To look at a different kind of field, we can use the same room, but replace the ice and stove with a wine glass and the proverbial opera soprano. This time the walls do not need to be insulating, but rather stiff, so that no sound created in the room gets outside it, and vice versa. For this system, the action-at-a-distance description is that the soprano makes the glass ring by singing the virtuoso note, until it eventually shatters. What is the corresponding field description?
The wine glass rings because its walls move. And we know that
things move when forces are placed on them. In this case, the
force is not from a point, like a fingertip, but a distributed
force all along the wall of the glass from the air in which it
is immersed. This force is related to the familiar property
called pressure. We recognize pressure from being under water.
The water is smooth and continuous, and at every place it
touches us, it presses uniformly. The amount of force
delivered, per chosen unit of area, is what we define as the
pressure.
It need not be the same at all places, which is
clear when we dive deeper and find that the pressure increases.
Whether it is obvious or not, air in Earth's atmosphere
also has pressure, due to the weight of all the rest of the
atmosphere above it. Like its temperature,
its pressure results from the motions of the molecules that move
and collide violently with one another and with any surface in
their way. The distributed force delivered by many such little
impacts is what we feel as pressure, and the total force
delivered is proportional to the area exposed, precisely because
the number of collisions we experience is proportional to that
area. As with any other force, pressure on a surface can
deliver momentum to what is pressed, but the pressure is
a property of the volume of fluid itself, and is not a
momentum. In fact, unless an object is placed there in the
first place, with different pressures on either side, and
allowed to move, no net momentum is being delivered, but the
pressure is still a property of the fluid.
We expect that pressure has something to do with the response of the wine glass, but the relation is less direct than the one for temperature. This is because the motion of air and things immersed in air, involving pressure, is much like the motion in the spring-mass system. We can see this as follows. If we put an object (like the wall of the glass) in the air, and there are different pressures on either side of the wall, it should feel some net force, and move in response. By the same token, if the pressures are the same on both sides, there should be no net force. (This is why children are often surprised to find that air has the same property of pressure as they know from being under water. Body cavities, like lungs and ears, start with a certain volume of air at the surface, and as the pressure increases, the air compresses and the body cavities have to deform. Thus they feel the pressure underwater. In air, though, the lungs, ears, etc., are freely refilled with the same air that is outside the body, and no deformation attends the fact that the pressure is nonetheless quite high.)
We could consider, though, not the wall of the glass, with air
on either side, but some small volume of air, with air on
either side. The same way a difference in pressure makes the
wine glass wall experience an overall force, it should make the
intermediate volume of air feel such a force. This volume of
air, which we could contain in a tiny imaginary box if we liked,
has a certain mass, just like any other object, so when it feels
a force it should accellerate according to . That turns
out to be exactly the correct law, but it relates differences in
pressure, not to a resulting pressure, but to an
altogether different field, motion, which we had not
considered.
It is convenient, because we have already,
to choose to represent the motion field by supposing that we can
start with completely still air, and measuring how much any
volume of air is displaced from its starting position when we
introduce different pressures at different places. (This is not
always the best way to describe fluids, but for the case we are
considering here, it will work, and it requires no new
notation.) The little intermediate volume of air continues to
be forced until it has moved out of the way and allowed the
pressures on either side to equalize. Implicit in this is yet
another observable fact about air; pressures result from packing
air into confined volumes. The more air we stuff into a bicycle
tire, the higher the pressure in that tire becomes, because the
stiff walls of the tire will not move out of the way to let the
air expand. This is why, when differences in pressure force an
object, and the object moves, it can allow the air that pressed
on either side to expand or be compressed until the pressures
are equalized. Thus, differences in pressure act like little
opposing, compressed springs pushing on either side of a volume
of air between them. This is shown in
fig. 7.3.
Like any other mass, though, the volume of air, once set in
motion by difference in pressure, can overshoot the position
where the pressures equalize, and compress the air in front of
its motion too much. At this point there develops an
excess of pressure in front, which both pushes the air in front
further out of the way, and pushes the over-shot air back in the
direction whence it came. This process of overshoot and
back-pressure is shown in fig. 7.3.
All this seems very complicated, but in fact it can be reduced to a cycle of three simple rules, once we divide a large roomful of air into very small volumes, and much as we did with temperature, talk about the behavior of the properties of each volume in response to what the volumes around it are doing. The rules are these.
Practice plugging into the laws, and seeing what kinds of behavior they predict, is given in the exercises.
We will find that, while there is a strong similarity of
pressure and motion in air to the spring-mass system, the common
characteristics are expressed differently. While the
spring-mass system can oscillate, the pressure-motion system of
air can be excited in waves.
A wave is a particular
pattern that we can recognize when we map out a field. Whether
we create waves or not, the pressure field in a roomful of air
exists, can be measured, and will give us some number at every
point. However, if we start with some smooth and constant
distribution of pressures, by shaking the air at some small
places, we can create a rippled pattern of deviations in
the pressure from the original constant value. These deviations
then travel out from where we excite them like ripples on a
pond. These are the waves.
Indeed, ripples on a pond are just another kind of wave. In
that case, the relevant field is the height of the surface
of the water. It doesn't matter from where we measure this
height. When the water is still, the height of every point on
its surface is the same. When we drop in a pebble, a rippled
pattern of deviations from that original height is created, and
spreads out from where the pebble excited them. The important
distinctions to note are these. Neither the physical substance,
the water, nor its surface, happens to be the field that we are
describing. The field is just a collection of measureable
heights, one for each point on the surface. Also, the field,
the thing we measure, is not the wave. The wave is a particular
recognizable pattern that the values of the field can take.
This will be important later, when we wish to create many waves,
as by throwing many pebbles into a pond. There is still only
one field, because the surface of the water still has only one
height at any given place. Yet there can recognizably be many
waves, which exist as patterns in that height field. We will be
concerned with how those wave interact and relate to each other.
Fig. 7.3, to illustrate their similarity, shows
the pattern of heights in a water wave and the pattern of pressures in
a pressure wave. Of course, the waves of pressure and motion that we
have been describing in air are called sound.
We note
two important things about sound, before going on to look at
other examples.