher a higher or lower alue depending upon he deails o he relaie moion beween (), () and he medium in which he sound waes propagae (usually Air (A)). In all ha ollow we will assume ha he elociy o sound is quoed relaie o he elociy o he medium, and ha all speeds o hings oher han sound waes are all less han he speed o sound regardless o he common reerence rame in which all o hem are quoed. -ais -ais Wae ron emied by ource () a and receied by eecor () a PF Creaed wih deskpf PF Wrier - rial :: hp://

2 he wae ron emied by he ource () a raels a oal disance o d in a oal ime o beore being receied by he eecor (). he sound wae raels along he - ais wih elociy rom () o (). his means ha d. Equae he epressions or d : Group ogeher all erms inoling : ( ) he wae ron emied by () a arries a () a. o be deinie le s imagine ha he wae ron we are alking abou is a cerain cres o he sound wae. he ery ne cres is hen emied by () a ime, he period o oscillaion o he source as measured relaie o is own res rame. Le be he ime when his ne wae ron is receied by (). -ais -ais Wae ron emied by ource () a receied by eecor () a and PF Creaed wih deskpf PF Wrier - rial :: hp://

3 3 Using eacly he same argumen as aboe, we can relae he siuaion a o he siuaion a. In he ime ineral he second wae ron raels a disance o d. Again, he sound wae raels wih a elociy along he -ais o so d. We can equae he epressions or d and sole or o ind he second wae ron hen arries a () a. Beween emiing he successie cress () moes hrough a displacemen along he - ais o, and in his same ime ineral () moes hrough a displacemen o. his means ha a he insan he second cres is emied by (), he separaion beween () and () is ( ). ubsiuing his epression ino he one or we obain ( ). o, according o () we hae a wae ron emied a ollowed by he ne equialen wae ron emied a. his is he deiniion o he period o he wae as obsered by (). I we ask () wha i hinks is own requency is, he answer will be. According o () he irs wae ron arries a and he ne equialen wae ron arries a. he period o he wae according o () is hen. We can clean his up by geing a common denominaor and simpliying. PF Creaed wih deskpf PF Wrier - rial :: hp://

4 Now, i we ask () or he requency o he receied sound wae, he answer will be. he resul describes he oppler requency shi due o relaie moion o he ource () and he eecor (): o use his resul consisenly, we mus deine an -ais upon along which () and () are moing relaie o one anoher. Our rame o reerence is se by he assumpion ha all elociies are quoed relaie o he medium in which he sound wae propagaes. In a momen we will see wha o do in he een ha we know he elociies o () and/or () relaie o he ground, he elociy o sound relaie o he medium, and he elociy o he medium relaie o he ground. Firs, le s igure ou how o ge he signs correc, hen we will allow he wind o blow. he power o using he elociy componens in deriing he oppler shi is ha i makes easy work o geing he signs righ. By answering a series o our simple quesions abou a gien siuaion, we can essenially assure a correc soluion. () Which direcion is he posiie direcion? () In which direcion is () moing? (i.e. wha is he sign o (3) In which direcion is () moing? (i.e. wha is he sign o (4) In which direcion mus sound rael o go rom () o ()? (i.e. wha is he sign o ome pecial Cases. uppose () moes away rom a saionary ource () a speed. Le he speed o sound relaie o he air be -ais 4 PF Creaed wih deskpf PF Wrier - rial :: hp://

5 () he posiie direcion is o he righ as shown. () () is no moing relaie o he air,. (3) () is moing in he posiie direcion relaie o he air, (4) he sound mus rael in he posiie direcion relaie o he air in order o go rom () o (),. I () emis a requency hen () deecs a requency. uppose () moes away rom a saionary ource () a speed. Le he speed o sound relaie o he air be. his ime le s pu () o he le o (). () he posiie direcion is o he righ as shown. () () is no moing relaie o he air,. -ais (3) () is moing in he negaie direcion relaie o he air, (4) he sound mus rael in he negaie direcion relaie o he air in order o go rom () o (),. I () emis a requency hen () deecs a requency ( ) ( ) As i mus be, his resul is he same as he preious one. Boh cases describe a eecor () receding rom he ource (). No maer wha direcion we label he posiie direcion () always measures a oppler shi o lower requency when i recedes rom (). 5 PF Creaed wih deskpf PF Wrier - rial :: hp://

6 3. uppose () and () muually approach one anoher wih speeds relaie o he air o and, respeciely. -ais () he posiie direcion is o he righ as shown. () () is moing in he negaie direcion relaie o he air,. (3) () is moing in he posiie direcion relaie o he air, (4) he sound mus rael in he negaie direcion relaie o he air in order o go rom () o (),. I () emis a requency hen () deecs a requency ( ) ( ) ( ) ( ) Wha i he medium in which he sound wae propagaes moes relaie o he ground? emember Physics 3 and how you learned o rea relaie elociies. Here we will be ineresed mainly in he componens o relaie elociies along a paricular ais which we will coninue o label as he - ais. As a reminder, consider wo objecs and moing along he - ais. -ais he -componen o elociy o objec () relaie o he Ground (G) is componen o elociy o objec () relaie o he Ground is elociy o objec relaie o objec is G G G G and -. he -componen o. We can relae hese elociies using. he elociy o () relaie o () lierally means he elociy ha () 6 PF Creaed wih deskpf PF Wrier - rial :: hp://

7 has when obsered rom he poin o iew o (). In considering wo objecs i is easy o swich he poin o iew rom one o he oher,. As ar as elociies and iew poins go, here is nohing special abou eiher objec, he ground, he air, or any oher poin o reerence. We can use his idea along wih he abiliy o swich reerence rame by inroducing a minus sign o deelop a simple rule o humb or relaing relaie elociies. We hae already saed ha. uppose we epress he second G erm on he righ side rom he poin o iew o () raher han he (G), resul is G. G G G. he he rule: he elociy o () relaie o () is gien by he elociy o () relaie o (G) PLU he elociy o (G) relaie o (). he subscrips o he resulan relaie elociy are he ouer subscrips o he ecor sum, he common reerence rame ha links he elociies o he objecs is labeled by he inner subscrips o he ecor sum. G common reerence rame G G Velociy o () relaie o () We wan o use his idea o rea he siuaion in which he medium (A or Air) hrough which he sound wae propagaes moes relaie o he Ground (G). A ypical siuaion is he one in which we are ineresed in he elociy o sound relaie o he Air (A), A, and he elociies o he ource () and eecor () relaie o he Ground (G), G and G, respeciely. he new wrinkle is ha he Air (A) is no necessarily a res relaie o he Ground (A). We mus connec he elociies o () and () wih ha o ound using he Air (A) as he common rame o reerence. We already hae a oppler ormula in which all elociies are epressed relaie o he Air (A), so as long as we careully coninue epress elociies relaie o (A), we can use he same ormula. In he deriaion o he oppler ormula we simply assumed ha AG, we are now going o see how o generalize ha condiion. uppose we know all o he ollowing: - he elociy o sound relaie o he Air (A), A - he elociy o he Air (A) relaie o he Ground (G), AG 7 PF Creaed wih deskpf PF Wrier - rial :: hp://

8 - he elociy o he ource () relaie o he Ground (G), G - he elociy o he eecor () relaie o he Ground (G), G - he requency emied by () as measured in is own reerence rame, We wish o know he requency as deeced by (),. I we were o epress all elociies relaie o (A), we could immediaely wrie down he answer: A A A A I is conenien o use he elociies o () and () relaie o (G), o do his we use he rule or relaie elociies, For he eecor (): A G GA A G AG For he ource (): A G GA A G AG he oppler eec in he presence o a moing medium is hen epressed in erms o he elociy o sound relaie o he medium AN he elociies o () and () relaie o he Ground (G) as A A [ ] G AG [ ] G AG Now, we should answer a series o ie simple quesions abou a gien siuaion i we wan o se eeryhing up correcly. () Which direcion is he posiie direcion? () In which direcion is () moing relaie o (G)? (i.e. wha is he sign o G (3) In which direcion is () moing relaie o (G)? (i.e. wha is he sign o G (4) In which direcion mus sound rael o go rom () o ()? (i.e. wha is he sign o A (5) In which direcion does he Air (A) rael relaie o he Ground (G) (i.e. wha is he sign o AG 8 PF Creaed wih deskpf PF Wrier - rial :: hp://