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Chapter 16: Superposition and Standing Waves ¨  Key Terms

¤  Interference ¤ Nodes ¤ Antinodes ¤ Mode number ¤  Fundamental frequency and harmonics ¤  Phase ¤  Path length difference ¤ Constructive ¤ Destructive ¤  Beats

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Principle of Superposition

¨  When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements due to each individual wave.

¨  The superposition of two waves is often called interference.

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Standing Waves

¨  Waves in which individual points on the string oscillate up and down, but the wave itself does not travel.

¨  The superposition of two waves with equal wavelength and amplitude that travel in opposite directions creates a standing wave.

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¨  Nodes are point of destructive interference that never move.

¨  Antinodes are points of constructive interference.

¨  Successive nodes (or antinodes) are half a wavelength apart.

Count the Nodes and Antinodes

Sample Problem #1 (ex. 16.1, page 511)

¨  Two children hold an elastic cord at each end. Each child shakes her end of the cord 2.0 times per second, sending waves at 3.0 m/s toward the middle, where two waves combine to create a standing wave. What is the distance between adjacent nodes?

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Standing Waves and String Instruments

¨  Guitars, pianos, and violins all have strings that are fixed at both ends and tightened to create tension.

¨  A disturbance is created by plucking, striking, or bowing the string.

¨  This disturbance creates a standing wave on the string.

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Sample Problem #2 (ex. 16.2, page 514)

¨  A 2.50 m long string vibrates as a 100 Hz standing wave with nodes at 1.00 m and 1.50 m from one end of the string and at no points in between these two. Which harmonic is this? What is the string’s fundamental frequency? And what is the speed of the traveling waves on the string?

Sample Problem #3 (ex. 16.3, page 515)

¨  A guitar has strings of a fixed length. Plucking a string makes a particular musical note (at the string’s fundamental frequency). A guitar player plucks a string to play a note. He then presses down on a fret to make the string shorter. Does the new note have a higher or lower frequency?

Sample Problem #4 (ex. 16.4, page 515)

¨  The fifth string on a guitar plays the musical note A, at a frequency of 110 Hz. On a typical guitar, this string is stretched between two fixed points 0.640 m apart, and this length of string has a mass of 2.86 g. What is the tension in the string.

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Standing Waves and Lasers

Sample Problem #5 (ex. 16.5, page 516)

¨  A helium-neon laser emits light of wavelength 633 nm. A typical cavity for such a laser is 15 cm long. What is the mode number of the standing waves in this cavity?

Standing Waves and Wind Instruments

¨  In instruments such as flutes, saxophones, and clarinets, air blown into the mouthpiece creates a standing sound wave inside a tube of air.

¨  The type of standing wave depends on whether the tube of air is open or closed at each end.

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Open-Open Columns

Closed-Closed Columns

Open-Closed Columns

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Sample Problem #6 (ex. 16.6, page 519) ¨  The eardrum, which transmits vibrations to the sensory organs

of your ear, lies at the end of the ear canal. The ear canal in adults is about 2.5 cm in length. What frequency standing waves can occur within the ear canal that are within the human range of hearing? The speed of sound in the warm air of the ear canal is 350 m/s.

Sample Problem #7 (ex. 16.7, page 520)

¨  A flute and a clarinet have about the same length, but the lowest note that can be played on a clarinet is much lower than the lowest note that can be played on a flute. Why is this?

Sample Problem #8 (ex. 16.8, page 520) ¨  Wind instruments have an adjustable joint to change the

tube length. Players know that they may need to adjust this joint to stay in tune. To see why, suppose a “cold” flute plays the note A at 440 Hz when the air temperature is 20°C. ¤ How long is the tube? ¤ As the player blows air through the flute, the air inside the

instrument warms up and the speed of sound increases to 350 m/s. What is the frequency?

¤ At the higher temperature, how must the length of the tube be changed to bring the frequency back to 440 Hz?

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Interference of Waves From Two Sources

¨  When waves of identical frequency, wavelength, and amplitude are emitted from two sources, any interference will depend upon the path-length difference and wavelengths.

Constructive Interference

¨  Two waves will be “in phase” and will produce constructive interference any time their path-length difference is a whole number of wavelengths.

Destructive Interference

¨  Two waves will be “out of phase” and will produce destructive interference any time their path-length difference is a whole number of wavelengths plus half a wavelength.

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¨  These principles of interference from two sources are valid for spherical waves as well.

Sample Problem #9 (ex. 16.11, page 524)

¨  Two loudspeakers 42.0 m apart and facing each other emit identical 115 Hz sinusoidal sound waves. Susan is walking along a line between the two speakers. As she walks, she finds herself moving through loud and quiet spots. If Susan stands 19.5 m from one speaker, is she standing at a quiet spot or a loud spot? Assume that the speed of sound is 345 m/s.

Sample Problem #10 (ex. 16.12, page 526)

¨  Two speakers are 3.0 m apart and play identical tones of frequency 170 Hz. Sam stands directly in front of one speaker at a distance of 4.0 m. Is this a loud spot or a quiet spot? Assume that the speed of sound is 340 m/s.

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Beats

¨  If the waves reaching the observer have different frequencies, “beats” are produced.