4.4 The Doppler Effect

4.4.1 Moving Source

**Figure 4.3:** Wave fronts emitted by a source moving with velocity $\vec{v}_s$ observed by a stationary observer . The points of origin of the wave fronts shown are marked with filled circles.
$\scalebox{0.7}{ \includegraphics{Doppler.eps} }$

Consider the spherical wave fronts emitted by a moving source illustrated in Fig. 4.3.

The difference $\delta$ in the path lengths, labeled

and

in Fig. 4.3, traversed by successive wave fronts to a distant observer

can be approximated by

$\begin{displaymath} \delta = r_1 - r_2 = v_s \tau \cos \theta \end{displaymath}$

(4.56)

where $\tau = \frac{2\pi}{\omega}$ is the period of the wave for $r_1,r_2 » v_s \tau$ .

The constraint $r_1,r_2 » v_s \tau$ is known as the far-field approximation.

$\delta = r_2 - r_1$ is not one wavelength due both to the motion of the source and to the location of the observer relative to the velocity axis.

The period $\tau'$ measured by observer

is the difference between the arrival times of successive wave fronts given by

$\begin{displaymath} \tau' = t_2 - t_1 = (\tau + \frac{r_2}{v}) - \frac{r_1}{v} = \tau - \frac{r_1 - r_2}{v} \end{displaymath}$

which with Eq. 4.56 yields

$\begin{displaymath} \tau' = \tau \left(1 - \frac{v_s}{v} \cos \theta \right) \end{displaymath}$

(4.57)

The Doppler shifted frequency is then

$\begin{displaymath} \omega' = \frac{2\pi}{\tau'} = \frac{2\pi}{\tau \left(1 - \frac{v_s}{v} \cos \theta \right)} \end{displaymath}$

which can be written

$\begin{displaymath} \omega' = \frac{\omega}{1 - \frac{v_s}{v} \cos \theta} \end{displaymath}$

(4.58)

4.4.2 Moving Observer

**Figure 4.4:** An observer moving with velocity $\vec{v}_o$ past two subsequent wave fronts from a stationary source. The two wave fronts are shown at the (different) times at which the observer meets them. is the phase velocity of sound in the medium.
$\scalebox{0.7}{ \includegraphics{DopplerO.eps} }$

An observer moving with respect to the medium does not observe a shift in wavelength. Instead the moving observer either chases or runs away from wave fronts moving in the medium, observing a decreased or increased apparent wave velocity, respectively.^4.3

Only the component of $\vec{v}_o$ parallel to the wave velocity, $v_o \cos \theta$ , affects the apparent wave velocity, so we have

$\begin{displaymath} v' = v - v_o \cos \theta \end{displaymath}$

(4.59)

We have made the approximation, in using an average $\theta$ for Eq. 4.59, that the observer is at a large distance from the source, relative to a single wavelength. The wavelength is exaggerated in Fig. 4.4 for convenience.

This shift in velocity relative to the observer gives a shifted frequency

$\begin{displaymath} \omega' = \frac{2\pi v'}{\lambda} = \frac{2 \pi v}{\lambda} \left(1 - \frac{v_o}{v} \cos \theta \right) \end{displaymath}$

$\begin{displaymath} \omega' = \omega \left(1 - \frac{v_o}{v} \cos \theta \right) \end{displaymath}$

(4.60)

4.4.3 Moving Source and Observer

For acoustic waves, there is an important asymmetry between moving sources and moving observers.^4.4

The motion of a source relative to the medium has consequences for all observers. (The wavelength and frequency depend on the position of any observer, moving or stationary.)
The motion of an observer has consequences for that observer only. (The wavelength and frequency depend on the observer's position and velocity relative to the medium.)

**Figure 4.5:** An observer moving with velocity $\vec{v}_o$ past two subsequent wave fronts from a source moving with velocity $\vec{v}_s$ . The two wave fronts are shown at the (different) times at which the observer meets them.
$\scalebox{0.7}{ \includegraphics{DopplerBoth.eps} }$

Fig. 4.5 shown an observer moving relative to a source which in turn moves relative to the medium. We now need two angles to describe the orientations of both $\vec{v}_o$ and $\vec{v}_s$ relative to the vector $\vec{v}$ , the velocity of the sound reaching the observer.

With these angles, we can apply the frequency shift for a moving observer (Eq. 4.60) to the shifted frequency for a moving source (Eq. 4.58) to find

$\begin{displaymath} \omega' = \frac{\omega}{1 - \frac{v_s}{v} \cos \theta_s} \left(1 - \frac{v_o}{v} \cos \theta_o \right) \end{displaymath}$

which simplifies to a general expression for the far-field Doppler shifted frequency

$\begin{displaymath} \omega' = \omega \left( \frac{v - v_o \cos \theta_o} {v - v_s \cos \theta_s} \right) \end{displaymath}$

(4.61)

Updated Wed Jan 18 09:51:28 2006