The Principle of Linear Superposition and Interference Phenomena Chapter 17 Lesson 1 The Principle of Linear Superposition

THE PRINCIPLE OF LINEAR SUPERPOSITION 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.

Check Your Understanding 1 The drawing shows two pulses traveling toward each other at t = 0 s. Each pulse has a constant speed of 1 cm/s. When t = 2 s, what is the height of the resultant pulse at (a) x = 2 cm, (b) x = 4 cm, and (c) x = 6 cm? (a) 0 cm, (b) -2 cm, (c) +2 cm

Constructive and Destructive Interference of Sound Waves Reading content When two waves always meet condensation-to-condensation and rarefaction-to-rarefaction (or crest-to-crest and trough-to-trough), they are said to be exactly in phase and to exhibit constructive interference.

When two waves always meet condensation-to-rarefaction (or crest-to-trough), they are said to be exactly out of phase and to exhibit destructive interference. In either case, this means that the wave patterns do not shift relative to one another as time passes. Sources that produce waves in this fashion are called coherent sources.

Destructive interference is the basis of a useful technique for reducing the loudness of undesirable sounds.  ,   ,    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.

Interference effects can also be detected if the two speakers are fixed in position and the listener moves about the room.

Example 1. What Does a Listener Hear? Two in-phase loudspeakers, A and B, are separated by 3.20 m. A listener is stationed at point C, which is 2.40 m in front of speaker B. The triangle ABC is a right triangle. Both speakers are playing identical 214-Hz tones, and the speed of sound is 343 m/s. Does the listener hear a loud sound or no sound? The listener hears a loud sound.

Diffraction

The bending of a wave around an obstacle or the edges of an opening is called diffraction. All kinds of waves exhibit diffraction.

Example 3. Designing a Loudspeaker for Wide Dispersion A 1500-Hz sound and a 8500-Hz sound each emerges from a loudspeaker through a circular opening whose diameter is 0.30 m . Assuming that the speed of sound in air is 343 m/s, find the diffraction angle q  for each sound.

Beats

The number of times per second that the loudness rises and falls is the beat frequency and is the difference between the two sound frequencies.

A 10-Hz sound wave and a 12-Hz sound wave, when added together, produce a wave with a beat frequency of 2 Hz. The drawings show the pressure patterns (in blue) of the individual waves and the pressure pattern (in red) that results when the two overlap. The time interval shown is one second.

Check Your Understanding 2 A tuning fork of unknown frequency and a tuning fork of frequency of 384 Hz produce 6 beats in 2 seconds. When a small piece of putty is attached to the tuning fork of unknown frequency, the beat frequency decreases. What is the frequency of that tuning fork? 387 HZ