Lord Rayleigh had a simple explanation for the ITD. 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:
Thus, the ITD is zero when the source is directly
ahead, and is a maximum of 
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.
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