To understand and apply radar principles to measure the frequency of a vibrating tuning fork using the Doppler effect.
Learn the fundamental concepts of Doppler radar and how electromagnetic waves interact with moving objects.
Study the vibration characteristics of tuning forks and their relationship to acoustic frequency.
Apply Doppler shift analysis to determine the vibration frequency of a tuning fork's prongs.
Develop experimental techniques for signal processing and data analysis in radar systems.
When a radar transmits an electromagnetic wave toward a moving target, the frequency of the reflected wave changes proportionally to the target's velocity. This phenomenon is called the Doppler effect.
A tuning fork consists of two prongs that vibrate symmetrically when struck. The prong tip moves sinusoidally:
The instantaneous velocity of the prong tip is the derivative of displacement:
When radar illuminates a vibrating tuning fork prong, the reflected signal experiences a time-varying Doppler shift. The received signal can be expressed as:
For small vibration amplitudes (β << 1), the spectrum contains the carrier frequency fc and sidebands at fc ± ft, fc ± 2ft, etc. The first sideband spacing directly gives the tuning fork frequency ft.
By analyzing the spectrum of the received radar signal:
Mount the CW Doppler radar module on a stable platform. Ensure the radar antenna is aligned perpendicular to the tuning fork prongs. Place the tuning fork approximately 30–50 cm from the radar antenna.
Attach a small piece of aluminum foil (≈ 1 cm²) to the tip of one prong to enhance radar reflectivity. Ensure the foil does not significantly alter the fork's mass or stiffness.
Power on the radar and verify the carrier frequency (10 GHz). Check the baseline spectrum without the tuning fork vibrating to ensure no spurious signals are present.
Strike the tuning fork gently with a rubber mallet or against a soft surface. Hold the fork by its stem, keeping the prongs free to vibrate. Position the fork so the prong tip moves toward and away from the radar.
Connect the radar IF output to the spectrum analyzer. Set the span to approximately ±5 kHz around the carrier. Observe the sidebands appearing as the fork vibrates.
Record the carrier frequency fc and the frequencies of the first upper and lower sidebands (fUSB and fLSB). Calculate the tuning fork frequency using:
Repeat the experiment 5 times with fresh strikes. Calculate the mean and standard deviation of the measured frequency. Compare with the manufacturer's specified frequency.
Repeat with tuning forks of different frequencies (440 Hz, 512 Hz, 1024 Hz). Observe how the sideband spacing changes. Try varying the radar-to-fork distance and note any amplitude changes.
Simulate the radar signal from a vibrating tuning fork and observe the Doppler sidebands.
Your lab report should be organized into the following sections:
Test your understanding with these 12 questions. Answers and explanations are provided below.
The Doppler effect describes the change in frequency of a wave in relation to an observer moving relative to the wave source. In this experiment, the vibrating prong acts as a moving target, causing a time-varying Doppler shift in the reflected radar signal.
The factor of 2 arises because the wave travels to the target and back. The Doppler shift occurs twice: once when the wave reaches the moving target, and again when the reflected wave returns from the moving target to the stationary radar.
Using vmax = 2πfA = 2π × 440 × 0.001 = 2.7646 m/s. The maximum velocity occurs when the cosine term equals 1 in v(t) = 2πfA·cos(2πft).
The received signal sr(t) = Ar·cos[2πfct + (4πA/λ)·sin(2πftt)] shows that the phase is modulated by the vibration. For small modulation index, PM and FM spectra are similar, both producing sidebands at fc ± nft.
ft = fUSB - fc = 10,000,440 - 10,000,000,000 = 440 Hz. The sideband spacing directly reveals the tuning fork's vibration frequency.
For small β, the Bessel function approximation gives J0(β) ≈ 1 (carrier), J1(β) ≈ β/2 (first sidebands), and higher-order Bessel functions are negligible. Thus only the carrier and first-order sidebands are significant.
A bare metal tuning fork prong has a small radar cross-section. Attaching a small piece of foil increases the reflective area, providing a stronger return signal for analysis. The foil must be small enough to not significantly alter the fork's mechanical properties.
Sideband amplitude is proportional to the Bessel function Jn(β), where β = 4πA/λ. As amplitude A increases, β increases, and the sideband amplitudes grow (up to a point, after which energy redistributes to higher-order sidebands).
The note A4 (the A above middle C) is standardized at 440 Hz. This is the most common tuning fork frequency used for musical instrument tuning and physics demonstrations.
The Doppler shift depends on the radial (line-of-sight) component of velocity: fd = 2vradialfc/c. When the prong moves perpendicular to the radar beam, vradial = 0, so there is no Doppler shift at that instant. This is why proper alignment is crucial.
The sidebands appear at fc ± ft, fc ± 2ft, fc ± 3ft, etc. Therefore, the spacing between any two adjacent sidebands (or between the carrier and first sideband) equals the tuning fork frequency ft.
A CW (Continuous Wave) radar transmits constantly, allowing continuous monitoring of the target's motion. A pulsed radar would miss parts of the vibration cycle between pulses. For measuring a continuously vibrating target, CW radar is ideal.