Nicholas J. Giordano
www.cengage.com/physics/giordano
• 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 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 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 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 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 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 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 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 The Equation of a Wave
• Assume the displacement generating the wave in the string vibrates as a simple harmonic oscillator with
yend = 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 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 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 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 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 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 Interpreting the Equation of a Periodic Wave
Displacement of the medium as a Frequency function of location (x) and time (t)
Amplitude Wavelength
Section 12.2 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 FT to keep the tension separate from the period • The speed of the wave is
Section 12.3 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 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 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 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 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 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 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 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 108 m/s • It is independent of the frequency of the wave Section 12.3 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 Water Waves
• A water wave can be generated by dropping a rock onto the surface • The waves propagate outward
Section 12.3 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 Wave Fronts: Spherical Waves
• A spherical wave travels away from its source in a three- dimensional fashion • The wave crests form concentric spheres centered on the source • The crests are also called wave fronts
Section 12.4 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 Intensity
• Intensity is the power carried by the wave over a unit area of the wave front • SI units of intensity is W/m2 • 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 Intensity, cont.
• At a distance r from the source, the surface area of the sphere is 4πr2 • The intensity is
• The intensity falls with distance as
Section 12.4 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
Intensity and Amplitude
• The intensity of a wave is related to its amplitude
• Spring example • The potential energy is ½ k x2 • 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 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 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 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 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 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 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 Example: Reflection of Light
• The light wave from a laser reflects from a mirror
Section 12.6 Reflection – Light Ray Details
• The rays make an initial angle of θi 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: θi = θr
Section 12.6 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 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 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 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 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 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 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 Harmonics
• The longest possible wavelength corresponds to the smallest possible frequency • This frequency is called the fundamental frequency, ƒ1 • The next longest wavelength is called the second harmonic • The pattern of wavelengths and frequencies is
Section 12.8 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 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 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 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 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 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