Problem Statement
A radar operates at a wavelength λ = 3.0 cm. Find the velocity of an approaching aircraft if the beat frequency between the transmitted signal and the signal reflected from the aircraft equals Δν = 1.0 kHz.
Given
- Radar wavelength: λ = 3.0 cm = 0.030 m
- Beat frequency: Δν = 1.0 kHz = 1.0 × 10³ Hz
- Speed of light: c = 3.0 × 10⁸ m/s
Concepts Used
Doppler effect for electromagnetic waves (radar):
When a radar transmits a signal of frequency ν₀ toward an approaching target moving with velocity v (v ≪ c), the reflected signal received back at the radar is Doppler-shifted. Because the wave travels to the target and back, the Doppler shift is doubled compared to a single-pass situation.
Frequency received by approaching aircraft (source stationary, observer approaching):
ν₁ = ν₀ · (c + v)/c
The aircraft reflects at ν₁. The radar now receives this from an approaching source (second shift):
ν₂ = ν₁ · (c + v)/c = ν₀ · (c + v)²/c²
Beat frequency: Δν = ν₂ − ν₀ = ν₀ · [(c + v)²/c² − 1]
For v ≪ c, to first order in v/c: Δν ≈ 2ν₀v/c = 2v/λ
Step-by-Step Solution
Step 1: The radar beat-frequency formula for an approaching target at velocity v:
Δν = 2v/λ
The factor of 2 arises because the Doppler shift accumulates on both the outgoing leg (transmitter to target) and the return leg (target to receiver).
Step 2: Solve for v:
v = Δν · λ / 2
Calculation
v = (1.0 × 10³ Hz × 0.030 m) / 2
v = 30 / 2 = 15 m/s
Answer
v = 15 m/s ≈ 54 km/h
Physical Interpretation
The radar Doppler technique is exceptionally sensitive because microwave frequencies ν₀ = c/λ ≈ 10¹⁰ Hz are so large. Even a modest aircraft speed of 15 m/s produces a detectable kilohertz-scale beat note measurable with standard electronics. This is the working principle behind police radar guns, air-traffic control systems, and weather Doppler radar. The double-pass nature of radar doubles the effective Doppler shift compared to a single-transit optical measurement, making the instrument twice as sensitive to target velocity as a one-way Doppler measurement would be.
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