Periodic waves in space and time

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Jocelyn Cannon
6 years ago
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1 Waves We talked about the motion of bodies, such as planets and baseballs. At the most fundamental level, one has elementary particles (electrons, quarks). They are characterized by their position, a single point that moves around in space. Waves exist at all points in space. A medium moves at all points simultaneously, for example air for sound waves, water for ocean waves, the electric and magnetic field for light waves, and people for a stadium wave. Ch

2 Periodic waves in space and time Fixed time: Fixed position: space x T time t Wavelength = Distance between valleys Period T = Time between valleys.

3 Wave velocity A wave advances by a wavelength in one oscillation period T.* x t Velocity v = = T * Distinguish the period T from the absolute temperature T.

4 Wave frequency The frequency f of a wave is the number of oscillations per time interval t. One has 1 oscillation per period T, that is: f = 1 T The unit of f is Hz (Hertz) = 1/s This gives a new expression for the wave velocity v: v = f While and f can vary widely, v is roughly constant. Larger frequency corresponds to smaller wavelength. Frequency is also denoted by the Greek letter (nu).

5 Example The speed of sound in air is 344 m/s. A tuning fork oscillating with the frequency of 440 Hz (standard A) produces the wavelength: = v f = 344 m/s = 0.78 m 440 1/s

6 Proportionality and inverse proportionality a and b are proportional to each other: a = const. b a b a and b are inversely proportional to each other: a = const. b a 1 b

7 A low frequency wave Frequency f = 1 oscillation per second = 1 Hz Remember the spacing between wave fronts.

8 A higher frequency wave Twice the frequency f Half the wavelength f goes like 1/

9 Types of waves Transverse wave: Each piece of the medium moves perpendicular to the propagation direction. Also true for light waves

10 Longitudinal waves Longitudinal wave: Each piece of the medium moves parallel to the propagation direction. Also true for sound waves

11 Sound waves The medium transporting the wave is air. The air is compressed at the starting point, then compresses air next to it, and so on.

12 Producing a sound save Using a tuning fork to produce a sound wave: As the tuning fork vibrates, it compresses and expands the air nearby. This produces alternating sheets of compressed and expanded air. These move away with the velocity of sound.

13 Sound from a tuning fork a) As the tuning fork continues to vibrate, sheets of compressed and expanded air move away from the tuning fork. b) An oscillating sine curve is produced when plotting the air pressure versus distance.

14 Picking up sound with a microphone + A microphone converts a pressure difference into an electric signal (a voltage difference). This is accomplished by a capacitor consisting of a fixed electrode (blue) and a flexible electrode (red). The voltage depends on the spacing of the electrodes. The pressure created by the sound wave moves the flexible electrode in and out.

15 Spherical and circular waves Plane waves move in a specific direction (see previous slide). Spherical waves (in 3D) and circular waves (2D) propagate radially from a center.

16 Electron wave in a quantum corral Light wave from a laser passing through a fine aperture

17 Shock waves and sonic booms The circles represent wave fronts emitted at different times by the moving source S, e.g. a boat or a plane. Successive waves 0,1,2 emitted from S 0, S 1, S 2 add up to a wave front. The angle of the shock wave is determined by the triangle v t,s 0,v s t. v= wavevelocity v S = source velocity t = time Front of the shock wave

18 Shock wave in water A ship going faster than the waves creates a V- shaped wake. The faster the ship, the narrower the V.

19 Supersonic boom When a plane moves faster than the speed of sound, it generates a conical shock wave. A small cloud is formed when humid air is cooled by rapid expansion behind the shock wave and condenses into water droplets. Some particle detectors use the same principle, but with light waves instead of sound waves (Cerenkov radiation). An example is the neutrino experiment at the South Pole.

20 Water waves The wave travels, but the water goes in circles. Focus your eyes on the upper yellow dot, and watch it circling around the crosshair.

21 Circular motion About half a wavelength deep Motion of water in a wave

22 Interference Two waves can weaken or reinforce each other, depending on their signs. This is interference, the hallmark of waves. Interference between the waves originating from two slits is often used as test for wave character. Two water droplets causing circular waves.

Chs. 16 and 17 Mechanical Waves

Chs. 16 and 17 Mechanical Waves The nature of waves A wave is a traveling disturbance that carries energy from one place to another, and even though matter may be disturbed as a wave travels through a