W9_parallel_resonance.eps

Week 11: Light 387

observes. If both motions are occurring, both shifts occur as a product. We show the

picture and quick derivation of each possibility below.

Moving Source

v Ts λ’

λ

Source Receiver

vs

Figure 153: Wave geometry for Doppler shift of moving source.

The source emits light waves that travel a distanceλ=cTin a single periodT. However,

in the timeTbetween wavefronts, the source moving at speedvstowards the receiver

travelsinto the wave it has emitted a distancevsT, reducing the distance at the time

of the next front toλ′=λ−vsT. This in turn reduces the timeT′between wavefronts

that cross the receiver (e.g. an eye or camera) and hence we can solve for the frequency

shift thus:

λ′ = λ−vsT

cT′ = cT−vsT

T′ = T

(

1 −

vs

c

)

1

T′

=

1

T

(^1

1 −vcs

)

f′ = ( f

1 −vcs

) (953)

For a source movingawayfrom the receiver the algebra and picture is the same, but the

wavelengthλ′=λ+vsTisincreased, so that:

f′=

( f

1 ∓vcs

) (954)

for an approaching (-) or receding (+) source describes the general moving source doppler

shift in the frequency/color detected by the receiver.

Note well that visible light sources moving away from the receiver areshifted towards the

redend of the spectrum, while sources moving towards the receiver are shifted towards the

violetend of the spectrum. Since spectral lines produced by atoms havesharp and well-

defined frequencies, this permits us to ascertain that the visible Universe isexpanding(as

all distant stars and galaxies are red-shifted). Since the velocity with which distant stars

are receding from the Earthincreases with distance, the red shift becomes ameter stick

permitting us to measure the size of the visible Cosmos. This is a small but significant

part of the physical evidence for theBig Bangcosmological model that so far seems best

to fit the data, and that suggests that the Big Bang occurred approximately 13.5 billion

years ago (give or take a billion years) so that the visible Cosmos is a sphere roughly

27 billion light years across, containing roughly a trillion galaxies containing order of a

trillion stars apiece. This is around Avogadro’s number ofstars.

With no boundaries visible in any direction, there is no particular reason for us to think

that we are in the exact center of the cosmos, save in the sense that every point is in the

middle of an infinite line. Sometimes small pieces of physics (such as theDoppler shift

of light) can haveenormousconsequences.