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Connors, Martin, Reading Notes on Vibrations and Waves. Athabasca, AB: Athabasca University,
2010.
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PHYS 302: Unit 13 Reading Notes – Athabasca University
PHYS 302: Vibrations and Waves – Unit 13 Reading Notes
French, A. P. Vibrations and Waves. New York: W.W. Norton & Company, 1971.
 Chapter 2: The Superposition of Periodic Motions (pp. 27–29, up to “Combination of Two Vibrations at
Right Angles”).
 Chapter 8: Boundary Effects and Interference (pp. 284–298).
Chapter 2: The Superposition of Periodic Motions
Many Superposed Vibrations of the Same Frequency (p. 27)
This section corresponds to the material at the beginning of the video lecture that leads to Eq. 2-7. The steps shown
in the viewing notes will help you follow the text discussion, but Fig. 2-7 shows more added vectors than the video
lecture does, so it is helpful in visualizing the extension to N slits.
Chapter 8: Boundary Effects and Interference
Multiple-Slit Interference (Diffraction Grating) (p. 284)
On p. 285, ignore the use of time delays and concentrate on phase. The equation at the bottom of the page shows the
displacement at P due to the various phase-delayed contributions from each slit. With respect to the original phase
ωt-φ1, these phase delays can be represented as angles, and the vectors can be added using Eq. 2-7. Fig. 8-19 shows
the vector sums. These are non-zero sums only in the case of collinear vectors (top) and where the white arrows are
shown. In each case, there is 10 vectors, but in many cases they overlap as they make more than one full circuit. As
page 287 describes, the collinear vectors take place at all angles where the two-slit system of the same separation
would have had maxima. These are called principal maxima. Most combinations sum to 0; those that do not define
subsidiary maxima.
Diffraction by a Single Slit (p. 288)
By Huygens’ principle, it is clear that all parts of a single slit act as sources for waves, much as individual narrow
slits are considered sources. Parts of a single slit will themselves be subject to phase differences, so they will
interfere with each other. As mentioned at the top of page 290, the phase difference (referred to as φ below)

between the edges of a slit of width b is 2 b sin
 , where the bsinθ is the path difference, and the ratio of the path
difference to the wavelength λ is the amount of a 2π phase advance arising from this difference. The procedure
followed is similar to that for multiple slits, but the width limit Δs of the imagined parts of the slit approaches 0,
while their number N approaches infinity. After this, the vector diagram approaches a smooth, circular arc.
Obtaining the length of the chord (p. 291) can be done by imagining two identical triangles making up the one that
spans the arc (Fig. 8-23). These triangles would each have angle φ/2 with opposite side A/2 and hypotenuse R. In
one of these it is clear that the “SOH” (sine-opposite-hypotenuse rule) gives sin
given in the text. Since 
 2
b sin 

, clearly

2

b sin 


2

A/ 2
R
, or A  2 R sin

2
as
. This is reminiscent of the importance attached to half the
slit width in the video lecture, where pairs half a slit-width apart always exist to match up with the same
interference. Since A0=Rφ (the small diagrams at the right in Fig. 8-23 show that this is not always “close” to A),
R
A0

and A  2 R sin

2
can be written as
A  2 A0 sin 2  A0
sin 2

, that is, Eq. 8-25. The notable feature of this
2
function, often called sinc, and identified here as a “Bessel function of order zero” (actually the Bessel function of
the first kind J0), is the recurrence of zeroes at intervals of π of the argument φ/2 (called α in Fig. 8-24). Since the
function is symmetrical, the first or central maximum is twice the overall width of the others. In Fig. 8-24(b), the
absolute value is plotted to show the resemblance to a diffraction grating. The text doesn’t mention that amplitudes
of electromagnetic waves add, and that the intensity (which we detect) is the square of the amplitude. Taking the
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PHYS 302: Unit 13 Reading Notes – Athabasca University
absolute value gives a curve that qualitatively resembles the intensity curve. As seen in the video lecture, lasers now
make it easy to see such a diffraction pattern.
On page 293, French establishes that diffraction is important if one’s distance from the slit is greater than b2/λ. Thus,
even a very wide slit shows diffraction if one is far enough away. For example, it is possible to detect the diffraction
from the edge of the moon as it passes in front of stars (this extreme example represents an edge, not a slit, but an
edge in fact gives a similar diffraction pattern).
Interference Patterns of Real Slit Systems (p. 294)
This section illustrates the various possible interference or diffraction patterns, most often with examples using
sound waves. Most of these patterns can be better illustrated using the coherent monochromatic light of a laser, as in
the video lecture. Fig. 8-27 gives an example of the diffraction pattern arising with sound, and introduces the useful
concept of displaying the pattern on a polar plot, a fairly natural thing to do with a function of angle. Polar plots are
often used to display the pattern of radiation from an antenna, which is also frequently determined by path
differences similar to those in diffraction. This section stresses that A2 is usually measured because it corresponds to
energy.
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