DOPPLER EFFECT IN ACOUSTICS PDF พิมพ์
  

 

Doppler effect describes the frequency shift of the signal in relation to the relative motion of a source and an observer. The wave generated by a source that moves away from an observer/receiver appears to him to be of lower frequency than the wave generated by a stationary source, or generated by a source moving toward the observer. The frequency of the signal detected by the receiver moving toward the still source is higher, compared to the frequency detected by the still receiver, or a receiver moving away from the source. This theory was formulated by Austrian physicist Ch.Doppler in 1842.1. Source moves, receiver is still.

Let us consider the source of wave moving toward the observer with velocity v and emitting the impulses with period T. At the moment t=0 the length between the source and the observer is L. The first impulse will reach the observer at time t=L/u, where u in the velocity of the waves. The second impulse will be sent to observer at time T. At this time the distance between the source and observer will be L1=L-vT, so the second impulse will reach the observer at time t1=T+(L-vT)/u. As a result the observer will detect the impulses with period

Tdop=t1-t= T(1- v/u)

And the frequency fdop registered by the observer equals:

fdop=f / (1-v/u) (source is moving toward the stationary observer)

where f is the frequency emitted by the source. We see from this formula that when the source is moving toward the observer the frequency is increased. The value of this increase is called Doppler shift. On the contrary, when source moves away from the receiver the frequency is diminished as follows:

fdop=f / (1+v/u) (source is moving away from the stationary observer)

In case when the source is moving and the observer is still, the Doppler shift appears because the wavelength is changes.

2. Receiver moves, source is still.

Next, we shall consider the case when observer moves and the source of the wave is still. In this case the wavelength is not changed and Doppler frequency shift appears because the velocity w of the wave relatively the observer is changed:

w = u + v (observer is moving toward the stationary source)

w = u - v (observer is moving away from the stationary source)

Because fdop=w/l , initial f=u/l0 and l =l0 we find that

fdop=f(1+v/u) (observer moves toward the stationary source)

fdop=f(1-v/u) (observer is moving away stationary source)

We can see from these conclusions that for acoustic waves the frequency shift will be different depending on what is moving: source or observer.  On the contrary, in the case of electro-magnetic wave the Doppler shift  depends only on the relative motion of source and receiver.

DOPPLER EFFECT 1

   Source is moving, the receiver is still.

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DOPPLER EFFECT 2

  Receiver is moving, the source is still

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FOURIER SERIES

Any periodic function f(t) with period T can be presented as a sum of sines and cosines of argument nwt, where n is an integer number, t is time, w =2p/T (so called Fourier series):

where          

where          


Fourier components of this series are called as harmonics. Any even function is expanded in a series of cosines and any odd function is expanded in a series of sines. In some cases the even harmonics are absent.

1. Let us consider meander with angular frequency w as it is shown in the figure (where a=b=T/2). This function is expanded in a series:

Animation below shows sum of the first 10 harmonics of Fourier composition of meander. We can see from this animation that first harmonic component is a sinus. Adding the higher harmonics we distort this sinus and, finally, the sum of first ten harmonics (the highest of which is of the frequency 19w ) gives the practically ideal meander.

2. Let us consider the other example of the function shown in the figure where T/b=4. This function can be expanded in Fourier series as follows:

image92.gif

Animations below shows the sum of the first 20 harmonics of the Fourier series. We can see in this figure that the function is mainly built by the first several harmonics. The high-order harmonics improve the sharpness of the fronts.

Animations below shows the sum of the first 20 harmonics of the Fourier series. We can see in this figure that the function is mainly built by the first several harmonics. The high-order harmonics improve the sharpness of the fronts.

impuls

FOURIER SERIES 1

   Sum of the first 10 harmonics of meander.

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FOURIER SERIES 2

    Sum of the first 20 harmonics of rectangular impulse with relative duration equal to 4

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Doppler effect describes the frequency shift of the signal in relation to the relative motion of a source and an observer. The wave generated by a source that moves away from an observer/receiver appears to him to be of lower frequency than the wave generated by a stationary source, or generated by a source moving toward the observer. The frequency of the signal detected by the receiver moving toward the still source is higher, compared to the frequency detected by the still receiver, or a receiver moving away from the source. This theory was formulated by Austrian physicist Ch.Doppler in 1842.1. Source moves, receiver is still.

Let us consider the source of wave moving toward the observer with velocity v and emitting the impulses with period T. At the moment t=0 the length between the source and the observer is L. The first impulse will reach the observer at time t=L/u, where u in the velocity of the waves. The second impulse will be sent to observer at time T. At this time the distance between the source and observer will be L1=L-vT, so the second impulse will reach the observer at time t1=T+(L-vT)/u. As a result the observer will detect the impulses with period

Tdop=t1-t= T(1- v/u)

And the frequency fdop registered by the observer equals:

fdop=f / (1-v/u) (source is moving toward the stationary observer)

where f is the frequency emitted by the source. We see from this formula that when the source is moving toward the observer the frequency is increased. The value of this increase is called Doppler shift. On the contrary, when source moves away from the receiver the frequency is diminished as follows:

fdop=f / (1+v/u) (source is moving away from the stationary observer)

In case when the source is moving and the observer is still, the Doppler shift appears because the wavelength is changes.

2. Receiver moves, source is still.

Next, we shall consider the case when observer moves and the source of the wave is still. In this case the wavelength is not changed and Doppler frequency shift appears because the velocity w of the wave relatively the observer is changed:

w = u + v (observer is moving toward the stationary source)

w = u - v (observer is moving away from the stationary source)

Because fdop=w/l , initial f=u/l0 and l =l0 we find that

fdop=f(1+v/u) (observer moves toward the stationary source)

fdop=f(1-v/u) (observer is moving away stationary source)

We can see from these conclusions that for acoustic waves the frequency shift will be different depending on what is moving: source or observer.  On the contrary, in the case of electro-magnetic wave the Doppler shift  depends only on the relative motion of source and receiver.

DOPPLER EFFECT 1

   Source is moving, the receiver is still.

คลิกค่ะ   download  จากฟิสิกส์ราชมงคล  


DOPPLER EFFECT 2

  Receiver is moving, the source is still

คลิกค่ะ   download  จากฟิสิกส์ราชมงคล  


FOURIER SERIES

Any periodic function f(t) with period T can be presented as a sum of sines and cosines of argument nwt, where n is an integer number, t is time, w =2p/T (so called Fourier series):

where          

where          


Fourier components of this series are called as harmonics. Any even function is expanded in a series of cosines and any odd function is expanded in a series of sines. In some cases the even harmonics are absent.

1. Let us consider meander with angular frequency w as it is shown in the figure (where a=b=T/2). This function is expanded in a series:

Animation below shows sum of the first 10 harmonics of Fourier composition of meander. We can see from this animation that first harmonic component is a sinus. Adding the higher harmonics we distort this sinus and, finally, the sum of first ten harmonics (the highest of which is of the frequency 19w ) gives the practically ideal meander.

2. Let us consider the other example of the function shown in the figure where T/b=4. This function can be expanded in Fourier series as follows:

image92.gif

Animations below shows the sum of the first 20 harmonics of the Fourier series. We can see in this figure that the function is mainly built by the first several harmonics. The high-order harmonics improve the sharpness of the fronts.

Animations below shows the sum of the first 20 harmonics of the Fourier series. We can see in this figure that the function is mainly built by the first several harmonics. The high-order harmonics improve the sharpness of the fronts.

impuls

FOURIER SERIES 1

   Sum of the first 10 harmonics of meander.

คลิกค่ะ   download  จากฟิสิกส์ราชมงคล  


FOURIER SERIES 2

    Sum of the first 20 harmonics of rectangular impulse with relative duration equal to 4

คลิกค่ะ   download  จากฟิสิกส์ราชมงคล  


 

 


 
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