Expression for the Apparent Frequency

 

Source, observer and the medium, all in motion

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Let S and O denote the initial positions of a source of sound and an observer. For the sake of simplicity, we shall assume that the source, the observer and the medium are all moving along the positive direction.

Let the velocity of sound in still air be = v

The velocity of the source = a

The velocity of the observer = b

and the velocity of the medium (wind blowing) = w

Let S^l and O^l represent the positions of the source and the observer after 1 second. Distance travelled in 1 second is nothing but the velocity.

The waves produced by the source travel a distance SA in 1 second, but as the wind is blowing with a velocity w, it carries the wavefront from A to A^l where A A^l= w. The distance travelled by the waves relative to the source in 1 second.

If f is the frequency of the waves produced by the source, then f waves are accommodated in a distance S^lA^l.

Since the observer recedes by a distance b in 1 second, the relative velocity of the waves with respect to the observer is (v + w - b). Therefore, the apparent frequency is given by the number of waves of wavelength l^l contained within the above distance.

Substituting for l^l from equation (i), we get

This is the general expression for the apparent frequency of the sound when the source of sound, observer and the medium are in motion, in the same direction.

Discussion of equation (1-38) for particular cases

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Case (i)

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Source moving towards a stationary observer

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Let us assume that the wind velocity is zero. Then w = 0 and b = 0. Equation (1-38) becomes

The apparent frequency will be greater than the actual frequency.

Case (ii)

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Source moving away from a stationary observer

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Assuming the wind velocity to be zero, putting -a in the place of a and b = 0 in equation (1-38), we get

The apparent frequency will be lesser than the actual frequency.

Case (iii)

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Observer moving away from a stationary source

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Putting w = 0 and a = 0 in equation (1-38), we get

The apparent frequency will be lesser than the actual frequency.

Case (iv)

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Observer moving towards a stationary source

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Putting w = 0, a = 0 and -b in the place of b in equation (1-38), we get

The apparent frequency will be greater than the actual frequency.

Case (v)

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Observer and source moving in the same direction as sound in a stationary medium

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Putting w = 0, in equation (1-38), we get

• If b < a, then (v - b) > (v - a) and f^l > f.

• If b > a, then (v - b) < (v - a) and f^l < f.

• If b = a, then f^l = f.

Thus, for a passenger sitting in a train the frequency of the whistle of the train appears to be the same, when the train moves, as when it was at rest.

Case (vi)

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Observer and source moving towards each other in a stationary medium

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Now the velocity of the observer is opposite to the velocity of sound, while that of the source is the same as that of sound. 'b' is to be replaced by -b in equation (1-38).

Case (vii)

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Observer and source moving away from each other in a stationary medium

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Considering the direction of the velocity of sound reaching the observer as positive, a is negative and b is positive. Then

The apparent frequency is less than the actual frequency.

Case (viii)

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Wind blowing opposite to the direction of the velocity of sound

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In this case, w is to be replaced by -w in equation (1-38)