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When a source of sound is in motion towards a stationary observer, the effect observed is ………… |
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Answer» Solution :WHENTHE source and theobserver arein relativemotionwithrespectto eachotherandto themediumin whichsoundpropagates, thefrequency of thesoundwaveobservedisdifferentfromthe FREQUENCY of thesource. Thisphenmenonis calledDopplerEffect (i )SOurcemovestowards the observer: supposea sourceS movesto theright( as shownin figure ) withavelocity`V_s` and letthe frequencyof thesoundwavesproducedby thesourcebe ` F_s `. Weassumethe VELOCITYOF SOUND in mediumis v.the compression( soundin thefigure. whenS is at`C_2` and similarlyfor `X_3` the compressionis at`C_1` . whenS isatposition`X_2` , thecompressionis at ` C_2`andsimilary for ` x_3` and `C_3`Assumethat if ` C_1`reachesthe pointC asshownin the figure. it is obviousto seethat thedistancebetweencompressions`C_2`and ` C_3`is shorterthan distancebetween`C_1and C_2`.thismeansthe wavelengthdecreases whenthe sourceS movestowardstheobserver O ( sincesoundtravels ) .ButFrequency isinverselyrelatedto wavelengthand therefore , frequency increases .  Let `lamda ` bethewavelengthof the sourceS asmeasured by theobderverwhen Ssi atposition`X_1 and lamda ` . bewavelengthof thesourceobserved by theobserver whenS movestopositionsourceto travelbetween`X_1and x_2`therefore, `lamda. = lamda -v_s t ` But`t=(lamda )/(v ) ` on substitutingequation(2 )in equation(3), we get `lamda.= lamda(1- (v_s) /(v) )` sincefrequencyis inversely proportionalto wavelength, wehave ` f. =(v_s )/(lamda. )andf=(v_s )/(lamda )` hence ` f. =(f )/( (1-(V_s )/(V)))` since`(v_s )/( v )lt lt1,`Weusethe binomialexpansion andretatiningonlyfirstorderin ` ( V_s)/( v ) `, weget ` f.= f (1+(v_s)/(v))v` (ii ) Sourcemovesawayfrom theobserver : sincethe velocityhereof thesourceis oppositein direction whencomparedto case(a) , therefore, changingthe signof thevelocityof thesourcein theabovecasei.e.,bysubstituting`( v_s to - V_s )` in equation(1) , we get ` f. =(f) /((1+(v_s)/(v)))` Usingbinomialexpansion again , we get , ` f. = f ( 1-(v_s)/(v))` |
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