Ch17 Linear superposition and interference.docx

LINEAR SUPERPOSITION AND INTERFERENCE When two or more sound waves are present at the same place at the same time we experience linear superposition. The principle of linear superposition states that the resultant effect of two or more concurrent waves is equal to the sum of the disturbances created by each of the individual waves. In the diagram above, the two waves produce a resultant disturbance that is found by adding the two individual displacements from equilibrium. When two waves add together to produce a disturbance that is larger than that of either individual wave, we call this constructive interference.When two waves add together to produce a disturbance that is smaller than either individual wave, we call this destructive interference. Waves travelling 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. If crests match up with troughs, the waves are said to be 180° out of phase and destructive interference results. 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. The red dots show locations where constructive interference occurs. The white dots indicate where destructive interference occurs. Example Loudspeakers A and B are set up on the X axis some distance apart. Both speakers vibrate in phase at 68.6 hz and the speed of sound is 343 m/s. Point C is 1.0 meter from speaker A and the line connecting A with C makes a 60 degree angle with the X axis. Find the minimum distance between the speakers that will produce destructive interference at point C. Diffraction 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. 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. 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. The angle ? in the drawing indicates the direction of the first minimum intensity due to destructive interference. It is measured relative to the direction straight through the opening. The equation used to find ? is: sin ? = ?/D where ? is the wavelength and D is the width of the opening. This is called a single slit minimum since only one opening is involved. If the opening in round(like a loudspeaker) instead of rectangular, the equation becones: sin ? = 1.22?/D If ?/D is small, the angle is small and the effect is called narrow dispersion. If ?/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. Example The entrance to a large room consists of two doors that cover one large opening. Each door is 0.700 m wide. If a sound with a frequency of 607 hz passes through the doors when the speed of sound is 343 m/s, find the diffraction angle with (a) one door open and (b) both doors open. Beat frequencies occur when two sounds interfere first constructively then destructively producing alternating loud and quiet sounds. The rate at which the beat frequency occurs is equal to the frequency difference between the two sounds. For 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. 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. Places within the medium which show little or no movement are called nodes. Places which show maximum movement are called antinodes. 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. Whole number multiples of this frequency are called overtones and are labeled second harmonic, third harmonic, etc. 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.