Nicholas J. Giordano. Chapter 12 Waves

Transcription

1 Nicholas J. Giordano Chapter 12 Waves

2 Wave Motion A wave is a moving disturbance that transports energy from one place to another without transporting matter Questions about waves What is being disturbed? How is it disturbed? The motion associated with a wave disturbance often has a repeating form, so wave motion has much in common with simple harmonic motion Introduction

3 Waves, String Example One example of a wave is a disturbance on a string Shaking the free end creates a disturbance that moves horizontally along the string A single shake creates a wave pulse If the end of the string is shaken up and down in simple harmonic motion, a periodic wave is produced Section 12.1

4 Waves, String Example cont. The disturbances are examples of waves Portions of the string are moving so there is kinetic energy associated with the wave There is elastic potential energy in the string as it stretches The wave carries this energy as it travels The wave does not carry matter as it travels Pieces of the string do not move from one end of the string to the other Section 12.1

5 Analysis of The Wave Pulse A single pulse propagates to the right The graph (part D in the figure shown) shows the displacement of point D on the string It is perpendicular to the direction of propagation The wave transports energy without transporting matter Section 12.1

6 Wave Terminology The thing being disturbed by the wave is its medium When the medium is a material substance, the wave is a mechanical wave In transverse waves the motion of the medium is perpendicular to the direction of the propagation of the wave The string was an example In longitudinal waves the motion of the medium is parallel to the direction of the propagation of the wave Section 12.1

7 Example: Longitudinal Wave The spring is shaken back and forth in the horizontal direction At some places the coils are compressed At other places the coils are stretched This motion produces a longitudinal wave Section 12.1

8 Describing Periodic Waves Assume a person is shaking the string so that the end is undergoing simple harmonic motion The crest is the maximum positive y displacement The trough is the maximum negative y displacement Section 12.2

9 Periodic vs. Nonperiodic Waves Nonperiodic waves The wave disturbance is limited to a small region of space Periodic waves The wave extends over a very wide region of space The displacement of the medium varies in a repeating and often sinusoidal pattern A periodic wave involves repeating motion as a function of both space and time Section 12.2

10 The Equation of a Wave Assume the displacement generating the wave in the string vibrates as a simple harmonic oscillator with y end = A sin (2 π ƒ t) The string s displacement is given by λ is the symbol for wavelength This is a mathematical description of a periodic wave It shows the transverse displacement y of a point on the string as it varies with time and location Section 12.2

11 More Wave Terminology Periodic waves have a frequency The frequency is related to the repeat time The period is the time that a point takes to go from a crest to the next crest in its motion Then ƒ = 1 / T Periodic waves have an amplitude Wave crests have y = + A Wave troughs have y = - A Section 12.2

12 Wavelength The wavelength is the repeat distance of the wave Start at a given value of y Advance x by a distance equal to the wavelength and y will be at the same value again Section 12.2

13 Periodic Wave, Summary Periodic waves have both a repeat time and a repeat distance A periodic wave is a combination of two simple harmonic motions One is a function of time The other is a function of space Section 12.2

14 Speed of a Wave The mathematical description of a wave contains frequency, wavelength and amplitude The speed of a wave is This is based on the definitions of period and wavelength Section 12.2

15 Direction of a Wave To determine the direction of the wave, you can focus on the motion of a crest As x becomes larger, the wave has moved to the right and the wave velocity is positive and its equation is The equation of a wave moving to the left and having a negative velocity is Section 12.2

16 Interpreting the Equation of a Periodic Wave Displacement of the medium as a function of location (x) and time (t) Frequency Amplitude Wavelength Section 12.2

17 Waves on a String Waves on a string are mechanical waves The medium that is disturbed is the string For a transverse wave on a string, the speed of the wave depends on the tension in the string and the string s mass per unit length Mass / length = μ Tension will be denoted as F T to keep the tension separate from the period The speed of the wave is Section 12.3

18 Waves on a String, cont. The speed of the wave is independent of the frequency of the wave The frequency will be determined by how rapidly the end of the string is shaken The speed of transverse waves on a string is the same for both periodic and nonperiodic waves Section 12.3

19 Sound Waves Sound is a mechanical wave that can travel through almost any material Travels in solids, liquids, and gases Assume a speaker is used to generate the waves Section 12.3

20 Sound Waves, cont. The speaker moves back and forth in the horizontal direction As it moves, it collides with nearby air molecules The x component of the velocity of the air molecules is affected by the speaker The displacement of the air molecules associated with the sound wave is also along the x direction The result is a longitudinal wave Section 12.3

21 Speed of Sound Waves The speed of sound depends on the properties of the medium At room temperature, the speed of sound in air is approximately 343 m/s The speed is independent of the frequency The speed applies to both periodic and nonperiodic waves Sound waves in a liquid or solid are also longitudinal The speed of sound is generally smallest for gases and highest for solids Section 12.3

22 Waves in a Solid Solids can support both longitudinal and transverse waves The longitudinal waves are considered sound waves The speed of the sound depends on the solid s elastic properties Section 12.3

23 Speed of Sound in a Solid For a thin bar of material, the speed of sound is given by The speed of a transverse wave is more complicated and depends on the shear modulus and other elastic constants In general, the speed of the transverse wave is slower than the speed of longitudinal waves Section 12.3

24 Transverse Waves Transverse waves can travel through solids They cannot travel through liquids or gases The displacements in transverse waves involve a shearing motion Liquids and gases flow and there is no restoring force to produce the oscillations necessary for a transverse wave Section 12.3

25 Electromagnetic Waves Electromagnetic (em) waves are not mechanical waves They are electric and magnetic disturbances that can propagate even in a vacuum No mechanical medium is required The electric and magnetic fields are always perpendicular to the direction of propagation So they are transverse waves EM waves are classified according to their frequency The speed of an em wave in a vacuum is 3.00 x 10 8 m/s It is independent of the frequency of the wave Section 12.3

26 Speed of Waves, Summary The speed of a wave depends on the properties of the medium through which it travels The speed varies widely From slow waves on a string To very fast em waves Generally, the wave speed is independent of both frequency and amplitude There are cases in light and optics where the speed does depend on the frequency The speed is the same for periodic and nonperiodic waves Section 12.3

27 Water Waves A water wave can be generated by dropping a rock onto the surface The waves propagate outward Section 12.3

28 Water Waves, cont. The motion of the water s surface is both transverse and longitudinal A bug on the surface moves up and down as well as backward and forward Section 12.3

29 Wave Fronts: Spherical Waves A spherical wave travels away from its source in a threedimensional fashion The wave crests form concentric spheres centered on the source The crests are also called wave fronts Section 12.4

30 Spherical Waves, cont. The direction of the wave propagation is always perpendicular to the surface of a wave front The direction is indicated by rays Each wave carries energy as it travels away from the source Power measures the energy emitted by the source per unit time Units of power are J/s = W W for Watt

31 Intensity Intensity is the power carried by the wave over a unit area of the wave front SI units of intensity is W/m 2 Once a wave front is emitted, its energy remains the same The intensity falls as the wave moves farther from the source The area is becoming larger Section 12.4

32 Intensity, cont. At a distance r from the source, the surface area of the sphere is 4πr 2 The intensity is The intensity falls with distance as Section 12.4

33 Plane Waves Wave fronts are not always spherical Another type is a plane wave In a perfect plane wave, each crest and trough extend over an infinite plane in space The intensity is approximately constant over long propagation distances Intensity is ideally independent of distance Section 12.4

34 Intensity and Amplitude The intensity of a wave is related to its amplitude Spring example The potential energy is ½ k x 2 For a wave on a spring, the displacement is proportional to the amplitude Therefore, the energy and intensity are proportional to the square of the amplitude Section 12.4

35 Superposition Waves generally propagate independently of one another A wave can travel though a particular region of space without affecting the motion of another wave traveling though the same region This is due to the Principle of Superposition When two (or more) waves are present, the displacement of the medium is equal to the sum of the displacements of the individual waves The presence of one wave does not affect the frequency, amplitude, or velocity of the other wave Section 12.5

36 Constructive Interference Two wave pulses are traveling toward each other They have equal and positive amplitudes At C, the two waves completely overlap and the amplitude is twice the amplitude of the individual waves The emerging pulses are unchanged This is an example of constructive interference Section 12.5

37 Destructive Interference Two pulses are traveling toward each other They have equal and opposite amplitudes At C, the two waves completely overlap, total displacement is zero The emerging pulses are unchanged This is an example of destructive interference Section 12.5

38 Interference Constructive interference causes the waves to produce a displacement that is larger than the displacements of either of the individual waves Destructive interference causes the waves to produce a displacement that is smaller than the displacements of either of the individual waves In either case, the energy of each wave is contained in the kinetic energy of the medium The waves can interfere, even destructively, and still carry energy independently Section 12.5

39 Interference of Periodic Waves The crests of the waves travel away from the initial source There is constructive interference where the wave crests overlap There is destructive interference where a crest and trough overlap The result shows an interference pattern with regions of constructive and destructive interference Section 12.5

40 Reflection Reflection changes the propagation direction of the wave Rays can be used to indicate the direction of energy flow The rays change direction when a wave reflects from the boundary of the medium The wave is inverted as it reflects from a fixed end Section 12.6

41 Example: Reflection of Light The light wave from a laser reflects from a mirror Section 12.6

42 Reflection Light Ray Details The rays make an initial angle of θ i with a line drawn perpendicular to the surface The perpendicular component of the wave s velocity reverses direction The parallel component of the wave s velocity is not affected by the reflection The angle of incidence will equal the angle of reflection: θ i = θ r Section 12.6

43 Reflection Free Surface The end of the string is attached to a ring that is free to move up and down When the wave is reflected, it is not inverted The properties of the medium at the boundary will determine if the reflected wave will be inverted or not Section 12.6

44 Radar An application of wave reflection is radar A radio wave pulse is sent from a transmitting antenna and reflects from some distant object A portion of the reflected wave will arrive back at the original transmitter, where it is detected Section 12.6

45 Radar, cont. Radar determines the distance to the object by measuring the time delay between the original and reflected signals By using a rotating antenna, the direction of the object can also be detected The amplitude of the reflected rays gives information about the size of the object A larger object reflects more of the wave energy and gives a larger signal at the detecting antenna Section 12.6

46 Refraction If the rays follow bent paths in a medium, they are said to be refracted The frequency of the wave stays the same It is determined by the source The change in direction of the wave is due to a change in its speed Section 12.7

47 Standing Waves Waves may travel back and forth along a string of length L If the string has both ends held in fixed positions, the displacement at both ends must be zero These conditions can be satisfied by a periodic wave only for certain wavelengths For these wavelengths, a standing wave can be produced It is called a standing wave because the outline of the wave appears stationary Section 12.8

48 Standing Waves, cont. The standing wave is obtained by the interference of two waves traveling in opposite directions The waves travel along the string and are reflected from the ends Section 12.8

49 Standing Waves, final Points where the string displacement is zero are called nodes Points where the displacement is largest are called antinodes Many standing waves may fit into the length of the string Section 12.8

50 Harmonics The longest possible wavelength corresponds to the smallest possible frequency This frequency is called the fundamental frequency, ƒ 1 The next longest wavelength is called the second harmonic The pattern of wavelengths and frequencies is Section 12.8

51 Harmonics, cont Combining the frequency and wavelength equations gives other expressions for the frequency: This is for standing waves on a string with fixed ends The allowed standing wave frequencies are integer multiples of the fundamental frequency Section 12.8

52 Musical Tones Many musical instruments use strings as a vibrating element Your fingers press down on the string and changes its length The string vibrates with all the possible standing wave pattern frequencies The pitch of note is determined by its fundamental frequency Two notes whose fundamental frequencies differ by a factor of 2 are said to be separated by an octave Section 12.8

53 Seismic Waves Seismic waves propagate through the Earth Their source can be any large mechanical disturbance such as an earthquake There are three types of seismic waves Section 12.9

54 Types of Seismic Waves S waves S for shear Transverse waves The displacement of the solid Earth is perpendicular to the direction of propagation P waves P for pressure Longitudinal sound waves Surface waves Similar to water waves but travel through the surface of the Earth Seismic waves can be detected by a seismograph Section 12.9

55 Structure of the Earth Seismic waves can help determine the interior structure of the Earth S waves do not propagate through the core So the core contains a liquid Both S and P waves are refracted Section 12.9

56 Structure of the Earth, cont. Analysis of the waves led to the following structure: Inner core Outer core Mantle Crust Many characteristics of these sections also were obtained from the study of seismic waves Section 12.9