Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
General Physics I Spring 2011 Traveling Waves and Sound 1 Mechanical Waves • A mechanical wave is a disturbance that propagates (travels) through a medium (solid, liquid, or gas). As the wave propagates, particles that compose the medium move about their equilibrium positions, but there is no overall transport of the medium with the wave. The wave travels at a fixed speed that depends on the properties of the medium. • Most waves fall into two categories: transverse and longitudinal. • A transverse wave is one in which the particles of the medium move transverse, or perpendicular, to the direction of propagation of the wave. A wave generated on a string is a transverse wave. • A longitudinal wave is one in which the particles of the medium move along the same direction as the wave travels. A sound wave is a longitudinal wave. 2 Mechanical Waves Water waves are neither purely transverse nor purely longitudinal. Each water element describes a circle as the wave passes it position. 3 Other Types of Waves • Electromagnetic waves are disturbances that consist of electric and magnetic fields and travel at the speed of light. These waves do not need a medium for propagation; they can travel in a vacuum. Examples are microwaves and visible light. • Matter waves are a representation of the behavior of material objects (usually very small, such as electrons) according to the subject of quantum mechanics. Quantum mechanics precisely and accurately explains the behavior of nuclei, atoms, and even large groups of atoms such as a solid material. • A common characteristic of all waves is that they transport energy. 4 Traveling Mechanical Waves • A mechanical wave propagates in a medium because adjacent pieces of the medium exert forces on each other. Consider a transverse wave pulse traveling to the right on a string. When the leading edge of the pulse reaches a given point in the string (see below), the tension forces acting on the tiny string piece (called a string element) at that point produce a net force and acceleration in the vertically upward direction. Thus, the element moves upward. At a later time when the curvature of the string at the position of the element is downward, the net force is downward and so the element slows down because its velocity is still upward. 5 Traveling Mechanical Waves • When the peak (crest) of the pulse arrives at the position of the string element, the element cannot go any higher, so its velocity must be zero. Because of the downward curvature of the string at that instant, the net force is downward and so the element will move downward immediately after the crest passes. The element continues to move down, speeding up initially then slowing down as the trailing edge of the pulse approaches. When the trailing edge passes, the element remains permanently at rest, as it was before the pulse arrived. 6 Speed of Transverse Waves on a String • If a transverse wave travels in an elastic string of mass m and length L in which the tension is Ts, the wave speed is given by v s tr in g where T = µs , µ = m. L is the linear mass density (mass per unit length) of the string. • We see that if the tension increases, the wave speed increases. This makes sense because a greater tension produces greater accelerations of string elements as they move up and down. Thus, the pulse passes any given point on the string faster. • If the mass per unit length increases, the wave speed decreases. This is because a greater mass gives smaller accelerations of string elements. Thus, the pulse moves more slowly. 7 Workbook: Chapter 15, Question 3, 4 8 Question 9 Graphical Description of Waves • Consider a wave pulse traveling to the right along an elastic string. The top figure shows a “snapshot” of the wave at a single instant in time. It shows the individual vertical displacements (y) of the elements of the string at their horizontal positions (x) along the direction of wave motion, all at the same instant in time. It shows a “profile” of the wave. The snapshot graph is a graph of vertical displacement of string elements versus horizontal position! • Each snapshot below the top one shows the pulse at a later time. Note that the pulse does not change its shape as it moves. 10 Graphical Description of Waves • Another way to graphically represent a wave is to plot the vertical displacement of a single string element as a function of time. Thus, the graph is a plot of y versus time (t) at a single value of x. This is precisely the motion that was explored on pages 5 and 6. A y versus t graph for a single particle is called a history graph. • The lower graph is the history graph corresponding to the string element at position x1 in the upper graph. 11 Graphical Description of Waves t=0s t=1s t=2s t=3s t=4s t=5s t=6s 12 Graphical Description of Waves • Note that the string element reaches the maximum displacement quickly because the front of the pulse rises steeply, so there is a short time interval between the arrival of the leading edge at x1 and the arrival of the peak. The displacement of the string element takes a longer time to go back to zero because the back of the pulse has a gentler slope than the front. • Remember that the history graph is a plot of vertical displacement of a string element versus time! 13 Workbook: Chapter 15, Questions 6(a), 7, 10 14 Solution 6 15 Solution 7. 16 Solution 10. 17 Snapshot and History Graph Problems • To generate a history graph given a snapshot graph: (1) Find the time at which the leading edge of the pulse arrives at the given position. (2) Convert the positions of the pulse in the snapshot graph to times using ∆t = ∆x / vwave. • To generate a snapshot graph given a history graph: (1) Find the position of the leading edge of the pulse at the given time. (2) Convert the times of the pulse in the history graph to positions using ∆x = vwave∆t. 18 Sinusoidal Waves • If one particle of a medium is disturbed by, e.g., applying a force, a wave disturbance will propagate outward from the source particle. If the applied force is a linear restoring force so that the particle moves with simple harmonic motion (SHM), then a sinusoidal wave will propagate. The passage of a sinusoidal wave causes a continuous periodic disturbance in the medium. All the particles of the medium where the wave exists will move with SHM. Waves Simulation (PhET) 19 Traveling Sinusoidal Waves • The top picture shows a history graph for one particle of a medium in which a sinusoidal wave is traveling. The amplitude (A) of the SHM (i.e., maximum displacement from equilibrium) is equal to the amplitude of the wave. The period of the SHM is equal to the period of the wave. It follows that the frequency of the SHM (f = 1/T) is equal to the frequency of the wave. • The lower picture shows a snapshot graph of a traveling sinusoidal wave in a medium (e.g., a string). We see that the vertical particle displacements also vary sinusoidally with the particle positions. (There is a particle at each position along the x axis.) 20 Traveling Sinusoidal Waves • The wavelength (λ) of the wave is the distance over which the wave repeats itself along the direction of travel. • Since the wave moves without changing its shape (like a rigid wire frame), the wave will travel a distance equal to one wavelength in the same amount of time it takes for the wave to repeat itself, which is one period. (The next slide shows this more clearly). • Since ∆x = vx∆t , we have λ = vT or, v = λ /T , which gives v = λ f . (Sinusoidal waves) 21 Traveling Sinusoidal Waves v • During one period of the wave, a crest moves a distance equal to one wavelength along the direction of wave motion. The particle at the position of the original crest has completed one cycle of SHM. (Remember that the particles of the medium do not move in the direction of wave motion! They simply oscillate as the wave passes their positions.) • Particles that are separated by a distance equal to one wavelength oscillate exactly in step (or in phase). They have exactly the same vertical displacements. Particles that are separated by a distance that is less than one wavelength will be at different stages in their SHM cycles at any given instant in time and so their vertical displacements will be different. 22 Mathematical Description • The displacement of a sinusoidal traveling wave moving along the positive x direction is given by 2 π x 2 π t y( x,t) = Acos − . T λ The equation gives the displacement of the particle located at position x at time instant t. The expression comes from the fact that a sinusoidal wave exhibits sinusoidal behavior as a function of both time t and position x. Note that a sine function could also have been used. • For a sinusoidal traveling wave moving in the negative x direction, the displacement is given by 2 π x 2 π t y( x,t) = Acos + . T λ 23 Workbook: Chapter 15, Questions 12, 13 Textbook: Chapter 15, Problem 62 24 Longitudinal Waves • A longitudinal wave is generated by applying a periodic force to a particle in a medium so that the particle oscillates with SHM with a displacement along the same line that neighboring particles are located on. • In a longitudinal wave, there are regions where the particles are closer to together than when the medium is undisturbed. These regions are called compressions. There are also regions where the particles are farther apart than when the medium is undisturbed. These regions are rarefactions. 25 Sound Waves • A sound wave is a longitudinal wave. As a sinusoidal sound wave propagates, the pressure in the medium varies sinusoidally with position. The high pressure areas (crests) are compressions and the low pressure areas (troughs) are rarefactions. Note that the equilibrium pressure is not zero. • Sound travels in solids, liquids, and gases. A medium is necessary for sound waves to exist. Sound waves generally have the greatest speeds in solids, where the atoms/molecules are closest together and interatomic forces are generally strongest. 26 Light Waves • Visible light is just one type of electromagnetic wave. Electromagnetic waves are due to oscillating electric and magnetic fields. These fields can exist in vacuum, so light does not need a medium to propagate. (This is fortunate; otherwise no sunlight would reach us across the vacuum of space.) Electromagnetic (EM) waves are transverse waves. In vacuum, all EM waves travel at the speed of light, which is 3.00×108 m/s. • The various types of electromagnetic waves comprise the electromagnetic spectrum, which is shown in the picture. Microwaves, x-rays, etc. correspond to different wavelength ranges. All types have the same speed in vacuum. 27