First, sound (the wave) is the field counterpart to oscillations of a mass on a spring. Where the oscillations describe the motion of a single object in one direction, the wave describes the continuous, smoothly connected motions of infinitely many individual quantities, such as pressures which could be measured at each place in the room. The two notions are similar, though, because neither exists unless there is both a `spring property', either the spring itself or the pressure, and an `inertia property', whether of an object or a little volume of air. We will see this again, when we discuss waves in the Electromagnetic Field. In that system, the electric and magnetic aspects of the field act toward one another much like a `spring property' and an `inertia property', though such a mechanical naming is neither particularly useful nor particularly appropriate. The important point is that their equations, like the equations for pressure and motion in air, relate the evolution of one side to the configuration of the other side. For air, the motion changes, according to what the pressure is around it. Then conversely, the pressure changes (because of compression or expansion of the small volumes) because of what the motion is around it. The interplay of these is what makes waves possible.

Next, we note the following. Each little molecule of air has some very fast motion, and pressure arises as a property of the air because the fast molecules can bounce off things and impart momentum to them. However, in a room full of still air, no net momentum is imparted to the glass, nor is any net energy, though the pressure field still exists and can take very high values. The glass does not spontaineously start to ring because `there is pressure', nor does it shatter. It is the waves of pressure and motion, the sound, that impart momentum to the glass walls, and energy to the whole glass. Clearly, this energy must come from the singer. Yet, like temperature and the heat from the stove, the waves that transport it to the glass are not momentum-waves or energy-waves. They are waves in the pressure and the position of the little volumes of air, measureable in their own right and distinct from what they transport to objects affected by them.

The other aspect of sound, which we will mention here and develop much more later, can be seen more mundanely in ripples on a pond, which share that property. If we create two ripples, with two pebbles, the ripples can travel separately, but when two ripples are affecting the height of the water at a single place, the height-difference of the water at that place is the sum of the height-difference that each ripple would have produced alone. This is shown in fig. 7.3,

and is so important it will be shown again in the first chapter of Part III. This summability of waves is called superposition.

Not all fields have superposition, just as not all of them have dynamics that create waves. The temperature field, with its averaging rule, does not have a property of superposition. For instance, we could consider two different configurations of the room with the ice and stove, and the associated stable temperature fields produced in each case. If we were to do experiments in a third room, which contained both pieces of ice and both stoves, placed as they were in the preceding cases, the resulting temperature value in this third room would not simply be the sum of the previously measured respective temperatures.

In contrast, we could do a similar experiment with two singers in a room, and we would find that we could still hear each singer's song just as it was before, though the other singer's would be present as well. Like ripples on a pond, sound waves add. We will look at one historically famous example of the importance of superposition, and treat its ramifications in much greater detail prior to introducing the quantum theory.


Thu Aug 31 12:01:42 CDT 1995