Eberhard Sengpiel — Interactive Tonmeister & Acoustical Physics Dashboard

Stereo and Microphone Arrays

Calculation of the Stereo Recording Angle (SRA), capsule sensitivity, M-S matrix coupling, and load impedance.

1. Coincident/Spaced Stereo Arrays (Williams SRA)

Angle between Mics (α) 90°

degrees

Distance between Mics (d) 17.0 cm

Capsule Polar Pattern

Stereo Recording Angle (SRA) 96.8°

Max Level Difference (ΔL) 12.5 dB

Max Time Difference (Δt) 0.495 ms

Williams SRA Equations: $$\text{SRA} = 2 \cdot \theta_{\text{limit}} \quad \text{where} \quad \frac{\Delta L(\theta_{\text{limit}})}{15} + \frac{\Delta t(\theta_{\text{limit}})}{1.5} = 1.0$$ $$\Delta t(\theta) = \frac{d \sin\theta}{c}, \quad \Delta L(\theta) = 20 \log_{10}\left(\frac{A(\theta - \alpha/2)}{A(\theta + \alpha/2)}\right)$$

2. Mid-Side Array (M-S Stereo)

Mid Microphone Pattern

Mid/Side Gain Ratio 1.0

M/S

Equiv. M-S Recording Angle 100.0°

Left/Right Sum Level L = Mid + Side, R = Mid - Side

M-S Stereo Equation: $$\text{SRA} = 2 \arccos\left(\frac{1-k}{k \cdot \text{Ratio}}\right)$$

3. Sensitivity & Impedance Loading Loss

Mic Sensitivity (mV/Pa) 20.0 mV

Mic Output Z (Z_src) 200 Ω

Preamp Input Z (Z_load) 2000 Ω

Sensitivity in dB (re 1V/Pa) -34.0 dB

Input Attenuation -0.83 dB

Impedance Loading and Sensitivity: $$\text{dB (re 1V/Pa)} = 20 \log_{10}\left(\frac{\text{Sens (mV)}}{1000}\right)$$ $$\text{Loss (dB)} = 20 \log_{10}\left(\frac{Z_{\text{load}}}{Z_{\text{src}} + Z_{\text{load}}}\right)$$

Room Acoustics and Normal Modes

Modal analysis in rectangular rooms (axial, tangential, and oblique modes) and critical distance.

4. Modal Distribution (Physical Resonances)

Room Length (L) 6.0 m

Room Width (W) 4.5 m

Room Height (H) 3.0 m

Volume (V) / Surface Area (S) 81.0 m³ / 117.0 m²

Rayleigh Wave Equation for Room Modes: $$f_{p,q,r} = \frac{c}{2} \sqrt{\left(\frac{p}{L}\right)^2 + \left(\frac{q}{W}\right)^2 + \left(\frac{r}{H}\right)^2}$$

Normal Modes Spectrum (< 250 Hz):

5. Critical Distance (Direct vs Reverberant Field)

RT60 Time 1.20 s

Directivity Factor (Q)

Critical Distance (Dc) 1.31 m

Equivalent Absorption Area 10.9 m²-sab

Critical Distance Formula: $$D_c = 0.057 \sqrt{\frac{Q \cdot V}{T_{60}}}$$

Phase, Time Delay & Comb Filtering

Relationship between frequency, wavelength, and comb filtering depth induced by time offsets.

6. Phase-to-Time / Wavelength Converter

Frequency (f) 1000 Hz

Phase Angle (θ) 90°

Ambient Temperature 20.0 °C

°C

Speed of Sound (c) 343.4 m/s

Wavelength (λ) 34.3 cm

Time Delay 0.250 ms

Distance Difference 8.6 cm

Speed of Sound, Wavelength and Phase Formulas: $$c = 331.3 \sqrt{1 + \frac{T_{\text{temp}}}{273.15}}\text{ m/s}, \quad \lambda = \frac{c}{f}$$ $$t_{\text{delay}} = \frac{\theta}{360 \cdot f}, \quad d_{\text{phase}} = \frac{\theta}{360} \cdot \lambda$$

7. Comb Filter Ripple Depth (Bob McCarthy Ch 3)

Level Difference (ΔL) 6.0 dB

Peak-to-Valley Amplitude 11.2 dB

Valley Attenuation -14.2 dB

Comb Filter Ripple and Cancellation: $$\text{Ripple} = 20 \log_{10}\left(\frac{1+a}{1-a}\right)\text{ dB}, \quad \text{Valley} = 20 \log_{10}(1-a)\text{ dB}$$ $$\text{where } a = 10^{-\Delta L/20}$$

8. Haas Effect (Precedence)

Arrival Delay (Δt) 15.0 ms

Level Difference (ΔL) 3.0 dB

Perceived Location Localized to Primary

Haas Effect (Precedence): $$\Delta L_{\text{limit}} = 10 \log_{10}(\Delta t / 4) \text{ dB} \quad (5\text{ ms} \lt \Delta t \lt 35\text{ ms})$$

Decibels, Voltages & Level Converter

Calculation of electrical levels dBu, dBV, dBm, and RMS voltages, coherent phase and incoherent summation.

9. Voltage & Electrical Power Converter

Input RMS Voltage 1.00 V

Load Impedance (R)

Electrical Level dBu (ref 0.775V) 2.22 dBu

Electrical Level dBV (ref 1.00V) 0.00 dBV

Power Level dBm (ref 1mW) 20.97 dBm

Real Disipated Power 0.125 W

Electrical Decibels and Power: $$\text{dBu} = 20 \log_{10}\left(\frac{V}{0.775}\right), \quad \text{dBV} = 20 \log_{10}(V)$$ $$\text{dBm} = 10 \log_{10}\left(\frac{V^2 / R}{0.001}\right), \quad P = \frac{V^2}{R}\text{ Watts}$$

10. Acoustic Summation (Coherent vs Incoherent)

Channel 1 Level 90 dB

Channel 2 Level 90 dB

Phase Difference (θ) 0°

Coherent Sum (Same origin) 96.0 dB

Incoherent Sum (Independent sources) 93.0 dB

Acoustic Summation (Pressure vs Power): $$L_{\text{coh}} = 20 \log_{10}\left(\sqrt{p_1^2 + p_2^2 + 2p_1p_2\cos\theta}\right)$$ $$L_{\text{incoh}} = 10 \log_{10}\left(10^{L_1/10} + 10^{L_2/10}\right)$$

System Optimizations & Propagation

Distance coverage calculation, array field limits, subwoofer spacing and cable damping factor.

11. SPL Propagation, Air & Inverse Square Law

Speaker Sensitivity 98.0 dB

Applied Power 500 W

Listener Distance 30 m

SPL at 1m (Max Power) 125.0 dB

Air Loss (1 kHz) -0.16 dB

SPL at Listener 95.3 dB

Distance Attenuation and Inverse Square Law: $$\text{SPL} = \text{Sens} + 10 \log_{10}(P) - 20 \log_{10}(d) - \alpha_{\text{aire}} \cdot d$$

12. Line Array Field Limits (McCarthy Ch 5)

Line Array Height (H) 4.0 m

Operating Frequency 2000 Hz

Near Field Limit (d_trans) 46.6 m

Attenuation at Listener Distance -8.7 dB

Near Field Limit (Cylindrical): $$d_{\text{trans}} = \frac{H^2 \cdot f}{2c}\text{ metros}$$

13. Subwoofer Spacing (Sub Aliasing)

High Cutoff Frequency 100 Hz

Desired Coverage Angle 60°

Maximum Center-to-Center Spacing 2.29 m

Anti-Aliasing Spacing Limit: $$d_{\text{max}} = \frac{c}{f (1 + \sin\theta_{\text{steer}})}$$

14. Damping Factor at the Speaker Terminals

Amplifier DF 200

Speaker Impedance (Z_spk) 8.0 Ω

Cable Length (Two-way run) 15 m

Cable Gauge (AWG)

Cable Resistance (R_cable) 0.248 Ω

Net DF at Speaker Terminals 27.8

Damping Factor and Cable Loss: $$DF_{\text{net}} = \frac{Z_{\text{spk}}}{R_{\text{amp}} + R_{\text{cable}}}, \quad R_{\text{cable}} = \frac{2 \cdot L \cdot 0.0172}{A_{\text{wire}}}$$