Linear Superposition and Interference.
1. When two waves add together to produce a disturbance that is larger
than that of either individual wave, we call this constructive interference.
2. When two waves add together to produce a disturbance that is smaller
than either individual wave, we call this destructive interference.
3. Waves traveling in the same direction can also exhibit constructive
interference. When crest corresponds to crest and trough to trough the
two waves are said to be in phase and interfere constructively.
4. If crests match up with troughs, the waves are said to be 180° out of
phase and destructive interference results.
5. When two sources of sound produce the same frequency and vibrate in
phase, constructive interference occurs if the observer's position is an
integral number of wavelengths farther from one source than the other.
If the difference in path lengths is a half integer number of wavelengths,
destructive interference occurs.
6. When a wave encounters an opening or an obstacle, the wave spreads
out in all directions after passing through the opening or by the edge of
the obstacle. This bending is called diffraction.
7. Bending occurs because each vibrating molecule in the opening acts
as a new source for a wave. Using the principle of linear superposition,
we can see that the waves from all of these molecules add together to
produce a resultant wave that spreads out in all directions.
8. Constructive and destructive interference occur at specific directions
of travel depending on the size of the opening and the wavelength of
the wave. The strongest constructive interference occurs directly in front
of the opening. Other areas of constructive and destructive interference
occur but become increasingly weaker as the direction moves away from
directly in front of the opening.
9. When a sound wave of wavelength λ passes through an opening, the
place where the intensity of the sound is a minimum relative to the
center of the opening is specified by the angle θ.
10. If the opening is a rectangular slit of width D, such as a doorway,
then the equation used to find θ is: sin θ = λ/D. This is called a single
slit minimum since only one opening is involved.
11. If the opening is round (like a loudspeaker) instead of rectangular,
the equation becomes: sin θ = 1.22λ/D.
12. If λ/D is small, the angle is small and the effect is called narrow
dispersion. If λ/D is relatively large, we get wide dispersion. Low
frequencies have long wavelengths and are heard over a large area
around a speaker. High frequencies have short wavelengths and often
are only heard directly in front of the speaker.
13. Beat frequencies occur when two sounds interfere first constructively
then destructively producing alternating loud and quiet sounds.
14. Two sound waves with almost the same frequency produce a
beat note, found by subtraction in absolute value, fB= |f1 - f2| .
For example, with 440 Hz and 438 Hz we would hear 2 beats per
second. This is how many stringed instruments are tuned. When
the beat frequency is zero, both the instrument and the tuner are
playing the same frequency.
15. An air column can resonate with a sound source increasing the
loudness of the source. A pipe closed on one end resonates at
odd multiples of λ/4 , while an open pipe resonates at all
multiples of λ/2 .
16. Standing wave patterns result from interference of waves that
have the same wavelength and are in a medium whose length is a
multiple of one half of that wavelength.
17. Places within the medium which show little or no movement
are called nodes. Places which show maximum movement are
18. Harmonics are the frequencies associated with the wavelengths
that can produce standing waves in a medium. The first harmonic
is called the fundamental tone and is the lowest pitched sound
that can be produced by a particular set of conditions.
19. Whole number multiples of this frequency are called overtones
and are labeled second harmonic, third harmonic, etc.
20. When musical instruments are played, it is possible to distinguish
one instrument from another because of the harmonic content that is
different for each instrument. Today that distinction is being reduced
by the use of synthesizers to produce waveforms that very closely
mimic any type of instrument.
21. Under certain conditions it is possible for all the vibrations in a light
beam to be confined to one plane at right angles to the beam. This
phenomenon is called Polarization. When this occurs, the beam is said
to be polarized. The French physicist E. L. Malus (1775-1812)
discovered this property of light.
22. When two pieces of polarizing material are used, one after the other,
the first is called the polarizer and the second, the analyzer. If the average
intensity of polarized light falling on the analyzer is So , the average
intensity of light leaving the analyzer, S, is given by Malus' Law, which is
S = Socos2(θ) , where θ is the angle between the transmission axes of the
polarizer and analyzer.
And still, we need these steps to solve any problem in Physics:
(i) read the problem and identify the given variables
(ii) determine what you are asked to solve for
(iii) find the correct equation to use
(iv) use Algebra, Trigonometry, and/or Calculus to isolate the unknown
(v) substitute-in the given information and simplify.