Doppler Effect

The Doppler effect describes the change in the observed frequency of a wave when there is relative motion between the source and the receiver.

🎯 Learning Objectives

Fundamental Idea of Doppler Effect:

Consider a source emitting successive wavefronts separated by period $T_s$. If the source moves toward the observer, the spacing between emitted wavefronts decreases. Thus the observed frequency $f_o$ differs from the emitted frequency $f_s$. This follows from a purely geometric argument about wavefront spacing.

Classical Cases:

Moving Observer, Stationary Source

Let the observer move with velocity $v_o$ toward a stationary source. Speed of sound is $c$.

The wave speed relative to the observer becomes $c + v_o$.

Observed frequency:

This case occurs in:

• Microphone on a drone
• Moving listener
• Automotive applications (drive-by tests)

Moving Source, Stationary observer

This is more subtle: the source motion compresses or dilates the emitted wavelength. If the source moves with velocity $v_s$ toward the observer:

If it moves away:

This asymmetry is essential: the Doppler shift is stronger for source motion than observer motion.

General Formula (Source + Observer Both Moving)

The general classical expression is:

Signs convention:

• $v_o > 0$ if observer moves toward the source
• $v_s > 0$ if source moves toward the observer

This formula is the one most used in engineering calculations.

Examples:

Stationary Sound Source

Source moving with $V<c$ (Mach 0.7)

Source moving with $V = c$ (Mach 1 - breaking the sound barrier)

Source moving with $V > c$ (Mach 1.4 - supersonic)

Interactive EXAMPLE

Doppler Effect in a Moving Medium (Wind):

If the medium moves with velocity $u$, wave speed becomes:

Thus:

This is essential for:

• Outdoor sound propagation
• Aviation and UAV acoustics
• Underwater acoustics (currents)

Mach Number and Supersonic Motion:

Define Mach number:

Subsonic Regime ($M < 1$)

Wavefronts compress in front of the source:

• frequency increases for approaching observer
• wavelength shortens
• sound appears more directional

Supersonic Regime ($M > 1$)

Source outruns its own wavefronts → Mach cone.

Cone angle

This is fundamental to:

• Sonic booms
• Aeroacoustics
• Shock waves

Shock-Based Doppler Effect:

If the source moves supersonically:

• No classical Doppler formula applies
• Pressure fronts accumulate into a shock
• Received signal is a single abrupt pulse with broadband content

Shock-front arrival time determines the observed “frequency” (really time interval).

Doppler Effect in Non-Uniform Media (Temperature Gradients):

Sound speed varies with temperature:

Meaning the Doppler effect depends on altitude.

Curved ray paths → geometric Doppler shift

Applications:

• Meteorology
• Atmospheric acoustics
• Long-distance sound propagation

Acoustic Doppler in Fluids (Navier–Stokes Perspective):

Wave propagation in a moving fluid satisfies:

The convective derivative $\mathbf{u} \cdot \nabla$ is responsible for the convective Doppler effect.

This formulation underlies:

• Computational Aeroacoustics (CAA)
• Lighthill analogy with moving flows
• Rotational Doppler effects

Doppler Sonography (Medical Ultrasound):

The received frequency shift from a moving reflector (blood flow):

The factor 2 appears because:

• The wave travels to the moving scatterer, and
• Back to the probe

Practical Examples:

Passing police car

High pitch approaching → low pitch receding.

Railway horn near station

Useful for measuring railway vehicle speed.

Rotorcraft & drones

Blade passing frequency (BPF) and convective Doppler.

Atmospheric acoustics

Wind shear bends rays → directional Doppler phenomena

🧪 Interactive Examples:

📝 Key Takeaways:

🧠 Quick Quiz: