Ohmic Audio

⚙️ ENGINEER LEVEL: Acoustics and SPL Theory

Sound Pressure Level Definition and Calculation

SPL is defined as:

SPL (dB) = 20 × log₁₀(P / P₀)

Where: - P = measured sound pressure (Pascals) - P₀ = reference pressure = 20 μPa (micropascals) - 20 μPa is the threshold of human hearing at 1 kHz

Alternative forms:

Intensity based:

SPL (dB) = 10 × log₁₀(I / I₀)

Where: - I = sound intensity (W/m²) - I₀ = reference intensity = 10⁻¹² W/m²

Power based (at fixed distance):

SPL (dB) = 10 × log₁₀(P_acoustic / P₀)

Relationship between pressure and intensity:

I = P² / (ρ × c)

Where: - ρ = air density ≈ 1.21 kg/m³ at sea level - c = speed of sound ≈ 343 m/s at 20°C - Product ρ×c ≈ 415 rayls (characteristic impedance of air)

Combining Multiple Sound Sources

When multiple sources play together, sound pressures add vectorially (considering phase):

For uncorrelated sources (different signals):

SPL_total = 10 × log₁₀(10^(SPL₁/10) + 10^(SPL₂/10) + ...)

Example: Two sources each producing 90 dB:

SPL_total = 10 × log₁₀(10^9 + 10^9)
SPL_total = 10 × log₁₀(2 × 10^9)
SPL_total = 10 × (log₁₀(2) + 9)
SPL_total = 10 × (0.301 + 9) = 93 dB

For correlated sources (same signal, in phase):

P_total = P₁ + P₂ + ...
SPL_total = 20 × log₁₀(P_total / P₀)

Example: Two identical sources each producing 90 dB, perfectly in phase:

P_total = 2 × P₁
SPL_total = 90 + 20 × log₁₀(2)
SPL_total = 90 + 6 = 96 dB

Key insight: Doubling power gives +3 dB, but doubling sources in phase gives +6 dB (you get more than just twice the power - you get coupling).

Distance and SPL: Inverse Square Law

In free field (no reflections), sound pressure decreases with distance:

SPL₂ = SPL₁ - 20 × log₁₀(r₂/r₁)

Where: - SPL₁ = sound level at distance r₁ - SPL₂ = sound level at distance r₂

For doubling distance:

SPL_loss = -20 × log₁₀(2) = -6 dB

However, in car cabin: - Free field equation doesn't apply - Reflections and standing waves dominate - SPL can actually increase with distance in some frequencies - Small enclosure acts more like pressure chamber than free field

Boundary conditions:

Car cabin is complex: - Below 80-100 Hz: pressure chamber (minimal distance loss) - 100-500 Hz: room modes dominate (complex pattern) - Above 500 Hz: approaches free field behavior

Cabin Gain and Transfer Function

Cabin gain is the increase in SPL due to the small, enclosed space of a vehicle interior.

Physical mechanism: At wavelengths comparable to or longer than cabin dimensions, the cabin acts as a pressure vessel rather than allowing sound to propagate away.

Wavelength calculation:

λ = c / f

Where: - λ = wavelength (meters) - c = speed of sound ≈ 343 m/s - f = frequency (Hz)

Example frequencies: - 20 Hz: λ = 17.2 m (56 feet) - 40 Hz: λ = 8.6 m (28 feet) - 80 Hz: λ = 4.3 m (14 feet) - 160 Hz: λ = 2.1 m (7 feet)

Since typical car cabin is 2-4 meters long, frequencies below ~80-100 Hz experience significant cabin gain.

Typical cabin gain curves: - Maximum gain: +10 to +15 dB (vehicle dependent) - Frequency range: 30-80 Hz - Sharp rolloff above transition frequency - Room modes create peaks and nulls

First-order model:

Cabin acoustic impedance:

Z_cabin = ρ × c² / (j × ω × V)

Where: - V = cabin volume (m³) - ω = angular frequency = 2πf - j = imaginary unit

Pressure gain relative to infinite baffle:

G_cabin(f) = 20 × log₁₀|1 + Z_cabin/Z_radiation|

This shows why small cabins have massive bass boost - the acoustic impedance presented by the cabin is much higher than radiation impedance, forcing more pressure development.

Room Modes and Standing Waves

Room modes are resonances that occur at specific frequencies determined by cabin dimensions.

Axial modes (dominant):

f_n = (n × c) / (2 × L)

Where: - n = mode number (1, 2, 3...) - L = dimension length - c = speed of sound

Example for 4-meter long cabin:

f₁ = (1 × 343) / (2 × 4) = 43 Hz (fundamental)
f₂ = (2 × 343) / (2 × 4) = 86 Hz (second harmonic)
f₃ = (3 × 343) / (2 × 4) = 129 Hz (third harmonic)

Tangential modes (involve two dimensions):

f_n,m = (c/2) × √[(n/L₁)² + (m/L₂)²]

Oblique modes (involve all three dimensions):

f_n,m,p = (c/2) × √[(n/L₁)² + (m/L₂)² + (p/L₃)²]

Practical implications: - Multiple overlapping modes create complex response - Peaks can be +10 to +15 dB - Nulls can be -20 dB or more - Position of subwoofer and listener affects which modes are excited - EQ and DSP required to flatten response

Q factor of room modes:

Modes have quality factor (Q) describing how "ringy" they are:

Q = f_resonance / Δf_3dB

Typical car cabin: Q = 5-15 (moderately to highly resonant)

High Q modes: - Sharp peaks in response - Long decay time - "Boomy" bass - Require narrow EQ cuts

Speaker Efficiency and Sensitivity

Sensitivity (dB @ 1W/1m) is the most common specification, but efficiency (percentage) is the fundamental property.

Efficiency definition:

η = P_acoustic / P_electrical

Typical loudspeakers: η = 0.1% to 3%

Converting efficiency to sensitivity:

Sensitivity (dB) = 112 + 10 × log₁₀(η)

Example: - 1% efficiency: 112 + 10 × log₁₀(0.01) = 112 - 20 = 92 dB - 0.5% efficiency: 112 + 10 × log₁₀(0.005) = 112 - 23 = 89 dB

Factors affecting efficiency:

  1. Radiation resistance: Higher for larger diaphragms and higher frequencies

  2. Electromechanical coupling:

    η_em = (Bl)² / (Re × Mms)

    Where:

    • Bl = force factor (T·m)
    • Re = voice coil resistance (Ω)
    • Mms = moving mass (kg)

  3. Acoustic impedance matching: Better at high frequencies (smaller wavelengths)

Practical sensitivity values:

Power needed for target SPL:

Starting from sensitivity (dB @ 1W/1m):

Required Power = 10^[(Target SPL - Sensitivity)/10]

Example: 88 dB sensitivity driver, target 118 dB:

P = 10^[(118-88)/10] = 10^3 = 1000 watts

Multiple drivers: Each doubling of drivers adds +3 dB (if uncorrelated) or +6 dB (if perfectly correlated):

Four identical 88 dB drivers: - Uncorrelated: 88 + 6 = 94 dB @ 1W total - Correlated: 88 + 12 = 100 dB @ 1W total

Loudness Perception and Psychoacoustics

Subjective loudness doesn't equal objective SPL measurement.

Sone scale: Perceptual loudness measurement where doubling sones = doubling perceived loudness.

Sones = 2^[(SPL - 40)/10]  (approximation for 1 kHz)

Phon scale: Equal-loudness contours defined by Fletcher-Munson (later ISO 226).

Key points at various SPL: - 40 phon: Bass appears 20-30 dB quieter than midrange - 80 phon: Bass appears 10 dB quieter than midrange - 100 phon: Nearly flat perception

Practical application: - Boost bass +6 to +10 dB for low-level listening - Flatten response for loud listening - This is why "loudness" controls boost bass/treble at low volume

Frequency-dependent loudness growth:

Bass frequencies have slower loudness growth than midrange: - 100 Hz: ~2× loudness per 10 dB SPL increase - 1 kHz: ~2× loudness per 10 dB SPL increase - 10 kHz: ~2× loudness per 10 dB SPL increase

But at equal SPL, 1 kHz sounds much louder than 100 Hz or 10 kHz.

Critical bands and masking:

Human hearing analyzes sound in critical bands (~1/3 octave wide).

Frequency masking: - Strong signal masks weaker signals within ±1/2 critical band - Upward spread of masking > downward spread - 80 dB signal at 1 kHz can mask 40 dB signal at 1.1 kHz

Temporal masking: - Forward masking: 50-200 ms after loud sound - Backward masking: 5-20 ms before loud sound - Important for transient reproduction

Implications for system design: - Crossover regions must be clean to avoid masking - Time alignment reduces masking between drivers - Distortion products above noise floor are audible even if masked

1.3 Types of Speakers, Amplifiers, and Subwoofers