<P> Since v s c ≪ 1 (\ displaystyle (\ frac (v_ (\ text (s))) (c)) \ ll 1) we can substitute the geometric expansion: </P> <P> 1 1 + v s c ≈ 1 − v s c (\ displaystyle (\ frac (1) (1 + (\ frac (v_ (\ text (s))) (c)))) \ approx 1 - (\ frac (v_ (\ text (s))) (c))) </P> <Table> <Tr> <Td> <Table> <Tr> <Td> </Td> </Tr> <Tr> <Td> Stationary sound source produces sound waves at a constant frequency f, and the wave - fronts propagate symmetrically away from the source at a constant speed c . The distance between wave - fronts is the wavelength . All observers will hear the same frequency, which will be equal to the actual frequency of the source where f = f . </Td> </Tr> </Table> <Table> <Tr> <Td> </Td> </Tr> <Tr> <Td> The same sound source is radiating sound waves at a constant frequency in the same medium . However, now the sound source is moving with a speed υ = 0.7 c . Since the source is moving, the centre of each new wavefront is now slightly displaced to the right . As a result, the wave - fronts begin to bunch up on the right side (in front of) and spread further apart on the left side (behind) of the source . An observer in front of the source will hear a higher frequency <P> f = c + 0 / c - 0.7c f = 3.33 f and an observer behind the source will hear a lower frequency </P> f = c - 0 / c + 0.7c f = 0.59 f . </Td> </Tr> </Table> <Table> <Tr> <Td> </Td> </Tr> <Tr> <Td> Now the source is moving at the speed of sound in the medium (υ = c). The wave fronts in front of the source are now all bunched up at the same point . As a result, an observer in front of the source will detect nothing until the source arrives where <P> f = c + 0 / c - c f = ∞ and an observer behind the source will hear a lower frequency </P> f = c - 0 / c + c f = 0.5 f . </Td> </Tr> </Table> <Table> <Tr> <Td> </Td> </Tr> <Tr> <Td> The sound source has now surpassed the speed of sound in the medium, and is traveling at 1.4 c . Since the source is moving faster than the sound waves it creates, it actually leads the advancing wavefront . The sound source will pass by a stationary observer before the observer hears the sound . As a result, an observer in front of the source will detect <P> f = c + 0 / c - 1.4c f = - 2.5 f and an observer behind the source will hear a lower frequency </P> f = c - 0 / c + 1.4c f = 0.42 f . </Td> </Tr> </Table> </Td> </Tr> </Table> <Tr> <Td> <Table> <Tr> <Td> </Td> </Tr> <Tr> <Td> Stationary sound source produces sound waves at a constant frequency f, and the wave - fronts propagate symmetrically away from the source at a constant speed c . The distance between wave - fronts is the wavelength . All observers will hear the same frequency, which will be equal to the actual frequency of the source where f = f . </Td> </Tr> </Table> <Table> <Tr> <Td> </Td> </Tr> <Tr> <Td> The same sound source is radiating sound waves at a constant frequency in the same medium . However, now the sound source is moving with a speed υ = 0.7 c . Since the source is moving, the centre of each new wavefront is now slightly displaced to the right . As a result, the wave - fronts begin to bunch up on the right side (in front of) and spread further apart on the left side (behind) of the source . An observer in front of the source will hear a higher frequency <P> f = c + 0 / c - 0.7c f = 3.33 f and an observer behind the source will hear a lower frequency </P> f = c - 0 / c + 0.7c f = 0.59 f . </Td> </Tr> </Table> <Table> <Tr> <Td> </Td> </Tr> <Tr> <Td> Now the source is moving at the speed of sound in the medium (υ = c). The wave fronts in front of the source are now all bunched up at the same point . As a result, an observer in front of the source will detect nothing until the source arrives where <P> f = c + 0 / c - c f = ∞ and an observer behind the source will hear a lower frequency </P> f = c - 0 / c + c f = 0.5 f . </Td> </Tr> </Table> <Table> <Tr> <Td> </Td> </Tr> <Tr> <Td> The sound source has now surpassed the speed of sound in the medium, and is traveling at 1.4 c . Since the source is moving faster than the sound waves it creates, it actually leads the advancing wavefront . The sound source will pass by a stationary observer before the observer hears the sound . As a result, an observer in front of the source will detect <P> f = c + 0 / c - 1.4c f = - 2.5 f and an observer behind the source will hear a lower frequency </P> f = c - 0 / c + 1.4c f = 0.42 f . </Td> </Tr> </Table> </Td> </Tr>

Both theory and history point to a close relationship between increased in