III Modeling Techniques based on Geometrical Acoustics

III.1 Introduction to Geometrical Acoustics
The main principle of geometrical acoustics (GA) is that sound is supposed to act as rays [1]. In the most basic form of GA, the sound is modeled as rays propagating in straight lines in the air, and whenever a ray hits a wall, it gets a new direction. This means, that the wave nature of sound is neglected, and all the wave-related phenomena are missing in the GA-based techniques. Especially, this means that there is no diffraction in the GA methods as opposed to the wave-based methods where all those phenomena are inherently included.

The limitations of geometrical acoustics are remarkable. For example, if we have free field conditions in which there is a small barrier between the source and the receiver that blocks the line-of-sight between those two points, the GA methods will predict that no sound will reach the receiver. However, this is not correct as the receiver will get most of the sound due to edge diffraction, and only the high frequency content is blocked. A similar, and more practical example is the orchestra pit in an opera hall. Most of the audience won't see the orchestra, but they will still hear the direct sound from the pit due to edge diffraction, in addition to all the reflections, while the GA methods are not able to reproduce this as such.

Despite these shortcomings, geometrical acoustics is widely applied in practice. It is an asymptotically correct model at high frequencies. At higher frequencies, wavelengths of sound are shorter, and wave-phenomena play a smaller role. For this reason, the results of GA-based predictions have least error at high frequencies. The main advantage of GA over wave-based methods is the relatively low computational cost that does not depend on the frequency. It is good to remember that the computational load of wave-based methods grows steeply at higher frequencies and, thus, these two approaches can be combined into a hybrid where they can compensate each other. Wave-based methods are at their best at low-frequencies as their computational load is still tolerable whereas high frequencies can be used for simulation of the high end as they are most accurate in that region with a decent computational load.

There is a direct correspondence between geometrical acoustics and modeling of light propagation that has been studied extensively in the area of computer graphics to produce photo-realistic images. The main difference between these two is that in modeling of behavior of light the propagation speed can be assumed infinite meaning that the light will reach everywhere immediately. Thus, the result of such a simulation is a static image where is acoustic modeling, the propagation speed is finite, has a key role in simulation and perception, and the final result of a simulation is typically an impulse response, or time-energy response, that describes the sound propagation from the sound to the receiver as function of time.

In the following, the main room acoustic modeling techniques and their basis are described. The text is supported by interactive illustrations that aim to help understanding the main concepts. Textual descriptions of the methods are supported also by an algorithmic pseudo-code snippets that show step-by-step how each technique operates. The 'Room Acoustic Rendering Equation' presents a mathematical framework that can be used to represent all the methods discussed in this section. If a reader prefers to start from the framework, and only after that proceed to each algorithm, that is a valid approach as well, and such readers are encouraged to start from that section before moving to the image-source technique. The current order of sections is recommended for those readers who prefer to see the algorithms first and dive deeper into the base only after learning the algorithmic structure of each technique.