[LAU] Where do the 60 degrees for stereo come from?

[Fons Adriaensen]

On Fri, Jun 17, 2011 at 10:57:57AM +0200, Philipp Überbacher wrote:
 
> After a little off-list discussion with Gabriel and refreshing my basic
> trigonometry a bit I understand your formula, but I still have no idea
> what it tells us. The main problem might be that I have no idea how
> velocity vectors relate to sound and what the decreasing magnitude tells
> us.

It's not easy to explain without some wave physics and
maths. But I will try :-) Note that I'm simplifying 
things a bit, and leaving out a lot of 'ifs' and 'whens'
- some of the things presented below are true only on
some conditions (which you can assume to be satisfied).

Sound consists of variations of air pressure that 
propagate as waves. In each point (x,y,z) in space 
we have a pressure that is a function of time. Written
as a function, we have P(x,y,z,t), which is called
the _pressure field_.

To generate those pressure variations some air must
move. In each point (x,y,z) the small volume of air
surrounding it has some velocity which is also a
function of time: V(x,y,z,t), _the velocity field_.

While P() is a single value for any given (x,y,z,t),
V() is a vector: it has not only a magnitude but also
a direction. In 3-D space, a vector can be seen as a
combination of 3 independent values, one for each of
the three cartesian coordinates.

One way to look at sound waves is to see them as the
result of the interaction between P and V: they sort
of generate each other, which is what makes the wave
propagate in space.

For a real single sound source the direction of the
vector V(t) is that towards the source, and P(t) and
V(t) in any given point are closely related. They are
of course measured in different units (Pascal, and
meters/second resp.), one is a scalar and the other
a vector, but they are proportional. So given P(t),
we know the magnitude of V(t) - they are the same
signal. An omni mic gives a signal proportional to
P(t), while a figure-of-eight mic gives a signal
proportional to the projection of V(t) on its axis.
For a single source they will produce the same signal
(if you point the bidirection mic to the source).

This is no longer true if we have the same signal
reproduced by two sources, e.g. two stereo speakers
driven by the same signal to generate a virtual
source at the center. The P() will add up, but the
V() add as vectors, so the sum will be shorter than
the sum of the magnitudes, by the cosine factor
mentioned before. So we do not longer have the
fixed relation between P and the magnitude of V.

At low frequencies (where the wavelenght is much
larger than the size of a human head), all the
information we have to determine the direction
of the source is the phase difference between
the signals at the two ears. This is not just a
single value, we can (and do) move our heads and
'explore' this phase difference in function of
those movements.

Now some wave physics and maths will show that
this phase difference depends only on the relative
magnitudes of P() and V() that would exist at the
point halfway between the ears if our head were not
there, and on the direction of our head w.r.t. to
that of the vector V() (and of course on frequency).
What we perceive as the direction of the source is
the direction of V(). But if the magnitudes of P()
and V() don't have the right ratio, the phase
difference will not be as expected by our brain,
and this will make the virtual source less stable
and convincing.

As said this is valid only for low frequencies.
At mid and high frequencies other mechanisms
take over, but these also can be analysed in
terms of the ratio between a scalar sum and
the magnitude of a vector sum, and lead to 
similar conclusions.


HTH,

-- 
FA