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, f_{B}= f_{1}  f_{2} . 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 called antinodes. 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 (17751812) 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 S_{o }, the average intensity of light leaving the analyzer, S, is given by Malus' Law, which is S = S_{o}cos^{2}(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) substitutein the given information and simplify. View Wave Slides. View Superposition Slides.

