Once you start learning about box design, you might hear the word 'alignment'. Often alignments are mentioned as something you must adhere to in order to obtain a certain level or quality of sound. Some books even recommend that you design to an alignment and then measure or verify that you attained it. What are these alignments all about? Well, suffice to say that these are types of filter responses that have useful properties such as flat response, minimum delay, etc, which are useful in certain circumstances, but don't actually correlate all that well to how something will sound. Next I will explain them in further detail for the different types of enclosures.
For sealed enclosures you hear about critically damped, Bessel, Butterworth and Chebychev. A sealed box is a relatively simple system, and these simply correlate to different values of Qtc, or sealed box system Q. Since sealed boxes are second order systems, the alignemt types Butterworth and Chebychev are often abbreviated as B2 or C2.
Critically Damped - Qtc = 0.5 - values of Qtc below this are called 'overdamped'
Bessel - Qtc = 0.57735 = 1/sqrt(3) - 'maximally flat delay'
Butterworth (B2) - Qtc = 0.7071 = 1/sqrt(2) - 'maximally flat response'
Butterworth is thus the break point in system Qtc's; Qtc's below 0.707 droop some above resonance, and Qtc's above this have some peaking above resonance.
Unnamed? Qtc = 1, the output at resonance is equal to the output in the passband.
Chebychev (C2) - Qtc=1.1 - said to have "max power handling" by some sources.
For Vented (4th order) enclosures you often hear about Bessel, Butterworth (B4), Quasi-third order Butterworth (QB3), Chebychev (C4), and Sub-Chebychev (SC4). A vented box is a more complicated (fourth order) system, and these functions only approximately fit the actual response. If you make some simplifying assumptions, such as lumping all the damping of the box into one term (Ql, typically has a value of ~7) you can apply these filter alignments to vented box design. Bessel and Butterworth functions correspond to flattest delay and flattest response, as they do for sealed boxes, but are actually fairly difficult to achieve in practice because they only work for one specific value of driver Qts. For example, the Butterworth B4 system when Ql=7 requires:
Qts = 0.4048
Vas/Vb = 1.06
Fb = F3 = Fs
Alignments are fairly complex to calculate, so the most expedient method is either to use design charts or tables, or to curve fit these to get a starting point, which is what Small and Keele did to get the cookbook box equations I give here. See the subpage below for some example alignment tables.
For 4th order bandpass enclosures, the alignments are the same as for vented boxes, but the driver Q is modified by the sealed enclosure, and it turns out another design procedure gives simpler results.
If you got this far, you must be fairly interested, so I will go into a little more detail, but briefly, and to get much out of it you probably need to have had some college calculus. Consider it more a motivation for further study. Alignments are mathematical abstractions using transfer functions that achieve certain specific goals, such as the flattest response, the lowest delay, the widest passband, etc.
A transfer function is an analysis of a physical system using laplace transforms to determine the response of the system for a given input. Generally it will be a rational function with a numerator and a denominator which are both composed of polynomials. The polynomial in the denominator is known as the characteristic polynomial. The roots of this polynomial are called the poles of the system. The coefficients of this polynomial determine where the poles lie on the complex frequency plane and the response shape. The alignment functions correspond to certain polynomials with interesting properties. If you use google you can find out what Butterworth, Bessel, Chebychev, polynomials are all about, and what specific coefficients are needed.
To actually calculate alignments, the best reference is Richard Small's series of articles which were published in the Journal of the Audio Engineering Society in the early 1970's.