Chapter 12 Vibrations and Waves. Section 12-1: Simple Harmonic Motion A repeated motion, such as that of an acrobat swinging on a trapeze, is called a.

Presentation on theme: "Chapter 12 Vibrations and Waves. Section 12-1: Simple Harmonic Motion A repeated motion, such as that of an acrobat swinging on a trapeze, is called a."— Presentation transcript:

1 Chapter 12 Vibrations and Waves

2 Section 12-1: Simple Harmonic Motion A repeated motion, such as that of an acrobat swinging on a trapeze, is called a periodic motion. Other periodic motions include those made by a child on a playground swing, a wrecking ball swaying to and fro, and the pendulum of a grandfather clock or a metronome. In each of these cases, the periodic motion is back and forth over the same path.

3 One of the simplest types of back-and-forth periodic motion is a mass attached to a spring. Let us assume that the mass moves on a frictionless horizontal surface. When the spring is stretched or compressed and then released, it vibrates back and forth around its unstretched position. In figure 12-1(a) on page 438, the spring is stretched away from its unstretched, or equilibrium, position (x= 0).

4 When released, the spring exerts a force on the mass toward the equilibrium position. This spring force decreases as the spring moves toward the equilibrium position, and it reaches zero at equilibrium. The force exerted on an object is zero at its equilibrium position, and its acceleration becomes zero at equilibrium. Though the spring force and acceleration decrease as the mass moves toward the equilibrium position, the velocity of the mass increases.

5 At the equilibrium position, when acceleration reaches zero, the velocity reaches a maximum. At that point, although no net force is acting on the mass, the mass’s momentum causes it to overshoot the equilibrium position and compress the spring. As the mass moves beyond equilibrium, the spring force and the acceleration increase

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7 Section 12-3: Properties of Waves Consider what happens to the surface of a pond when you drop a pebble into the water. The disturbance created by the pebble generates water waves that travel away from the disturbance. If you examined the motion of a leaf floating near the disturbance, you would see that the leaf moves up and down in reference to its original position. However, the leaf does not undergo any net displacement from the motion of the waves.

8 The leaf’s motion indicates the motion of the particles in the water. The water molecules move locally, like the leaf does, but they do not travel across the pond. The wave moves from one place to another, but the water itself is not carried with it.

9 A wave is the motion of a disturbance. Ripple waves in a pond start with a disturbance at some point in the water. This disturbance causes water on the surface near that point to move, which in turn causes points farther away to move. In this way, the waves travel outward in a circular pattern away from the original disturbance.

10 In this example, the water in the pond is the medium through which the disturbance travels. Medium– the material through which a disturbance travels. Particles in the medium– in this case, water molecules– vibrate up and down as waves pass. Note that the medium does not actually travel with the waves. After the wave passes, the medium returns to its original position.

11 Waves of ALMOST every kind require a material medium in which to travel. Sound waves cannot travel through outer space because space is very nearly a vacuum. In order for sound waves to travel, they must have a medium such as air or water. Mechanical waves– waves that require a material medium for transmission.

12 Not all wave propagation requires a medium. Electromagnetic waves do not require a material medium for transmission. Waves such as visible light, radio waves, microwaves, and X rays can travel through a vacuum.

13 Pulse wave– a wave that consists of a single traveling pulse. It s caused by a single, nonperiodic disturbance. Periodic wave– a wave whose source is some form of periodic motion.

14 Transverse wave– a wave in which the particles of the medium vibrate perpendicularly to the direction of wave motion. A picture of a wave is sometimes called a waveform. A waveform for a transverse wave can be obtained by plotting displacement versus time. Crest– the highest point above the equilibrium position. Trough– the lowest point below the equilibrium position.

15 Amplitude– the maximum displacement from equilibrium. Wavelength– the distance between two adjacent similar points of the wave, such as from crest to crest or from trough to trough. The wavelength is the distance the wave travels during one cycle.

16 Longitudinal wave– a wave in which the particles of the medium vibrate parallel to the direction of wave motion. Longitudinal waves consist of series of compressions and rarefactions. Compression– the region of a longitudinal wave in which the density of the medium is greater than normal. Rarefaction– the region of a longitudinal wave in which the density of the medium is less than normal.

17 A waveform for a longitudinal wave can be obtained by plotting density versus time. The crests correspond to compressions and troughs correspond to rarefactions. Compressions correspond to regions of high density and rarefactions correspond to regions of low density.

18 The vibrating object that causes a sine wave always has a characteristic frequency. Frequency– the number of cycles or vibrations per unit time. Because this motion is transferred to the particles in the wave, the frequency of vibration of the particles is equal to the frequency of the source. When the vibrating particles of the medium complete one full cycle, one complete wavelength passes any given point. Wave frequency describes the number of crests or troughs that pass a given point in a unit of time.

19 The period of a wave is the amount of time required for one complete vibration of the particles of the medium. Period—the time it takes to execute a complete cycle of motion. As the particles of the medium complete one full cycle of vibration at any point of the wave, one wavelength passes by that same point. The period of a wave describes the time it takes for a complete wavelength to pass a given point.

20 The period of a wave is inversely related to its frequency. f = 1/T or T = 1/f The SI unit for frequency is Hertz (Hz). The SI unit for period is seconds (s).

21 Wave Speed Equals Frequency Times Wavelength v = fλ The speed of a mechanical wave is constant for any given medium. For example, at a concert, sound waves from different instruments reach your ears at the same moment, even when the frequencies of the sound waves are different. Although the frequencies and wavelengths of the sounds produced by each instrument may be different, the product of the two is always the same at the same temperature.

22 When a mechanical wave’s frequency is increased, its wavelength must decrease in order for its speed to remain constant. The speed of a wave changes only when the wave moves from one medium to another or when certain properties of the medium are varied.

23 Waves Transfer Energy When a pebble is dropped into a pond, the water wave that is produced carries a certain amount of energy. As the wave spreads to other parts of the pond, the energy likewise moves across the pond. The wave transfers energy from one place in the pond to another while the water remains in essentially the same place. Waves transfer energy NOT matter.

24 The rate at which a wave transfers energy depends on the amplitude at which the particles of the medium are vibrating. The greater the amplitude, the more energy a wave carries in a given time interval. For a mechanical wave, the energy transferred is proportional to the square of the waves amplitude. 2 x amplitude = 4 x energy ½ amplitude = ¼ energy

25 Section 12-4: Wave Interactions Waves can pass through one another. Superposition– the combination of two overlapping waves. Mechanical and electromagnetic waves undergo superposition to form interference patterns. According to the superposition principle, when two or more waves travel through a medium, the resultant wave is the sum of the displacements of the individual waves at each point.

26 The superposition principle holds true for all types of waves, both mechanical and electromagnetic. Each wave maintains its own characteristics after interference. Constructive interference– interference in which individual displacements on the same side of the equilibrium position are added together to form the resultant wave.

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28 Destructive interference – interference in which individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave. Complete destructive interference occurs when two waves of equal amplitude but of opposite signs coincide. The resultant wave has a displacement of zero. The waves cancel each other out.

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30 Reflection Reflection– the turning back of a wave at a boundary. At a free boundary, waves are reflected. Consider a pulse wave traveling on a stretched rope whose end forms a ring around a post. Assume that the ring is free to slide along the post without friction.

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32 At a fixed boundary, waves are reflected and inverted. Consider a pulse traveling on a stretched rope that is fixed at one end.

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34 Standing Waves Standing wave– a wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere. The resultant wave pattern does not move along the string. The standing wave consists of alternating regions of constructive and destructive interference.

35 The points at which two waves cancel are called nodes. Node– a point in a standing wave that always undergoes complete destructive interference and is therefore stationary. Antinode– a point in a standing wave, halfway between two nodes, at which the largest amplitude occurs. Only certain frequencies of vibration produce standing wave patterns.

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37 Figure 12-23 shows different possible standing waves for a given string length. In each case, the curves represent the position of the string at different instants of time. If the string were vibrating rapidly, the several positions would blur together and give the appearance of loops. A single loop corresponds to either a crest or trough alone, while two loops correspond to a crest and trough together, or one wavelength.

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39 The ends of the string must be nodes because these points cannot vibrate. Standing waves can be produced for any wavelength that allows both ends of the string to be nodes.