Concept Summary

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Topic | Discussion | Learning Aids
Principle of linear superposition | 17.1 THE PRINCIPLE OF LINEAR SUPERPOSITION The principle of linear superposition states that when two or more waves are present simultaneously at the same place, the resultant disturbance is the sum of the disturbances from the individual waves. |
Constructive and destructive interference | 17.2 CONSTRUCTIVE AND DESTRUCTIVE INTERFERENCE OF SOUND WAVES Constructive interference occurs at a point when two waves meet there crest-to-crest and trough-to-trough, thus reinforcing each other. Destructive interference occurs when the waves meet crest-to-trough and cancel each other. |
In-phase and out-of-phase waves | When waves meet crest-to-crest and trough-to-trough, they are exactly in phase. When they meet crest-to-trough, they are exactly out of phase. |
Conditions for constructive and destructive interference | For two wave sources vibrating in phase, a difference in path lengths that is zero or an integer number (1, 2, 3, …) of wavelengths leads to constructive interference; a difference in path lengths that is a half-integer number of wavelengths leads to destructive interference. |
Diffraction | 17.3 DIFFRACTION Diffraction is the bending of a wave around an obstacle or the edges of an opening. The angle through which the wave bends depends on the ratio of the wavelength of the wave to the width D of the opening; the greater the ratio , the greater the angle.
When a sound wave of wavelength passes through an opening, the first place where the intensity of the sound is a minimum relative to the center of the opening is specified by the angle . If the opening is a rectangular slit of width D, such as a doorway, the angle is
Single slit—first minimum |

| (17.1)

| If the opening is a circular opening of diameter D, such as that in a loudspeaker, the angle is |
Circular opening–first minimum |

| (17.2)

Beats Beat frequency | 17.4 BEATS Beats are the periodic variations in amplitude that arise from the linear superposition of two waves that have slightly different frequencies. When the waves are sound waves, the variations in amplitude cause the loudness to vary at the beat frequency, which is the difference between the frequencies of the waves. |

Use Self-Assessment Test 17.1 to evaluate your understanding of 17.1, 17.2, 17.3 and 17.4

Standing waves Nodes and antinodes | 17.5 TRANSVERSE STANDING WAVES A standing wave is the pattern of disturbance that results when oppositely traveling waves of the same frequency and amplitude pass through each other. A standing wave has places of minimum and maximum vibration called, respectively, nodes and antinodes. |
Natural frequencies Harmonics | Under resonance conditions, standing waves can be established only at certain natural frequencies. The frequencies in this series (f1, 2f1, 3f1, etc.) are called harmonics. The lowest frequency f1 is called the first harmonic, the next frequency 2f1 is the second harmonic, and so on.
For a string that is fixed at both ends and has a length L, the natural frequencies are
String fixed at both ends |

| (17.3)

where v is the speed of the wave on the string and n is a positive integer.
17.6 LONGITUDINAL STANDING WAVES For a gas in a cylindrical tube open at both ends, the natural frequencies of vibration are
Tube open at both ends |

| (17.4)

where v is the speed of sound in the gas and L is the length of the tube.
For a gas in a cylindrical tube open at only one end, the natural frequencies of vibration are
Tube open at only one end |

| (17.5)

Use Self-Assessment Test 17.2 to evaluate your understanding of 17.5 and 17.6.

Complex sound wave | 17.7 COMPLEX SOUND WAVES A complex sound wave consists of a mixture of a fundamental frequency and overtone frequencies. |

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