⚙️ ENGINEER LEVEL: Acoustic Optimization Theory
Cabin Transfer Function Modeling
The vehicle cabin is not a rigid piston but a complex resonant cavity. The transfer function from driver volume velocity to cabin pressure involves:
Modal analysis — each cabin dimension supports standing wave modes:
f_lmn = (c/2) × √[(l/Lx)² + (m/Ly)² + (n/Lz)²]
For a sedan (Lx=1.2m, Ly=1.5m, Lz=4.0m interior dimensions):
f_100 = (343/2) × (1/1.2) = 143 Hz (width mode)
f_010 = (343/2) × (1/1.5) = 114 Hz (height mode)
f_001 = (343/2) × (1/4.0) = 43 Hz (length mode)
At 40-50 Hz, the wavelength (7-8.5m) is longer than cabin length. The cabin pressurizes uniformly — this is ideal for SPL. No nodes or antinodes to worry about. Pure pressure gain.
Cabin gain formula (simplified for low frequencies):
G_cabin = 10 × log₁₀[(4π × V_cabin) / λ²]
At 50 Hz, λ = 6.86m, V_cabin = 3 m³:
G_cabin = 10 × log₁₀[(4π × 3) / 6.86²] = 10 × log₁₀(0.801) = -0.96 dB
Wait, that's a loss, not gain. Let me reconsider the cabin gain model.
Correct approach — pressure doubling from boundary loading:
In a sealed cabin at frequencies well below first mode, pressure increases because the air cannot escape. The effective radiation impedance seen by the driver increases, causing more mechanical-to-acoustic power conversion.
Practical cabin gain: 10–20 dB depending on vehicle size and sealing quality. Smaller, well-sealed vehicles exhibit more gain.
Thermal Power Compression in Competition
Voice coil temperature during burping:
Assume 15-second burp at 10,000W into a driver with Re = 1Ω (cold).
Power dissipation in voice coil:
P_heat = I² × Re = (√(10000/1))² × 1 = 10,000W
All 10,000W becomes heat in the voice coil (100% conversion to thermal at DC and low frequencies).
Temperature rise (using thermal mass of typical 3" voice coil):
Thermal mass ≈ 50 J/°C (aluminum former, copper wire)
ΔT = (P × t) / C_thermal = (10000 × 15) / 50 = 3000°C theoretical
Obviously impossible — the coil would vaporize. What actually happens:
- Voice coil heats rapidly to 300-500°C
- Resistance increases (Rhot = Rcold × [1 + α×ΔT])
- Current decreases for same voltage
- Power decreases (power compression)
- After ~5-10 seconds, thermal equilibrium at reduced power
Competition strategy: Burp duration kept under 15 seconds. Cooling between runs (forced air, liquid cooling on extreme builds).
Liquid-cooled voice coils (exotic competition tech):
Some championship builds use subwoofers with hollow voice coil formers. Coolant (water or glycol) circulates through the former, removing heat directly. Allows sustained high power. Adds complexity and cost.