The Doppler effect causes shifts in wavelengths of sound and light.

This effect means that the frequency experienced by an observer depends on the relative motion of the object emitting the wave.

If the source of the wave is moving towards a stationary observer with a constant velocity, then the movement means that the wavefronts are being emitted from a position closer to the stationary observer each time. This means that the observer will receive more wavefronts in a given time, as the speed of the wave remains constant but the distance from the wavefronts to the stationary observer decreases, so they will experience a higher frequency than the one emitted by the source.

If the source of the wave is moving away from a stationary observer with a constant velocity, then the movement means that the wavefronts are being emitted from a position farther away from the stationary observer each time. This means that the observer will receive fewer wavefronts in a given time, as the speed of the wave remains constant but the distance from the wavefronts to the stationary observer increases, so they will experience a lower frequency than the one emitted by the source.

The following relationship describes the Doppler effect for sound and can be used to solve problems involving the observed frequency, source frequency, source speed and wave speed.

$$ f_o=f_s[\dfrac{v}{v \space \pm \space v_s }] $$

→ for a source travelling towards the observer, $[\dfrac{v}{v \space \pm \space v_s }]$ should be greater than 1 to make the observed frequency greater than the source frequency, so the denominator of the fraction will be $v-v_s$, to make the fraction $v$ divided by something smaller than $v$, which makes it more than 1.

→ for a source travelling away from, the observer, $[\dfrac{v}{v \space \pm \space v_s }]$ should be less than 1 to make the observed frequency less than the source frequency, so the denominator of the fraction will be $v+v_s$, to make the fraction $v$ divided by something greater than $v$, which makes it less than 1.

The Doppler effect has implications for light travelling to Earth from faraway celestial objects, as they are in motion relative to Earth.

The light from objects moving away from us is shifted to longer wavelengths (redshift).

As the frequency experienced is smaller, as per the Doppler effect, and the speed is constant, the wavelength will be longer.

The inverse is true for objects moving towards us — the light from these objects is shift to smaller wavelengths (a blueshift or a negative redshift).

The degree to which a galaxy is ‘redshifted’ can be quantify using the a value called the redshift.

The redshift of a galaxy is the change in wavelength divided by the emitted wavelength.

This is described the the following relationship.

$$ z=\dfrac{\lambda_{observed}-\lambda_{rest}}{\lambda_{rest}} $$

→ $z$ is positive when the $\lambda_{observed}-\lambda_{rest}$ value is positive, which is when the $\lambda_{observed}>\lambda_{rest}$. This is true of galaxies moving away from Earth, so that their frequencies are observed to be lower and their wavelengths observed to be greater — nearer to the ‘red’ side of the spectrum, i.e. light from these galaxies are red-shifted.

→ $z$ is negative when the $\lambda_{observed}-\lambda_{rest}$ value is negative, which is when the $\lambda_{observed}<\lambda_{rest}$. This is true of galaxies moving towards Earth, so that their frequencies are observed to be greater and their wavelengths observed to be smaller — nearer to the ‘blue’ side of the spectrum, i.e. light from these galaxies are blue-shifted.

For slowly moving galaxies, redshift is also the ratio of the recessional velocity of the galaxy to the velocity of light.

$$ z=\dfrac{v}{c} $$

→ The ‘recessional value’ is negative for galaxies moving towards the Earth.

The above relationships can be used to solve problems involving redshift, observed wavelength, emitted wavelength, and recessional velocity.

The following relationship shows that the the recessional velocity of a galaxy is directly proportional to its distance from us, where $H_o$ is constant.

$$ v=H_od $$

→ This is known as Hubble’s Law or the Hubble-Lemaître Law.

The Hubble-Lemaître Law allows us to estimate the age of the Universe.

This method places the age of the Universe at roughly $1.4 \times 10^{10}$ years.

Measurements of the velocities of galaxies and their distance from us lead to the theory of the expanding Universe**.**

This is shown by the Hubble-Lemaître Law which demonstrates that objects that are further away (have a greater distance from Earth, $d$) are receding with a greater recessional velocities, $v$. Working backwards from this leads to the conclusion that the Universe started out as a single point and then all of the matter expanded outwards and continues to expand.

Now we know that the rate of this expansion is increasing, which is evidence for energy present in the Universe that is driving this accelerated rate of expansion, which energy cannot be detected through regular means.