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Fig. 23-11 is the resonance curve for this one. 'Phis resonance curve turns out to
be very interesting. It is the resonance found in a certain reaction among strange
--- Trang 424 ---
particles, a reaction in which a K— and a proton interact. “The resonance 1s
detected by seeing how many of some kinds of particles come out, and depending
on what and how many come out, one gets diferent curves, but of the same
shape and with the peak at the same energy. We thus determine that there is a
resonance at a certain energy for the K— meson. That presumably means that
there is some kind oŸ a state, or condition, corresponding to this resonance, which
can be attained by putting together a K— and a proton. This is a new particle,
or resonance. Today we do not know whether to call a bump like this a “particle”
or simply a resonance. When there is a very shørp resonance, it corresponds to a
very đefinaiie energu, just as though there were a particle of that energy present
in nature. When the resonance gets wider, then we do not know whether to
say there is a particle which does not last very long, or simply a resonance in
the reaction probability. In the second chapter, this point is made about the
particles, but when the second chapter was written this resonance was not known,
so our chart should now have still another partiele in itl
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TT-drtSf©reÉs
24-1 The energy of an oscillator
Although this chapter is entitled “transients,” certain parts of i% are, in a
way, part of the last chapter on forced oscillation. One of the features of a forced
oscillation which we have not yet discussed is the energy in the oscillation. Let
us now consider that energy.
In a mechanical oscillator, how much kinetic energy is there? lt is proportional
to the square of the velocity. NÑow we come to an important point. Consider an
arbitrary quantity A, which may be the velocity or something else that we want
to discuss. When we write A = Âc 1t, a complex number, the true and honest
A, in the physical world, is only the real part; therefore if, for some reason, we
want to use the sguøre of A, i% is not right to square the complex number and
then take the real part, because the real part of the square of a complex number
1s not just the square of the real part, but also involves the Z#maginarw part. So
when we wish to fñnd the energy we have to get away from the complex notation
for a while to see what the inner workings are.
Now the true physical 4 is the real part of Agef«++^) that is, A = Ao cos (£+
A), where Â, the complex number, is written as Aoe?^. Now the square of this
real physical quantity is 4? = 4ä cos” („#+ A). The square of the quantity, then,
goes up and down from a maximum to zero, like the square of the cosine. 'Phe
square of the cosine has a maximum of 1 and a minimum of 0, and is average
value is 1/2.
In many circumstances we are not interested in the energy at any specifc
moment during the oscillation; for a large number of applications we merely
want the average of 42, the mmean of the square of A over a period of time large
compared with the period of oscillation. In those circumstances, the average
of the cosine squared may be used, so we have the following theorem: if A is
represented by a complex number, then the mean of .4” is equal to 3.43. NÑow 4ã
--- Trang 426 ---
1s the square of the magnitude of the complex Â. (This can be written in many
ways—some people like to write |A|?; others write, 44”, 4 times its complex
conjugate.) We shall use this theorem several tỉmes.
Now let us consider the energy in a forced oscillator. The equation for the
forced oscillator is
m d”z/dtÊ + m dz/dt + mua = F1). (24.1)
In our problem, of course, #'(£) is a cosine function of ¿. NÑow let us analyze the
situation: how much work is done by the outside force †'? "The work done by the
force per second, ¡.e., the power, is the force times the velocity. (W© know that
the diferemtial work in a tỉme đý is f'dz+, and the power is F'dz+/di.) Thus
da: d+z d2z d+z dz\
P=F-.= — g5 —— —|- 24.2
dt m7) (mm) s2 m)|+an ( n) (242)
But the ñrst two terms on the right can also be written as d/df[Sm(dz/dt)? +
smœf+2], as is immediately verifled by differentiating. That is to say, the term in
brackets is a pure derivative oŸ ©wo terms that are easy to understand——one is the
kinetic energy of motion, and the other is the potential energy oŸ the spring. Let
us call this quantity the sứored energu, that is, the energy stored in the oscillation.
uppose that we want the average power over many cycles when the oscillator is
beiïng forced and has been running for a long time. In the long run, the stored
energy does not change——its derivative gives zero average efect. In other words,
1 we average the power in the long run, đÌÏ the energu ulttmatelU ends up ín the
resistiue term +m(dz/df)?. There is some energy stored in the oscillation, but
that does not change with time, if we average over many cycles. Therefore the
mean power (P) is
(P) = (wém(dz/d£)). (24.3)
Using our method of writing complex numbers, and our theorem that (42) =
34ã, we may find this mean power. Thus if z = êc”“”, then dœ/dt = iuêâc”*!,
'Therefore, in these circumstances, the average power could be written as
(?)= 3m đg. (24.4)
In the notation for electrical circuits, đø/đ# is replaced by the current ï (T is
dq/dt, where q corresponds to +), and zw⁄ corresponds to the resistance Ÿ. Thus
--- Trang 427 ---
the rate of the energy loss—the power used up by the forcing function——is the
resistance in the circuit times the average square of the current:
(P) = RỊ?) = R- 313. (24.5)
This energy, of course, goes into heating the resistor; i% is sometimes called the
heating loss or the Joule heating.
Another interesting feature to discuss is how much energy is sfored. 'That is
not the same as the power, because although power was at first used to store up
some energy, after that the system keeps on absorbing power, insofar as there
are any heating (resistive) losses. At any moment there is a certain amount of
sbored energy, so we would like to calculate the mean stored energy (2) also. We
have already calculated what the average of (dz/đ#)2 is, so we find
(E) = ‡m((de/4)®) + ÿmeŠ(v?) 016)
= 3m(0Ÿ + 08) šzg.
Now, when an oscillator is very efficient, and IÝ œ is near œọ, so that |Ê{ is large,
the stored energy is very high—we can get a large stored energy from a relatively
smaill force. 'Phe force does a great deal of work in getting the oscillation goiïng,
but then to keep it steady, all it has to do is to fight the friction. The oscillator
can have a great deal of energy if the friction is very low, and even though it is
oscillating strongly, not much energy is being lost. The eficiency of an oscillator