When the source and the observer are in relative motion with respect to each other and to the medium in which sound propagates , the frequency of the sound wave observed is different from the frequency of the source . This phenmenon is called Doppler Effect
(i ) SOurce moves towards the observer : suppose a source S moves to the right ( as shown in figure ) with a velocity `V_s` and let the frequency of the sound waves produced by the source be ` F_s ` . We assume the velocity of sound in medium is v. the compression ( sound in the figure . when S is at `C_2` and similarly for `X_3` the compression is at `C_1` . when S is at position `X_2` , the compression is at ` C_2`and similary for ` x_3` and `C_3` Assume that if ` C_1` reaches the point C as shown in the figure . it is obvious to see that the distance between compressions `C_2` and ` C_3` is shorter than distance between `C_1 and C_2` .this means the wavelength decreases when the source S moves towards the observer O ( since sound travels ) .But Frequency is inversely related to wavelength and therefore , frequency increases .
Let `lamda ` be the wavelength of the source S as measured by the obderver when S si at position `X_1 and lamda ` . be wavelength of the source observed by the observer when S moves to position source to travel between `X_1 and x_2` therefore ,
`lamda. = lamda -v_s t `
But `t=(lamda )/(v ) `
on substituting equation (2 ) in equation (3) , we get
` lamda. = lamda (1- (v_s) /(v) )`
since frequency is inversely proportional to wavelength , we have
` f. =(v_s )/(lamda. ) and f=(v_s )/(lamda )`
hence ` f. =(f )/( (1-(V_s )/(V)))`
since `(v_s )/( v )lt lt 1,` We use the binomial expansion and retatining only first order in ` ( V_s)/( v ) `, we get
` f.= f (1+(v_s)/(v))v`
(ii ) Source moves away from the observer : since the velocity here of the source is opposite in direction when compared to case (a) , therefore , changing the sign of the velocity of the source in the above case i.e., by substituting `( v_s to - V_s )` in equation (1) , we get
` f. =(f) /((1+(v_s)/(v)))`
Using binomial expansion again , we get ,
` f. = f ( 1- (v_s)/(v))`
