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of „# can be considered as the horizontal projectlon of a rofating uector. Suppose
there were a vector ¡ of length 4 rotating with time, so that its angle with the
horizontal axis is œ‡ + ởị. (WS shall leave out the œ# in a minute, and see that it
makes no diference.) Suppose that we take a snapshot at the tìme £ = 0, although,
in fact, the picture is rotating with angular velocity œ (Fig. 29-9). The projection
of Ai along the horizontal axis is precisely Ai cos (ð£ + ở). Now at £ =0 the
second wave could be represented by another vector, 4a, of length 4a and at
an angle ós, and also rotating. Phey are both rotating with the same angular
velocity œ, and therefore the relafiue positions of the two are fxed. The system
goes around like a rigid body. The horizontal projection oŸ Áa is 4a cos (0£ + da).
But we know from the theory of vectors that if we add the bwo vectors in the
ordinary way, by the parallelogram rule, and draw the resultant vector Án, the
#-component of the resultant is the sum of the #z-components of the other two
vectors. hat solves our problem. It is easy to check that this gives the correct
--- Trang 509 ---
result for the special case we treated above, where Ái = 4a = A. In this case,
we see from Fig. 29-9 that Áp lies midway between 4+ and 4a and makes an
angle 3(Óa — ới) with each. Therefore we see that Áp = 2Ácos 3(s — ới), a8
before. Also, as we see from the triangle, the phase of Ág, as it goes around, is
the average angle of Áq and 4s when the two amplitudes are equal. Clearly, we
can also solve for the case where the amplitudes are not equal, Just as easily. We
can call that the geometrical way oŸ solving the problem.
There is still another way of solving the problem, and that is the ønalfical
way. hat is, instead of having actually to draw a picture like Fig. 29-9, we
can write something down which says the same thing as the picture: instead of
drawing the vectors, we write a complez mxwmber to represent each of the vectors.
'The real parts of the complex numbers are the actual physical quantities. So in
our particular case the waves could be written in this way: Aieff†1) [the real
part of this is Ai cos (ø£ + ởi)| and Asef@f†22), Ñow we can add the two:
h = Aiei@etrói) + Aasei6et92) = (Aie2t + Aac192)c«t (29.13)
Ñ= Aic? + Aac!?2 = Ancl6n, (29.14)
'This solves the problem that we wanted to solve, because it represents the result
as a complex number of magnitude Áp and phase ón.
To see how this method works, let us ñnd the amplitude An which is the
“length” of f. To get the “length” of a complex quantity, we always multiply the
quantity by its complex conjugate, which gives the length squared. he complex
conjugate is the same expression, but with the sign of the 7's reversed. 'Phus we
A? = (Aic'? + Aac??2)(Aie"??! + Aae”12), (29.15)
In multiplying this out, we get 4ƒ + 443 (here the es caneel), and for the cross
terms we have
Ai4Aa(cft®i=4) + cit02~91)),
e9 + e~?? = cosØ + isỉn Ø + cos Ø — ?sỉn 6.
That is to say, e'? + e~? = 2cosØ. Qur fnal result is therefore
4a = 4? + A2 + 2AI4a COS (Óa — Ị). (29.16)
--- Trang 510 ---
As we seo, this agrees with the length of Áp in Eig. 29-9, using the rules of
trigonometry.
Thus the sum of the two efects has the intensity 4? we would get with one
of them alone, plus the intensity 43 we would get with the other one alone,
plus a correction. 'Phis correction we call the mterƒference effect. It is really
only the diference bebween what we get simply by adding the intensities, and
what actually happens. We call it interference whether it is positive or negative.
(Interference in ordinary language usually suggests opposition or hindranee, but
in physics we often do not use language the way it was originally designedl) TỶ the
Interference term is positive, we call that case construcfzue interference, horrible
though it may sound to anybody other than a physicistl The opposite case is
called des‡ructzue interference.
Now let us see how to apply our general formula (29.16) for the case of Ewo
oscillators to the special situations which we have discussed qualitatively. To
apply this general formula, it is only necessary to fnd what phase diference,
Ó1 — đa, ©exists between the signals arriving at a given point. (It depends only on
the phase difference, of course, and not on the phase itself.) So let us consider
the case where the two oscillators, of equal amplitude, are separated by some
distance đ and have an intrinsic relative phase œ. (When one is at phase zero, the
phase of the other is œ.) Then we ask what the intensity will be in some azimuth
direction Ø from the E—W line. [Note that this is mof the same Ø as appears
in (29.1). We are torn between using an unconventional symbol like lý, or the
conventional symbol Ø (Fig. 29-10).| The phase relationship is found by noting
that the diference in distance from ? to the two oscillators is đsin Ø, so that the
phase diference contribution from this is the number of wavelengths in đsin 6,
multiplied by 2z. (Those who are more sophisticated might want to multiply the
wave number k, which is the rate of change of phase with distance, by đsin;
Aell0tta) To Point P
AeetZ đsin80
Fig. 29-10. 'Iwo oscillators of equal amplitude, with a phase differ-
ence œ between them.
--- Trang 511 ---
1b is exactly the same.) The phase diference due to the distance difference is
thus 2zdsin Ø/^À, but, due to the timing of the oscillators, there is an additional
phase œ. So the phase diference at arrival would be
Óa — Ôi = œ+ 2mdsin 0/À. (29.17)
'This takes care of all the cases. 'Thus all we have to do is substitute this expression
into (29.16) for the case 4 = 4a, and we can calculate all the various results for
two antennas of equal intensity.
Now let us see what happens in our various cases. The reason we know, for
example, that the intensity is 2 at 30° in Eig. 29-5 is the following: the two
oscillators are ¿À apart, so at 30°, dsin Ø = À/4. Thus ó¿ — ởị = 2mÀ/4ÀA = m/2,
and so the interference term is zero. (We are adding two vectors at 909.) The
result is the hypotenuse of a 45° right-angle triangle, which is v⁄2 times the unit
amplitude; squaring it, we get ©wice the intensity of one oscillator alone. All the
other cases can be worked out in this same way.
--- Trang 512 ---
})rffr-(rcff©ore
30-1 The resultant amplitude due to ?øw equal oscillators
'This chapter is a direct continuation of the previous one, although the name
has been changed om /n#erference to Diffraction. No one has ever been able to
defñne the diference between interference and difraction satisfactorily. It is just a
question of usage, and there is no specife, important physical diference between
them. The best we can do, roughly speaking, is to say that when there are only
a Ífew sources, say ©wo, interfering, then the result is usually called interference,