sound waves strike the ear, an object may emit sound waves at a
constant frequency, yet may produce different pitches in ears
differently situated. Such a case is not usual, but an example of it
will serve a useful purpose in fixing certain facts as to pitch.
Conceive two railroad trains to pass each other, running in opposite
directions, the engine bells of both trains ringing. Passengers on
each train will hear the bell of the other, first as a _rising_ pitch,
then as a _falling_ one. Passengers on each train will hear the bell
of their own train at a _constant_ pitch.

The difference in the observations in such a case is due to relative
positions between the ear and the source of the sound. As to the bell
of their own train, the passengers are a fixed distance from it,
whether the train moves or stands; as to the bell of the other train,
the passengers first rapidly approach it, then pass it, then recede
from it. The distances at which it is heard vary as the secants of a
circle, the radius in this case being a length which is the closest
approach of the ear to the bell.

If the bell have a constant intrinsic fundamental pitch of 200 waves
per second (a wave-length of about 5.5 feet), it first will be heard
at a pitch of about 200 waves per second. But this pitch rises
rapidly, as if the bell were changing its own pitch, which bells do
not do. The rising pitch is heard because the ear is rushing down the
wave-train, every instant nearer to the source. At a speed of 45 miles
an hour, the pitch rises rapidly, about 12 vibrations per second. If
the _rate of approach_ between the ear and the bell were constant, the
pitch of the bell would be heard at 212 waves per second. But suddenly
the ear passes the bell, hears the pitch stop rising and begin to
fall; and the tone drops 12 waves per second as it had risen. Such a