There are only a few other three-dimensional orthogonal coordinate systems that lead to practical horn profiles. Of these are spheroidal coordinates, which come in two flavors: prolate and oblate. Although they are too complicated to deal with in this text, they are worth mentioning. Spheroidal coordinates are constructed from overlapping families of ellipses and hyperbolas, which share two focal points. If the ellipses are rotated about an axis passing through the focal points, they become prolate spheroids (cigar shaped) and we have a prolate–spheroidal coordinate system. Then any one of the rotated hyperbolas can be chosen as a horn profile. Such a profile looks parabolic near the throat but becomes more conical as the distance from the throat increases. Similarly, if the ellipses are rotated about an axis passing between the two focal points, they become oblate spheroids (flying saucer shaped) and we have an oblate–spheroidal coordinate system. Again, any one of the rotated hyperbolas can be chosen as a horn profile [3]. In this case, however, the profile looks hyperbolic near the throat but becomes more conical as the distance from the throat increases. However, spheroidal wave functions [4,20], unlike Bessel and Legendre functions, are not frequency independent. The fact that a whole series of harmonics must be calculated at each frequency step somewhat complicates the analysis. Ellipsoidal coordinates lead to similar horn profiles but with cross sections that are not circular.

Some other three-dimensional coordinate systems are simply two-dimensional systems translated through parallel planes. For example, elliptical–cylindrical coordinates are formed by translating the ellipses and hyperbolas of the spheroidal system. From this we can form horn profiles having two straight parallel walls and two curved walls that are hyperbolas. Again, we have the problem that the resulting Mathieu functions [4] are not frequency independent.

A rigorous treatment of a horn profile would involve solving the wave equation in three dimensions with the correct boundary conditions at the throat, walls, and mouth, but the analysis would be somewhat complicated. It is much simpler if we can reduce the wave equation down to one dimension by assuming that pressure variations over the cross section of the horn are minimal. In practice, the errors produced by such an assumption are fairly small. We have already introduced Webster's equation [21], Eq. (2.27), which is one-dimensional, but allows for a number of different functions S(x) to describe the variation of cross-sectional area S with distance x along its length. However, this equation assumes that the wave front does not change shape as it progresses along the length of the horn; otherwise it is not truly one-dimensional. In the case of a parabolic horn (with two parallel walls and two nonparallel) or conical horn, this assumption is generally true. However, we shall also consider exponential and hyperbolic horns in which case the wave front starts off substantially planar near the throat and becomes more curved as it progresses along the length of the horn. As a result, the infinite horn exhibits an abrupt cutoff frequency below which no power is transmitted. However, for a finite horn, the errors produced by this one-dimensional assumption are not too bad. It should be noted that there is no orthogonal coordinate system for an exponential or hyperbolic horn that leads to a separable wave equation with an exact solution, but proposals have been made to improve Webster's one-dimensional theory that include recasting it [5], applying expansions [6] or correction factors [7,8], and smoothing the cutoff discontinuity with a complex wave number [9].

First we shall consider infinite horns, as these provide the simplest solutions for the throat impedance and hence radiated power under idealized conditions. If the horn is a number of wavelengths long and if the mouth circumference is larger than the wavelength, we may call it “infinite” in length. This simplification leads to equations that are easy to understand and are generally useful in design. Then we shall develop 2-port transmission matrices for finite horns, which can be used as part of an overall loudspeaker system design. Our analysis will be limited to parabolic, conical, exponential, and hyperbolic horns.

For a horn to be a satisfactory transformer, its cross-sectional area near the throat end should increase gradually with axial distance x. If it does, the transformation ratio remains reasonably constant with frequency over a wide range. Exponential and hyperbolic horns are closer to this ideal, but the more gradual cutoff of a conical horn makes it easier to integrate into a loudspeaker system when used as a high-frequency unit or tweeter. The parabolic horn is often used in reverse as a transmission line because it is the easiest to construct.

We already mentioned that the directivity is largely defined by the flare angle at the mouth. This is certainly the case for parabolic and conical horns when the wavelength is smaller than the diameter of the mouth. The mouth of a conical horn behaves somewhat like a spherical cap in a sphere because it produces spherical waves that are largely confined within the angle of the apex of the cone at high frequencies. A conical horn may not necessarily have a circular cross-section though. A rectangular cross-section enables different angles of dispersion in the horizontal and vertical planes. It also produces a smoother on-axis response. Pulsating spherical and rectangular caps in spheres are discussed in Example 9.4 and covered in detail in Section 12.7. The high-frequency directivity factor is