Auto Transfer Function (Cabin Gain Explained)
Typically there’s no such thing as a free lunch in the world especially audio. All except for one --- low frequency cabin gain. Even then it’s not really “gain” as there’s no amplifier involved but it is an unexpected and seemingly magical phenomenon for car and home sound.
Here’s the scoop. In an anechoic environment sound pressure - including bass - falls off at 6 dB for every doubling of distance. Things are louder when you are closer to them. Furthermore, a sealed box woofer falls off at 12 dB per octave below the resonant frequency (Fsb) of the system and bass reflex systems fall off at 24 dB per octave below their tuned frequency. Bandpass systems respond in a similar manner depending on whether the rear chamber is sealed or ported.
The situation is further complicated because the necessary speaker displacement (a function of the piston area times the stroke of the woofer and the amplifier power) quadruples for every halving of frequency. So if it requires a given displacement to deliver 120 dB SPL at 62 Hz it requires 4 times that displacement to do 120 dB at 32 Hz and four times that (16 times) to do 120 dB SPL at 16 Hz. Therefore, getting sufficient sound pressure at low frequencies is a big deal requiring large, long stroke woofers and big amplifiers.
Now add an enclosed space (room or car) to the equation and you introduce interesting phenomena. Because the walls stop the sound pressure from disappearing into the atmosphere they help maintain sound pressure although, depending on frequency, they can also affect the distribution of sound pressure in the space. In what they call the modal region standing waves are formed that cause hot and dead spots related to the wavelength of frequencies being reproduced. In a living room standing waves are typically formed between roughly 30 and 300 Hz. In a space the size of a Corvette interior the standing wave region is pushed up roughly an octave to 60 to 600 Hz.
But in the region below the lowest axial mode (related to the longest dimension) in any given space there is a wonderful re-balancing effect. Ever wonder why your voice sounds so wonderfully baritone when you sing in the shower? That’s because your bathroom is an acoustically “small” space. At frequencies below the lowest axial room mode (the half wavelength of the length of the frequency being reproduced) there is an apparent reinforcement of low frequency sound pressure of 12 dB per octave as frequency falls. In other words, containing the sound pressure at low frequencies offsets the natural falloff of a sealed woofer system below resonance nearly perfectly. Further in a small enough space you’re always closer to the pressure generator and there are fewer air molecules requiring excitation compared to a larger space.
If this is so then why doesn’t every small space have incredibly deep powerful bass down to DC? Won’t everything sound like an approaching locomotive? Well small spaces will produce deep, powerful bass, but you first have to supply sufficient speaker displacement to generate the sound pressure at subwoofer frequencies. Your bathroom helps maintain the lower part of your vocal range giving it an increased sense of baritone but it isn’t subwoofer quality because your voice system just can’t generate the pressure, even softly, at very low frequencies.
When you can generate pressure at subwoofer frequencies it is possible get 120 dB SPL at 10 Hz (unweighted) in a Corvette using a 10-inch woofer in a small enclosure and a reasonably sized amplifier. To get a similar result in my large home theater listening room requires eight 15-inch (23.4 mm Xmax) woofers and 5000-watts. At home the lowest axial room mode occurs at 16 Hz and so cabin gain in my home theater doesn’t even start until my car has been already bassin’ out for two octaves.
The graph depicts a 15-inch woofer in a 1.1-cubic foot enclosure measured near-field (microphone within ½-inch of the dust cap) and compared with the same system placed in the hatch area of a 2001 Corvette C5 Coupe with the microphone at ear height in the driver’s seat and adjusted for distance. It’s easy to see the 28 dB of cabin reinforcement at 16 Hz and over 36 dB at 10 Hz. The in-car trace has been equalized to smoothen it out as much as possible but below 60 Hz ripples in the response are a function of sound pressure cabin leaks and not standing waves or other acoustical phenomenon. Because this system will use a low pass filter no EQ above 150 Hz was used. Other items are also apparent. The Lab trace (solid) shows the break-up modes that happen with woofers in their upper range and modal affects above 60 Hz are also apparent in the In-Car trace (dots). I used this 15-inch particular system to illustrate the effect but I’ve often obtained this level of performance with a long stroke 10-inch woofer.
You may also ask whether sound pressure levels of 120 dB are dangerous. Well they are if the frequency is relatively high. 120 dB SPL at 2-3000 Hz where our hearing sensitivity is greatest is both painful and dangerous. But 120 dB SPL at 10 or 20 Hz is neither. In fact you often receive loud low frequency sound when a train arrives at the station or mother nature treats you to a thunderstorm. It doesn’t seem all that loud even if the sound pressure is measurably loud. And it’s not dangerous to your hearing.
In fact, few home theater subwoofers deliver enough volume displacement (Vd) to generate satisfactory SPL at frequencies below 25 to 30 Hz While it’s easier to do in a car it’s not dangerous if the frequency remains low. Loud cars may often be dangerous to listeners in the car because the music doesn’t contain only low frequencies. However the boom-boom you hear from the low-rider in the next lane may be annoying but it’s not dangerous to bystanders because the windows and car body blocks the higher frequency more dangerous sounds.