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Doppler Effect

When the source emitting waves, and the observer receiving them, are in relative motion with respect to the medium in which the waves propagate, the frequency of the waves received by the observer is different from the frequency of the vibrating source. This phenomenon was first noticed by C.J. Doppler in 1842 in sound waves and is known as the Doppler effect. This phenomenon of the apparent change in the observed frequency of the wave, when there is a relative motion between the source and the observer is exhibited by all types of waves, including light waves which do not require a medium for their propagation. We shall discuss the Doppler effect in sound waves.

The Doppler effect in sound waves may be readily observed by a person at a railway platform. When a through train, blowing its whistle, approaches the observer, the pitch of the whistle appears to rise and it suddenly appears to drop when the engine has passed by. For waves which require a medium for propagation, the apparent change in the frequency depends on which of the three—the source, the observer or the medium is moving. Let us now compute this change in frequency for the following cases.

Source in Uniform Motion—Observer and Medium Stationary

To fix our idea let us take the source to be a tuning fork of frequency v. It produces sound (compressional) waves of frequency v, which travel through the medium with a velocity , where B is modulus of elasticity and ρ , the density of the medium. Let us be the velocity with which the source, i.e. the fork, moves bodily towards or away from a stationary observer stationed at O. As shown in Figure, the wavelength of the waves in the medium, depends on the magnitude and direction of us. In Figure we consider the source approaching the observer. Let, at an instant of time t = 0 (a), the prongs of the fork move towards producing a compression C1 in the medium close to the prong.

Figure-Tuning Fork

This compression will travel in the medium with the velocity v of the wave of compression. After one full period, i.e. at t = T, where T (=1/v) is the period of vibration of the fork, the second compression C2 is produced because at t = T, the prongs will once again move outwards. If the fork is at rest, the distance λ (b) between two consecutive compressions (which is the distance travelled by the compression in one period) is, by definition, the wavelength of the wave in the medium.
 

If, however, the fork is moving with velocity us (c) it travels a distance us T in time T, towards the observer. At this position the fork produces the second compression C2, the distance between two consecutive compressions having been reduced to λ 1, which is the wavelength of the wave when the fork is in motion. Obviously λ and λ 1 are related a
                                

 

Now, since wave velocity υ is fixed (being determined by the properties of the medium, namely, B and ρ ), the frequency of the wave of wavelength λ 1 is given by

               -----------(i)

Hence, the frequency of the waves received by the observer is v1, which is greater than v. If the source is moving away from a stationary observer, the apparent wavelength , in which case, the apparent frequency as observed by the observer is given by (since us change sign)

                    ------------(ii)

In this case, . The cause of the change in frequency is the increase or decrease of the wavelength in the medium brought about by the motion of the source. In obtaining Equations (i) and (ii) we have assumed that us < v.

 

Note
A special situation occurs in the case of supersonic sources which move with velocities greater than that of sound. The relations (i) and (ii) do not hold because then in a given time the source advances more than the wave. The resultant wave motion is a conical wave called a shock wave which is sudden and violent sound we hear when a supersonic jet plane flies by.

Observer in Uniform Motion—Source and Medium Stationary

The effect is different when the source is stationary and the observer moves through the medium. Let the observer move away from a source with a uniform velocity u0. If v is the frequency of the vibrating source it emits v waves in one second. If the observer is at rest, he receives these waves in one second. Therefore, the frequency of the wave he hears is v. If he now starts running away from the source, he will miss the waves which occupy the distance he travels. In one second he moves a distance of u0 away from the source. Therefore, the number of waves missed by him in one second = number of waves contained in the distance u0 he moves in one second = u0/λ , since, in a distance of wavelength λ , there is exactly one wave, by definition. Thus, out of v waves, he misses u0/λ waves in one second.

Therefore, the number of waves received by him in one second = v – u0/λ or since λ =
υ/v, the frequency v2 of sound heard by him is given by

                      -------------(iii)

 

On the other hand, if the observer were approaching the source, he would receive u0/λ additional waves in each sound and would hear a sound of frequency given by

                           ------------(iv)

 

Figure illustrates how the abserver hears a sound of higher frequency when he moves towards it. Notice that the apparent change in the frequency of sound heard by the observer results from his intercepting more waves (when he approaches) or fewer waves (when he recedes) each second.

Figure - Spherical wave from a point source S travelling to the right. Imagine the observer to stay at O for one second and then move suddenly to P. He now receives more waves than if he had stayed at O and hence hears a higher frequency

In obtaining the equations (iii) and (iv) we have assumed that u0 < υ. These equations can also be obtained as follows. We have seen that, if the source of sound is in motion and the observer is stationary, the apparent change in frequency of sound received by the observer is due to a change in the wavelength (due to the motion of the source); the speed of sound relative to the observer remaining unaltered at υ. On the other hand, if the source of the sound is stationary and the observer is in motion, the apparent change in frequency of sound received by the observer is due to a change in the speed of sound relative to the observer; the wavelength of sound heard by the observer remaining unaltered at λ = υ/v. Thus the physics of the change in frequency in the two cases is different.

If the observer approaches a source with a speed u0, the speed of sound relative to him will be (υ + u0). Hence the frequency of sound heard by him will be

         

 

Since λ = υ/v, we have

         

 

On the other hand, if the observer recedes from a source with a speed u0 the speed of sound relative to him is (υ – u0) and the frequency of sound heard by him will be

        
 

 

Source and Observer both in Uniform Motion—Medium Stationary
We shall compute the apparent frequency in the following possible situations

Source Approaching a Receding Observer

If the source approaches a stationary observe with a velocity us, the wavelength of the waves reduces to and the frequency of these waves is v1. The observer receives v1 waves in one second. If the observer is receding with a velocity u0, he will miss u0/λ 1 waves each second. In one second the number of waves received by him will be given by

 

   

Using Equation (i), the expression for the apparent frequency becomes

    

If u0 = us, v3 = v, since then there is no relative motion between the source and the observer.
 

Both the Source and Observer Receding from Each other
The apparent frequency , in this case, is obtained by reversing the sign of us

      
 

Both the Source and the Observer Approaching Each Other Reversing the sign of u0 in Eq (13.53) the apparent frequency in this case is

      
 

Observer Approaching a Receding Source In this case the signs of u0 and us in are reversed and the apparent frequency v4', is given by

    

Notice that v4'= v, if u0 = us.
 

Effect of the Motion of the Medium
The velocity of material waves is affected by the motion of the medium. If the medium is moving with a velocity um in the direction of propagation of sound, the effective velocity of sound in increased from
υ to (υ + um). In the relations υ is replaced by (υ + um). On the other hand, if the medium is moving in the opposite direction, υ is replaced by (υ – um).





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