Doppler Effect :
The Doppler effect is the apparent change of frequency when a source or observer move relative to each other.
The effect is observed in the following cases : In the change of pitch of a train risen as it passes through a station, in the use of by aircraft radars in police speed traps etc. There is a simple analogy. Imagine that you are working in a chocolate factory packing chocolates that come to you down a steadily moving conveyer belt. AI the other end of the belt another person puts the chocolates on the belt at a steady rate. The chocolates therefore reach you at the same steady rate that they were put on.
Now the other person starts to walk slowly towards you alongside the conveyer belt, still putting chocolates on at the original steady rate. You can see that you will receive the chocolates at a faster rate because after putting one chocolate as, your partner walks after it and when the next chocolate is put on the belt, it will be closer to the first chocolate then if he or she had not moved.
You will also receive the chocolates faster if you walk towards the other end of the conveyer belt collecting the chocolates as you go.
Now, the rate at which chocolates are put on the belt corresponds to the original frequency of the source of the waves, the velocity of the belt corresponds to the wave velocity (which is constant and is unaffected by the rnotion of either the source or the observer) and the rate at which you receive them corresponds to the observed frequency.
We will now consider the Doppler effect in wave motion. Consider a source `5` moving from left to right as shown. Initially it is at position `1` and sometime later it will be at positions `2` and `3`. If it is emitting a wave then the three circles represent the position or wave emitted at `1, 2` and `3` same time after the source was at position `3`.
You will see that the waves on the right are closed together than those on the left, if the source is approaching an observer the wavelength will therefore be decreased and the frequency increased, while if it is moving away the reverse will be true.
`text( Moving source ):`
Consider a source moving at a velocity `v_s` , towards an observer `O` . Let the source emit `f_0`, waves per second. In one second the waves will move forward a distance `c`, and the source will move forward `v_s` , . Therefore `f_0` waves will be contained in a distance of `(c - v _s )`. Hence the new wavelength
`lambda=((c-v_s)/f_0)`
Thus apparent frequency heard by an observer (listener)
`f'=c/(lambda')`
or `f'=(c/(c-v_s))`
Sign Convention :
While solving problems, we will take the source to listener (observer) direction as positive or the SL. axis as positive. Thus speed of sound will always be positive as it always travels from source to listener. The appropriate signs to the velocity of the source or listener is put based on this convention
`text(Moving observer :)`
In one second [waves will pass a Stationary observer but an observer (Listener) who is moving forward will pass through more waves per second and thus the observed frequency will increase. Let the observer move from `0` to `O` in one second.
Now, `OO' =v` as `t = 1 s`, and so the extra number of waves received per second is `v/lambda` .Hence the new frequency
`f'=f+v/lambda=f+(v*f)/c;`
`f'=f(1+v/c)`
If the observer were receding the
`f'=f(1+v/c)`
Now this `v` is the positive `v_2` if we take `S-L` axis as positive
`->f' = (1-v_L/c)f=(c-v_L/c)f`
Now. the `f` here is the `f'` of the previous case. where the source is moving.
Hence, if both source and observer are moving.
`f'=(c-v_L/c)[c/(c-v_L)]f_0=>f'=((c-v_L)/(c-v_L))*f_0`
If the source or observer's velocity is not parallel to the `S-L` axis. then `v_s` , corresponding to the component of the source's velocity along `SL` axis. `v_L` corresponds to the component of listener's velocity along the `S-L` axis.
If the wind is blowing, then the component of wind velocity along the `SL` axis is added to the speed of sound to get `c`.
The effect is observed in the following cases : In the change of pitch of a train risen as it passes through a station, in the use of by aircraft radars in police speed traps etc. There is a simple analogy. Imagine that you are working in a chocolate factory packing chocolates that come to you down a steadily moving conveyer belt. AI the other end of the belt another person puts the chocolates on the belt at a steady rate. The chocolates therefore reach you at the same steady rate that they were put on.
Now the other person starts to walk slowly towards you alongside the conveyer belt, still putting chocolates on at the original steady rate. You can see that you will receive the chocolates at a faster rate because after putting one chocolate as, your partner walks after it and when the next chocolate is put on the belt, it will be closer to the first chocolate then if he or she had not moved.
You will also receive the chocolates faster if you walk towards the other end of the conveyer belt collecting the chocolates as you go.
Now, the rate at which chocolates are put on the belt corresponds to the original frequency of the source of the waves, the velocity of the belt corresponds to the wave velocity (which is constant and is unaffected by the rnotion of either the source or the observer) and the rate at which you receive them corresponds to the observed frequency.
We will now consider the Doppler effect in wave motion. Consider a source `5` moving from left to right as shown. Initially it is at position `1` and sometime later it will be at positions `2` and `3`. If it is emitting a wave then the three circles represent the position or wave emitted at `1, 2` and `3` same time after the source was at position `3`.
You will see that the waves on the right are closed together than those on the left, if the source is approaching an observer the wavelength will therefore be decreased and the frequency increased, while if it is moving away the reverse will be true.
`text( Moving source ):`
Consider a source moving at a velocity `v_s` , towards an observer `O` . Let the source emit `f_0`, waves per second. In one second the waves will move forward a distance `c`, and the source will move forward `v_s` , . Therefore `f_0` waves will be contained in a distance of `(c - v _s )`. Hence the new wavelength
`lambda=((c-v_s)/f_0)`
Thus apparent frequency heard by an observer (listener)
`f'=c/(lambda')`
or `f'=(c/(c-v_s))`
Sign Convention :
While solving problems, we will take the source to listener (observer) direction as positive or the SL. axis as positive. Thus speed of sound will always be positive as it always travels from source to listener. The appropriate signs to the velocity of the source or listener is put based on this convention
`text(Moving observer :)`
In one second [waves will pass a Stationary observer but an observer (Listener) who is moving forward will pass through more waves per second and thus the observed frequency will increase. Let the observer move from `0` to `O` in one second.
Now, `OO' =v` as `t = 1 s`, and so the extra number of waves received per second is `v/lambda` .Hence the new frequency
`f'=f+v/lambda=f+(v*f)/c;`
`f'=f(1+v/c)`
If the observer were receding the
`f'=f(1+v/c)`
Now this `v` is the positive `v_2` if we take `S-L` axis as positive
`->f' = (1-v_L/c)f=(c-v_L/c)f`
Now. the `f` here is the `f'` of the previous case. where the source is moving.
Hence, if both source and observer are moving.
`f'=(c-v_L/c)[c/(c-v_L)]f_0=>f'=((c-v_L)/(c-v_L))*f_0`
If the source or observer's velocity is not parallel to the `S-L` axis. then `v_s` , corresponding to the component of the source's velocity along `SL` axis. `v_L` corresponds to the component of listener's velocity along the `S-L` axis.
If the wind is blowing, then the component of wind velocity along the `SL` axis is added to the speed of sound to get `c`.