Advanced Sonar Beam Pattern Analysis Tool

Educational tool for understanding linear array acoustics, beam steering, and spatial aliasing

📚 Theory and Mathematics

🎯 Beam Steering with Phased Arrays

Beam steering is achieved by applying progressive phase delays across array elements. To steer the beam to angle θ₀, element n receives phase delay:

φₙ = -n × k × d × sin(θ₀)

Where:

φₙ = phase delay for element n (radians)
k = 2π/λ (wave number)
d = element spacing
θ₀ = desired steering angle

Physical Implementation: In sonar systems, these phase delays are implemented using variable delay lines or digital signal processing to create the desired beam direction.

⚠️ Spatial Aliasing and the Nyquist Criterion

Spatial aliasing occurs when element spacing exceeds λ/2, creating grating lobes - unwanted copies of the main beam that can cause false targets.

Nyquist Criterion: d ≤ λ/2

Why λ/2? This comes from spatial sampling theory:

Acoustic waves create spatial patterns that must be "sampled" by array elements
To avoid aliasing, sampling rate must be ≥ 2 × highest spatial frequency
For plane waves, highest spatial frequency = 1/λ
Therefore: element spacing ≤ λ/2

Consequences of Violation:

Grating Lobes: Additional peaks at predictable angles
False Targets: Sonar may detect objects that aren't there
Reduced Performance: Energy spreads across multiple lobes
Steering Limitations: Grating lobes become visible during steering

🎯 What is a Linear Array?

A linear array is a series of acoustic transducers arranged in a straight line. By controlling both amplitude weights and phase delays, we can shape both the beam pattern and steer the acoustic beam electronically without mechanical movement.

📐 Enhanced Array Factor Formula

With both amplitude shading and beam steering, the Array Factor becomes:

AF(θ) = Σ(n=0 to N-1) wₙ × e^(j×n×k×d×[sin(θ) - sin(θ₀)])

Where:

wₙ = amplitude weight for element n (shading)
θ₀ = steering angle (beam direction)
sin(θ) - sin(θ₀) = steering transformation

🔬 Advanced Concepts:

🎛️ Phased Array Advantages:

Electronic Steering: No mechanical movement required
Fast Response: Beam can be redirected in microseconds
Multiple Beams: Can form several beams simultaneously
Adaptive Processing: Real-time optimization possible

⚠️ Design Considerations:

Steering Range: Limited by grating lobe appearance
Beam Distortion: Pattern changes with steering angle
Hardware Complexity: Requires precise phase control
Cost: More expensive than fixed arrays

💡 Real-World Applications

Naval Sonar: Modern naval vessels use large phased arrays for 360° coverage and rapid target tracking without mechanical steering.

Medical Ultrasound: Phased array transducers allow real-time imaging by electronically focusing and steering the acoustic beam.

Radar Systems: Phased array radars can track multiple targets while maintaining surveillance coverage.

🔊 Advanced Sonar Beam Pattern Analysis Tool

📊 Array Parameters

📐 Beam Characteristics

📡 Element Phase Distribution (Beam Steering)

📈 Element Amplitude Weights