Doppler Effect Formulas

Why This Matters

The Doppler Effect is one of those physics concepts that shows up everywhere—from the changing pitch of an ambulance siren to measuring how fast distant galaxies are racing away from us. In your intro physics course, you're being tested on your ability to select the right formula for different scenarios and correctly apply sign conventions. The tricky part isn't the math itself; it's knowing when to use which version of the equation and whether to add or subtract velocities.

Understanding these formulas also connects to broader themes you'll encounter: wave behavior, relative motion, and the transition from classical to relativistic physics. The Doppler Effect beautifully illustrates how the same phenomenon requires different mathematical treatments depending on the speeds involved. Don't just memorize formulas—know what physical situation each one describes and why the math changes when you switch from sound waves to light waves traveling at relativistic speeds.

---

Classical Doppler Effect: Sound Waves

These formulas apply when waves travel through a medium (like sound in air) and speeds are much slower than the wave speed. The key insight is that the medium provides an absolute reference frame, so it matters whether the source, observer, or both are moving.

General Doppler Effect Formula

• $f' = f \cdot \frac{v \pm v_o}{v \mp v_s}$—the master equation that handles any combination of source and observer motion
• Sign convention is critical: use the upper signs (+ in numerator, − in denominator) when source and observer approach each other; reverse for separation
• $v$ is the wave speed in the medium—for sound in air at room temperature, use $v \approx 343 \text{ m/s}$

Moving Observer, Stationary Source

• $f' = f \cdot \frac{v \pm v_o}{v}$—simplified formula when only the observer moves
• Add $v_o$ when the observer moves toward the source (higher frequency); subtract when moving away
• Classic example: you're in a car driving toward a stationary horn—you hear a higher pitch than someone standing still

Moving Source, Stationary Observer

• $f' = f \cdot \frac{v}{v \mp v_s}$—use this when only the source is in motion
• Subtract $v_s$ in the denominator when the source approaches (frequency increases); add when it recedes
• This formula explains the siren effect—an ambulance sounds higher-pitched approaching you, lower-pitched driving away

---

Relativistic Doppler Effect: Electromagnetic Waves

When dealing with light or any electromagnetic wave, there's no medium—and when speeds approach $c$, you must account for time dilation. These formulas incorporate special relativity and are symmetric between source and observer motion.

Relativistic Doppler Formula

• $f' = f \cdot \sqrt{\frac{1 - \beta}{1 + \beta}}$—where $\beta = v/c$ is the velocity as a fraction of the speed of light
• No medium means no preferred reference frame—only relative motion between source and observer matters
• The square root comes from time dilation—moving clocks run slow, which affects the observed frequency

Electromagnetic Wave Doppler Shift

• Same formula as above, but often written for the case where source and observer are receding (causing redshift)
• For approaching motion, flip the signs: $f' = f \cdot \sqrt{\frac{1 + \beta}{1 - \beta}}$ gives blueshift
• Essential for astrophysics—this is how we measure stellar velocities and confirm the universe's expansion

---

Quantifying the Shift: Frequency and Wavelength Changes

These formulas help you calculate and interpret how much a wave's properties change. Frequency and wavelength are inversely related, so a shift in one automatically means an opposite shift in the other.

Frequency Shift Formula

• $\Delta f = f' - f$—the difference between observed and emitted frequency
• Positive $\Delta f$ means blueshift (higher frequency, observer and source approaching)
• Negative $\Delta f$ means redshift (lower frequency, observer and source separating)

Wavelength Shift Formula

• $\Delta \lambda = \lambda' - \lambda$—the change in wavelength from emitted to observed
• Positive $\Delta \lambda$ indicates redshift (wavelength stretched, objects moving apart)
• Inverse relationship: since $v = f\lambda$, when frequency increases, wavelength decreases proportionally

---

Redshift and Blueshift Parameters

These dimensionless ratios are the standard way to express Doppler shifts in astronomy and allow easy comparison across different wavelengths.

Redshift Parameter (z)

• $z = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}}$—a dimensionless measure of wavelength increase
• Positive $z$ means the object is receding—used extensively to measure galaxy distances and cosmic expansion
• For small velocities: $z \approx v/c$, giving a quick estimate of recession speed

Blueshift Parameter

• $z = \frac{f_{observed} - f_{emitted}}{f_{emitted}}$—can also be expressed in terms of frequency
• Negative $z$ (or positive when defined this way) indicates approach—the object is moving toward the observer
• Rare in cosmology since most distant objects are redshifted due to universal expansion

---

Quick Reference Table

---

Self-Check Questions

1. When would you use the classical Doppler formula versus the relativistic formula? What physical feature determines your choice?

2. An ambulance approaches you, then passes and drives away. Using the moving-source formula, explain why the pitch drops as it passes—what happens to the denominator?

3. Compare the formulas for a moving observer with a stationary source versus a moving source with a stationary observer. Why aren't they mathematically identical, and what does this tell you about the role of the medium?

4. If a star has a redshift of $z = 0.1$, is it moving toward or away from Earth? Estimate its velocity as a fraction of the speed of light.

5. A problem states that both the source and observer are moving toward each other. Write out the general formula with the correct signs, and explain how you determined which sign to use in the numerator and denominator.