DELTAE deals with one-dimensional strings of acoustic and
thermoacoustic elements (see Chapter IV for branches), so the one-dimensional
wave equation is of the greatest importance. We always assume a time
dependence of 
, so the wave equation is a second-order
differential equation for the complex pressure amplitude p(x)
(III.1)
More complexity can be added, as needed, for given geometries. For example, in a lossy duct, the wave equation is
where A is the cross-sectional area of the duct, 
is its perimeter, and
is a correction for thermal properties of the duct wall that is
usually negligible. In a shallow lossy horn, where A and themselves
depend on
x, the wave equation is
In a stack, we use Rott's wave equation:
In DELTAE, the computation uses the wave equation that is appropriate for each segment. Within each segment, wave propagation is controlled by local parameters such as area and perimeter. Although DELTAE uses analytic solutions to the wave equation for some of the simplest segment types, it often must integrate the wave equation numerically, so it is generally correct to imagine DELTAE beginning at the BEGIN segment and numerically integrating the wave equation through each segment, in turn, to the HARDEnd or SOFTEnd, using local parameters, such as area and perimeter, as it goes.
It is sometimes easier to think of the second-order wave equation as two coupled first-order equations in pressure p and volume velocity U:
From this point of view it is easier to understand DELTAE's use of continuity of p and U to pass from the end of one segment to the beginning of the next.
Either way, however, it is clear that the solution p(x), U(x)is only determined uniquely if four real boundary conditions are imposed because the governing equation is second order in a complex variable. This is true whether considering a single segment or a one-dimensional string of segments with each joined to its neighbor(s) by continuity of p and U. If all four boundary conditions are given at one end of the apparatus (i.e., if we know the complex p and complex U at the BEGIN segment) then the integration is utterly straightforward. But usually some of the boundary conditions are given elsewhere. For example, in the plane-wave resonator in the previous Chapter, the boundary conditions were U = (0.01, 0) m/s at the BEGIN segment, and U = (0,0) at the HARDEnd. In such conditions DELTAE uses a shooting method, by guessing any unknowns among the four numbers defining p and U at the BEGIN segment, integrating to the HARDEnd, comparing the results with the boundary conditions imposed at the HARDEnd, and adjusting its guesses until it comes out right.
The boundary conditions imposed at the HARDEnd are in DELTAE's TARGET vector. The unknown conditions at the BEGINning, which DELTAE is supposed to find, are in DELTAE's GUESS vector. The number of elements in the TARGET vector must equal the number of elements in the GUESS vector; otherwise the system is over- or under-determined.
One of DELTAE's most powerful features is that the elements of the GUESS vector are not limited to the conventional choices consisting of real and imaginary parts of p and U at the BEGINning. Any variables that have an effect on the TARGET vector variables can be used. This enables DELTAE to calculate resonance frequencies, geometrical dimensions, temperatures, or even concentration in binary gas mixtures in order to satisfy given boundary conditions.
To add thermoacoustic computation ability to this linear acoustic picture, only one more equation is required, i.e., that for the temperature Tm(x). As for p(x) and U(x), the equation for Tm(x) depends on the type of segment, and continuity of Tm(x) is imposed at the junctions between segments. Most segments, such as isothermal ducts and cones, obey simply dT(x)/dx = 0. Stacks have a more complicated, but still only first-order, differential equation for Tm(x):
So, for thermoacoustic calculations, DELTAE integrates from BEGINning to HARDEnd, with respect to five real variables: real Tm(x), complex p(x), and complex U(x). It uses the appropriate wave equation and temperature equation for each segment. Within each segment, wave propagation is controlled by local parameters, such as area and perimeter, and by global parameters, such as frequency and mean pressure. Spatial evolution of temperature profile is also controlled by such local parameters, which include enthalpy flow. (Enthalpy flow is a conceptually difficult parameter because it depends on the heat flows into adjacent heat exchangers and on work flowing along the apparatus. It is therefore like the frequency in a resonant system in that it is a parameter that controls wave propagation in a segment but whose value is determined by geometry elsewhere.)
