Click here to purchase the entire book in PDF format. Specular Reflections and Snell's Law The discussion in Section 3.2.1 assumes that the wave propagation is normal, or perpendicular, to the surface boundary. In most instances, however, the angle of incidence - an angle subtended by a normal to the boundary4.3 and the incident sound ray - is an oblique angle. If the reflective surface is large and flat relative to the wavelength of the reflected sound, there exists a simple relationship between the angle of incidence and the angle of reflection, subtended by the reflected ray of sound and the normal to the reflective surface. SnellÕs law describes this relationship as is shown in Figure 3.24 and Equation 3.31 [Isaacs, 1990]. Figure 3.24: Relationship between the angles of incidence and reflection in the case of a specular reflector. $$\sin(\vartheta_{i}) = \sin(\vartheta_{r})$$ (4.31) and therefore, in most cases: $$\vartheta_{i} = \vartheta_{r}$$ (4.32) This is exactly the same as the light that bounces off a mirror. The light hits the mirror and then is reflected off at an angle that is equal to the angle of incidence. As a result, the reflections looks like a light bulb that appears to be behind the mirror. There is one interesting thing to note here - the point on the mirror where the light is reflected is dependent on the locations of the light, the mirror and the viewer. If the viewer moves, then the location of the reflection does as well. If you don't believe me, go get a light and a mirror and see for yourself. Since this type of reflection is most commonly investigated as it applies to visual media and thus reflected light, it is usually considered only in the spatial domain as is shown in the above diagram. The study of specular reflections in acoustic environments also requires that we consider the response in the time domain as well. This is not an issue in visual media since the speed of light is effectively infinite in human perception. If the surface is a perfect specular reflector with an infinite impedance, then the reflected pressure wave is an exact copy of the incident pressure wave. As a result, the reflection is equivalent to a simple delay with an attenuation determined by the propagation distance of the reflection as is shown in Figure 3.25 and the corresponding impulse response in Figure 3.26. Figure 3.25: Drawing showing the relationship in space between the sound source, the microphone (or listening position), the incoming direct sound, and the reflected sound. Figure 3.26: Impulse response of direct sound and specular reflection. Note that the Time is referenced to the moment when the impulse is emitted by the sound source, hence the delay in the time of arrival of the initial direct sound.