One of the pioneers in spatial hearing research was John Strutt, who is better known as Lord Rayleigh. About 100 years ago, he developed his so-called Duplex Theory. According to this theory, there are two primary cues for azimuth:
Interaural Time Difference (ITD)
Interaural Level Difference (ILD)
Lord Rayleigh had a simple explanation for the ITD ^{4}^{4}Lord Rayleigh. https://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh.. Sound travels at a speed c of about 343 m/s. Consider a sound wave from a distant source that strikes a spherical head of radius a from a direction specified by the azimuth angle. Clearly, the sound arrives at the right ear before the left, since it has to travel the extra distance to reach the left ear. Dividing that by the speed of sound, we obtain the following simple (and surprisingly accurate) formula for the interaural time difference:
$$ITD=\frac{a}{c}(\alpha +sin\theta ),-{90}^{\circ}\le \theta \ge +{90}^{\circ}$$ |
The ITD is zero when the source is directly ahead
The ITD has a maximum of $(a/c)(\pi /2+1)$ when the source is off to one side
This represents a difference of arrival time of about 0.7 ms for a typical size human head, and is easily perceived
Lord Rayleigh also observed that the incident sound waves are diffracted by the head. He actually solved the wave equation to show how a plane wave is diffracted by a rigid sphere. His solution showed that in addition to the time difference there was also a significant difference between the signal levels at the two ears — the ILD.
As you might expect, the ILD is highly frequency dependent. At low frequencies, where the wavelength of the sound is long relative to the head diameter, there is hardly any difference in sound pressure at the two ears. However, at high frequencies, where the wavelength is short, there may well be a 20-dB or greater difference. This is called the head-shadow effect, where the far ear is in the sound shadow of the head.
The Duplex Theory asserts that the ILD and the ITD are complementary. At low frequencies (below about 1.5 kHz), there is little ILD information, but the ITD shifts the waveform a fraction of a cycle, which is easily detected. At high frequencies (above about 1.5 kHz), there is ambiguity in the ITD, since there are several cycles of shift, but the ILD resolves this directional ambiguity. Rayleigh’s Duplex Theory says that the ILD and ITD taken together provide localization information throughout the audible frequency range.