Principle of Linear Superposition and Interference Phenomena Chapter 17

Combining waves Consider two wave pulses moving toward each other with two upward pulses. When the pulses merge, the wave assumes a shape that is the sum of the shapes of the individual pulses. Wave pulses pass through each other and continue moving.

Combining waves Consider two wave moving toward each other, but with one upward pulse and one downward pulse. When the pulses merge, the wave shape is the sum of the shapes of the individual pulses. In this case, the pulses momentarily cancel as they pass each other. Wave pulses pass through each other and continue moving. Wave interference demonstration on YouTube

Principle of linear superposition The disturbance created by two waves is the sum of the disturbances of the two waves.

Constructive wave interference In-phase waves reinforce each other. In phase Peaks line up with peaks, valleys line up with valleys.

Destructive wave interference Out of phase waves cancel each other. The waves cancel each other out. Out of phase Peaks line up with valleys, valleys line up with peaks.

Destructive wave interference application Noise canceling headphones use destructive interference to cancel out the noise.

Coherent wave sources Coherent waves maintain the same phase pattern.

ΔL = |L1 - L2| Path length difference ΔL Path length difference ΔL is the absolute value of the difference in the distances from each source to a point being considered. ΔL = |L1 - L2| L1 L2 path 1 path 2

Path length difference Find the path length difference ΔL for points 1, 2, 3, 4, 5, and 6. Express the path length difference as a number of wavelengths. For example: ΔL = 3½ wavelengths 1 2 3 4 5 6

Peaks reinforce peaks and valleys reinforce valleys. Path length difference: constructive interference Constructive interference occurs when the difference is path length is zero or an integer number of wavelengths. ΔL = 0, 1λ, 2λ, 3λ, etc. Peaks reinforce peaks and valleys reinforce valleys.

Peaks and valleys cancel each other. Path length difference: destructive interference Destructive interference occurs when the difference is path length is a half integer number of wavelengths. ΔL = ½λ, 1½λ, 2½λ, 3½λ, etc. Peaks and valleys cancel each other.

Constructive and destructive interference Constructive interference ΔL = 0 3λ Destructive interference ΔL = ½λ 3½λ 3λ

Example 1 What Does a Listener Hear? Two in-phase loudspeakers A and B, are separated by 3.2 m. A listener is stationed at C, which is 2.4 m in front of speaker B. 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? Critical thinking step: Constructive interference would produce a loud sound. Destructive interference would produce no sound (sound cancelation). What is the path length difference for the waves from speakers A and B to the listener at C?

Example 1 What Does a Listener Hear? Calculate the path length difference. 4 m Calculate the wavelength. Therefore ΔL=1 λ Because the path length difference is equal to an integer number of wavelengths, the waves will constructively interference and the sound is loud.

Conceptual Example 2 Out-Of-Phase Speakers To make speakers operate, two wires must be connected between each speaker and the amplifier. The two speakers vibrate in phase when the connections are exactly the same way on each speaker. The speakers are out of phase with each other when one speaker is connected backwards.

Diffraction Diffraction is the bending of a wave when it interacts with the edges of an opening or an obstacle.

Diffraction Smaller openings have more diffraction. Larger openings have less diffraction. Maximum diffraction happens when the wavelength is the same size as the opening.

Diffraction Circular opening diffraction equation short wavelength high frequencies long wavelength low frequencies width D of the speaker opening Circular opening diffraction equation

Beat frequency tuning forks Two overlapping waves traveling in the same direction with slightly different frequencies create sound beats. Beat frequency demo 1 on YouTube Beat frequency demo 2 on YouTube

Beat frequency Beat frequency is equal to the difference between the two frequencies.

Waves are reflected when they reach a boundary. Reflected waves Waves are reflected when they reach a boundary. 1 3 4 2 forward motion backward motion Reflected waves are upside down (inverted). When the timing is right, the incoming wave and the reflected wave can form standing waves.

Transverse standing waves f1 2 f1 3 f1 fundamental 2nd harmonic 3rd harmonic 1st harmonic node antinode Harmonic frequencies are multiples of a fundamental frequency. No motion at the nodes. Maximum motion at the antinodes. Strings produce all harmonic frequencies. Ends of the string can't move so they are always nodes . Standing wave demo on YouTube

Transverse standing waves An integer number (n) of half-wavelengths must exactly fit between the fixed ends of the string. (The integer n is the number of loops.) ½ λ Fundamental vibration mode has just 1 half- wavelength. Strings can produce all the harmonic frequencies.

How long is the string? Guitar string vibrates at a fundamental frequency of 164.8 Hz. The string tension is 226 N. The string linear density is 0.00528 kg/m3. n = 1 for the fundamental frequency. Side calculation for the wave speed on the stretched string.

Longitudinal standing waves Longitudinal standing wave pattern on a spring. node: minimum vibration antinode:maximum vibration

Sound wave for a tube open at both ends Air molecules have maximum motion at the tube's open ends so the open ends are antinodes. The distance from one antinode to the next antinode is ½ λ. The natural modes of vibration occur when an integer number (n) of half-wavelengths exactly fit between the open ends of the tube. All the harmonic frequencies can be produced.

Sound wave for a tube open only at one end Air molecules have maximum motion at the tube's open end and no motion at the closed end so the open end is an antinode and the closed end is a node. The distance from one antinode to the next node is ¼ λ. The natural modes of vibration occur when an odd number of quarter-wavelengths exactly fit between the open ends of the tube. Only the odd harmonic frequencies can be produced.

Example - A tube that is 2 m long open at both ends open end and closed end

The End