The Doppler effect describes the change in observed frequency (or wavelength) of a wave when the source and observer are in relative motion. The classic example is the changing pitch of a siren as an ambulance approaches and then moves away.

If a stationary source emits waves of frequency fs (source frequency), and the observer moves with speed vO directly toward the source (in the same line), the observed frequency f′ satisfies:
f' = fs(v + vO) / v = fs(1 + vO/v),
If the observer moves away from the source,
f' = fs(v - vO) / v.

If the source moves toward the observer with speed vS, it emits waves into an already moving medium of compressed wavefronts in the observer’s direction. The observed frequency f′ is given by:
f' = fs(v / (v - vS)),
If the source moves away from the observer:
f' = fs(v / (v + vS)).

In the most general one-dimensional case where the observer moves with speed vO (positive if moving toward the source) and the source moves with speed vS (positive if moving toward the observer), the observed frequency is:
f' = fs(v + vO) / (v - vS).

1. Redshift and Blueshift.
   ○ Light or other electromagnetic waves from a celestial object moving away from Earth are observed at longer wavelengths (lower frequency) than emitted; this is called redshift. Conversely, if an object is approaching, its light is blueshifted (shorter wavelength, higher frequency).
   ○ For velocities v much less than the speed of light c, the fractional change in wavelength Δλ/λ relates to radial velocity vr by:
   Δλ/λ = vr/c.
2. Spectral Lines.
   ○ By comparing spectral lines (e.g., hydrogen emission or absorption lines) from laboratory measurements to those observed in starlight, one can determine if the total spectrum is shifted. A uniform shift of all lines indicates motion along the line of sight.

1. Doppler Ultrasound.
   ○ In medical ultrasound, high-frequency sound waves reflect off moving blood cells. Because red blood cells are carried by the bloodstream, the reflected waves experience a shift in frequency.
   ○ Let fs be the frequency of the ultrasound transducer. Blood cells moving at velocity vb toward or away from the probe perceive a Doppler-shifted frequency. The net effect (transmitter→blood cell→receiver) doubles the Doppler shift. For small blood velocities vb ≪ vsound,
   Δf ≈ 2 fs vb cos θ / vsound,
   where θ is the angle between the ultrasound beam and the direction of blood flow.
2. Echocardiography.
   ○ A specialized Doppler ultrasound technique used to assess heart function; it measures flow across heart valves, ventricular inflow, and outflow, providing information about stenosis or regurgitation.