Doppler Shift of Light
Lab vs observed spectral lines
80-2200 nm
Doppler formulas for light
Compare the non-relativistic approximation with the exact relativistic relation and watch where they diverge.
Non-relativistic wavelength and frequency
$$\lambda_{\rm obs} = \lambda_0\left(1 + \frac{v_r}{c}\right),\qquad \nu_{\rm obs} = \frac{\nu_0}{1 + v_r/c}$$
- Good for small $|v_r|/c$.
- Wavelength and frequency shifts are not exact sign mirrors at finite speed.
Relativistic Doppler
$$\lambda_{\rm obs} = \lambda_0\sqrt{\frac{1+\beta}{1-\beta}},\qquad \nu_{\rm obs} = \nu_0\sqrt{\frac{1-\beta}{1+\beta}},\qquad \beta = \frac{v_r}{c}$$
- Required as $|v_r|$ approaches a meaningful fraction of $c$.
- The wavelength and frequency factors are exact reciprocals.
Redshift and inverse velocity
$$z = \frac{\lambda_{\rm obs} - \lambda_0}{\lambda_0},\qquad \beta = \frac{(1+z)^2 - 1}{(1+z)^2 + 1}$$
- $z > 0$ means redshift (receding).
- $z < 0$ means blueshift (approaching).
Model boundary
- This demo models kinematic Doppler shift only.
- Cosmological and gravitational redshift are distinct mechanisms.
- Line catalogs are empirical rest wavelengths measured in air (standard lab values); hydrogen lines computed from the Bohr model are vacuum wavelengths.
Why don't waves bunch near the source?
Sound (medium)
Ripples compress ahead and stretch behind because air carries the disturbance.
Light (no medium)
Each crest travels at $c$ in vacuum. The spacing is set at emission and remains uniform when observed.
How to read these panels
Connect relative motion to shifted spectral lines with synchronized wave spacing and lab-vs-observed comparison strips.
Wave spacing. Observed crests are uniformly spaced at $\lambda_{\rm obs}$. Lab vs observed. Connectors show how each rest line shifts under $v_r$.
What to notice
- Wave crests stay uniformly spaced at the observer; only spacing changes with velocity.
- Connectors tie each lab line to its observed position so shifts are visually traceable.
- Non-rel and relativistic formulas agree at low speed but diverge at higher $|v_r|$.
- Hydrogen uses model-computed lines; heavier elements use empirical catalogs.
Model notes
- Speed of light: $c = 299,792.458\ \text{km/s}$ (exact).
- Positive velocity means receding (redshift), negative means approaching (blueshift).
- The physical state couples $v_r$ and $z$ through the relativistic inverse relation.
- Vacuum wavelengths are used throughout.
- This instrument models kinematic Doppler shift, not cosmological expansion or gravitational redshift.
Explore further
- Spectral Lines: Review how rest wavelengths are set by atomic transitions. Open Spectral Lines →
- Galaxy Rotation: Apply Doppler readouts to rotation curves and dark-matter inference. Open Galaxy Rotation →