Periodic Functions and Waves

Transcription

1 Ron Ferril SBCC Physics 101 Chapter ul06A Page 1 of 14 Chapter 06 Waves Periodic Functions and Waves Consider a function f of one variable x. That is, we consider a rule by which a number f(x) is computed for any given number x. As an example, we could choose the function squaring with f(x) = x 2. This function squares any given number x. For example f(2) = 2 2 = 4 and f(3) = 3 2 = 9. A particular kind of function of great interest to scientists and engineers is the periodic functions type. A periodic function has the interesting property that there is some constant number T, called the period of the function, such that f(x+t) = f(x) for any value of x. Whatever value f(x) the function has for given x, the function will again have the same value at x+t. In other words, a periodic function repeats as x increases. Also, the function repeats as x decreases. A little mathematical trick called substitution of variables can be used to show that we can also correctly write f(x T) = f(x) to show that the function repeats as we decrease x. Thus, the function f(x) for a given x has the same value at x T. An example of a periodic function is provided by a swinging pendulum. Let a number x, a length, represent the position of the pendulum at any time. Thus, x will be a function of time t and I write x(t). I start a pendulum swinging by first holding the pendulum at a displacement from its resting position. Let us call that position x=0. To keep the example simple and make the motion of the pendulum periodic, I will ignore friction and viscosity of air. The pendulum swings out from where I started it, passes the rest point where x=0, and the position x increases until it reaches a maximum value. Then the pendulum swings back with the value of x decreasing below the x=0 value until it reaches a minimum value. Then the pendulum swings with increasing x value until it reaches the place where the swinging was started. Thus the pendulum has completed an entire swing or an entire cycle. During that

2 Ron Ferril SBCC Physics 101 Chapter ul06A Page 2 of 14 cycle it was at the starting position three times: when the swing started, when the value x was decreasing and it passed the starting position, and when it finally reached the starting position again to end the cycle. The position was also at x=0 at least twice once when x was increasing and again when x was decreasing. The time that was required for the pendulum to complete a cycle is the period T of the function x(t). In this example, the period T was a time value. For a periodic function f(x) of a variable x with period T, mathematicians use the term period for T regardless of the physical nature of the variable. That is, a mathematician would still call T the period regardless of whether the variable x is a time, a temperature, a spacial coordinate giving a position, or some other physical quantity. Scientists and engineers often also use the term in the same way. However, physicists and engineers often use a restricted meaning for the term period in which the variable is specifically time. Such a restriction is useful when dealing with waves, but we first should broaden our view of periodic functions in order to understand waves. Thus far, we considered periodic functions of only one variable. The concept of periodic nature can also be applied to functions f(x, y, z...) of more than one variable. A specific example of this, commonly encountered in physics and engineering, is a function f(x,t) periodic both in a spacial coordinate x with period λ and in a time t with period T. f(x+λ, t) = f(x, t+t) = f(x, t) = f(x+λ, t+t) Such a function describes the behavior of waves. Physicists commonly call λ the wavelength and T is simply called the period. This is an example of a situation in which physicists use the more restricted meaning of the term period in which the variable is time. Periods for spacial coordinates like x are called wavelengths. This distinction between time and the spacial coordinates is often useful. To get a physical feel for the connection between periodic functions and waves, consider a continuous set of waves (often called a continuous wave ) on the surface of water. If we sit at the edge of the water and notice the rate at which the waves arrive, we can measure the period T as the time between arrivals of the waves. If we look down on the water from above, we can measure the wavelength λ as the distance between crests of the waves. The following figure shows the wavelength measurement.

3 Ron Ferril SBCC Physics 101 Chapter ul06A Page 3 of 14 From this example of waves on water, you can see that the wave is periodic in both time and spacial coordinate. The wave repeats for every period T in time or every wavelength λ in length along the direction in which the wave is traveling. Our textbook distinguishes between a vibration and a wave by describing a vibration as a wiggle in time and a wave as a wiggle in both space and time. Thus, a vibration can be viewed as a function f(t) periodic in one variable, time t, and a wave can be viewed as a function periodic in spacial variables such as length, width and height for example and time. However, periodic functions also describe other physical phenomena such as the swinging pendulum. The periodic nature of the functions distinguish waves from mere pulses. A pulse is physically like a wave in that it is a function of time and maybe also spacial coordinates, but lacks the periodic nature. In a sense, a wave is a continuous periodic progression of pulses. Some indirect studies of waves focus on pulses rather than on complete waves. For example, a physics teacher often sends a pulse along some medium so students can see the behavior of a single pulse and determine the behavior of a complete wave composed of a continuous stream of such pulses. Waves move in the sense that part of the wave, such as the maximum value of f(x,t), can move along the x direction with advancing time t. f(x,t) changes as either x is varied or t is varied, but it is possible to change x and t simultaneously so that f(x,t) remains constant. Thus, the varying values of x and t can continue to give the coordinates of a part of the wave as it moves. This changing of x and t but leaving f(x,t) constant represents the motion of the wave. It is difficult to see that f(x,t) indicates the wave is moving because the function f is very abstract. To better show the wave motion, some books prefer to represent a wave by a function g of a single variable as in g(x-vt) where v is the component of velocity of the wave along a chosen direction and the

4 Ron Ferril SBCC Physics 101 Chapter ul06A Page 4 of 14 value x-vt can be viewed as a single variable. This single variable x-vt is actually a function of position x and time t. f(x,t) = g(x-vt) The main idea for this notation g(x-vt) is that it allows people to easily see that the function can remain constant even though both x and t are changing. To show such a function periodic in x and t, v is written in terms of changes Δx in position and Δt in time as v = Δx / Δt Δx = v Δt The value of v can be positive, zero or negative. A positive value indicates the wave moves along the chosen direction but a negative value indicates motion in the opposite direction. Next, changing x by Δx and t by Δt gives g(x+δx - v[t+δt]) = g(x+vδt vt vδt) = g(x-vt) This function g has wavelength λ and period T given by Since we already said we get λ = Δx = v Δt T = Δt f(x,t) = g(x-vt) f(x, t) = f(x+λ, t+t) g(x+λ -v[t+t]) = g(x-vt) so the function g(x-vt) is not changed by moving x as a function of time with velocity component v (changing x by Δx and t by Δt) along the x direction. This property of the value of the function g(x-vt) remaining constant, as x is moved at speed v, is the property we want for representing a moving wave. It shows that part of the wave, such as either the maximum or the minimum value of g(x-vt) for example, moves with velocity component v along the x direction. Pendulums, Bobs and Characteristics of Waves A simple pendulum has a mass M hung on the end of a string of negligible mass. One end of the string is attached to a rigid support and the other end is attached to the mass. When the pendulum is at rest,

5 Ron Ferril SBCC Physics 101 Chapter ul06A Page 5 of 14 the mass hangs directly under the string and rigid support. We will call this point, where the mass is at rest, the equilibrium point of the pendulum. Equilibrium refers to a state in which the net force is zero so there is no acceleration. To start the oscillation of the pendulum, the mass is pulled to one side a distance A, with the string remaining taught, and then released. The pendulum swings with the mass going from a distance A on one side of the equilibrium point to a distance A on the other side of the equilibrium point. This distance A is called the amplitude of the oscillation. The mass swings a distance 2A (twice the amplitude) during half a period, and then swings back the same distance 2A for the next half a period. The period T of the oscillation depends on the length of the string but not on the mass M. If the amplitude A is small, the mass appears to be traveling horizontally and the period is roughly independent of the amplitude A. For small oscillations, an approximation can be used to show that the period T is proportional to the square root of the length of the string. T 1/(2π) [L/g] (for small oscillation) For larger amplitudes, the vertical motion of the mass becomes significant and the dependence of the period on amplitude becomes significant. A bob is a mass that undergoes a vertical motion similar to the horizontal motion of the mass of the simple pendulum. An easy example of such a bob is a mass hung from a spring. The bob is initially at rest at a position we will call the equilibrium position. We start the oscillation of the bob by vertically displacing it from its equilibrium position by a distance A which will be the amplitude of the oscillation we are about to start, and then releasing the bob so it oscillates. The bob travels periodically between A below its equilibrium position and A above. In half a period it travels from A below to A above, and in the next half a period travels from A above to A below the equilibrium position. In both example, that of the pendulum and that of the bob, the oscillation had an amplitude A and a period T. These were examples of oscillations. They can be represented by periodic functions of time t. Waves can be represented by periodic functions of time and spatial coordinates. Waves have amplitude A and period T like oscillations do, but also have a wavelength λ. For either an oscillation or a wave, the frequency f is the reciprocal of the period T in time t. f=1/t. If time is measured in seconds then frequency is expressed in Hertz (Hz). 1 Hz = 1/s

6 Ron Ferril SBCC Physics 101 Chapter ul06A Page 6 of 14 People often say a Hertz (Hz) is a cycle per second because they consider the frequency as a rate at which cycles of oscillation are completed, but really the unit can be applied to other rates such as the rate at which subatomic particles are incident on a surface. Thus, I find it better to say Hz=1/s. Thus far, we have seen the quantities amplitude, period and frequency for oscillations or waves, and the wavelength for for waves. However, a wave travels with a speed v. The speed depends on the frequency and the wavelength. v = f λ = λ/t f = v/λ λ = v/f = vt T = λ/v Those equations are useful for calculating certain quantities when two other quantities are known. It is important to notice that this wave speed is the speed of the wave or of the crests of the wave, not the speed of matter. Waves can transport energy but without transporting matter. As an example, a boat on the ocean far from land is lifted and dropped by waves so the boat moves in a small loop, but the boat is not transported by the waves. (Ocean water currents can transport the boat.) Now we have the following quantities for oscillations and waves. The amplitude A (for an oscillation or a wave) is half the difference of the maximum and minimum of the periodic function f(x,t). The period T (for an oscillation or a wave) is the time for a complete cycle to occur. The wavelength λ is the distance between crests of a wave. Crests are the parts of the wave where the periodic function reaches its maximum. In the general mathematical sense, both T and λ are periods. The frequency f (for an oscillation or a wave) is the reciprocal of the period and is the rate at which cycles are completed. The speed v is the speed of the crests of the wave. Actually, it is the speed of any part of the wave (crest, trough, steep rise or steep fall). These quantities describe some major characteristics of a wave.

7 Ron Ferril SBCC Physics 101 Chapter ul06A Page 7 of 14 Types of Waves We have seen that a wave is a traveling oscillation. It is represented by a function that is periodic in time and distance. However, we should consider what is oscillating. Various types of waves are distinguished by what is oscillating. In other words, the wave type is specified by specifying what physical property the periodic function represents. Sometimes the physical quantity represented by the periodic function has a direction with respect to the direction the wave travels. For example, the function may represent a vibration along the direction of the wave travel or a vibration to the sides of the direction of travel. A wave for which the periodic physical quantity has a direction perpendicular to the line along which the wave travels is called a transverse wave. Waves for which the property's direction is along the line are called longitudinal waves. Your instructor should demonstrate both types of wave in class with a long spring. When he shakes the spring sideways, a transverse pulse (a piece of a wave) is seen to travel down the spring. When he thrusts the end of the spring forward or backward, a longitudinal pulse travels down the spring. Examples of transverse waves include water waves on the surface of water and light waves since the electromagnetic fields of light are perpendicular to the direction of travel. Examples of longitudinal waves include sound in which the small movements of molecules along the line along which the waves travel cause compressions and rarefactions. (Notice the compressions and rarefactions don't have a direction, but the movements of molecules do. It is these microscopic movements that allow sound waves to be considered longitudinal.) Interference We saw that waves can be represented by functions f(x,t) that are periodic in both time t and spatial coordinates x. Interference refers to the effects of two waves of the same type existing in the same space. Most types of waves obey the superposition principle which says the result of two waves, of the same type, with oscillating the physical quantity represented by functions f 1 (x,t) and f 2 (x,t) occupying the same space is a single periodic function f(x,t) for the physical quantity where f(x,t) = f 1 (x,t) + f 2 (x,t). This addition of functions representing two waves is called superposition. People often paraphrase the superposition principle as saying the two waves add. A subtle point of the expression is that the two waves can subtract. f(x,t)=f 1 (x,t)+f 2 (x,t)

8 Ron Ferril SBCC Physics 101 Chapter ul06A Page 8 of 14 As an example, consider the two waves having the same frequency and going in the same direction. If the two waves are such that the function f 1 (x,t) rises when and where f 2 (x,t) rises and f 1 (x,t) falls when and where f 2 (x,t) falls, then the two waves are said to be in phase and the waves add to form a wave with a larger amplitude than either wave individually had. This is called constructive interference because the waves reinforce each other to yield a larger amplitude. However, if f 1 (x,t) rises when and where f 2 (x,t) falls, and f 1 (x,t) falls when and where f 2 (x,t) rises, then the two waves are said to be out of phase and they subtract to form a wave with a smaller amplitude than the larger amplitude of the two original waves. This kind of interference is called destructive interference because the amplitude was lower than at least one of the original two waves. Suppose the two waves are identical except going in opposite directions. In some places the waves will be in phase and constructive interference will occur, but in some other places the waves will be too far out of phase and destructive interference will occur. The result is oscillations at different places, and these oscillations don't travel. In some places, called nodes, the destructive interference will result in minimal amplitude, zero amplitude in this example. The places where the waves add to form a result with maximum amplitude, twice the amplitude as the original waves had in this example, are called anti-nodes or antinodes (two spellings of the same word). This combination of nodes, antinodes and all the other places where the interference occurs results in what is called a standing wave because the result is a set of oscillations that do not travel. The nodes and antinodes don't move. That is the case for two waves that are identical except for their directions. If the two waves superposed have slightly different frequencies, then the nodes and antinodes do move. Your instructor should use a spring to demonstrate standing waves in class. The waves sent along the spring reflect from an end and travel back so the situation has waves of the same frequency moving in both directions along the spring. Since the spring has a set finite length, certain frequencies result in standing waves but standing waves (with nodes and antinodes that don't move or change with time) don't form for other frequencies. The lowest frequency of a standing wave that can exist on the spring is called the fundamental frequency. Other standing waves on the spring have frequencies that are integer multiples of the fundamental frequency. These frequencies, or the standing wave with these frequencies, are called harmonics. The first harmonic is the fundamental frequency. The nth harmonic has frequency n times the fundamental frequency and wavelength 1/n of that of the fundamental frequency. This nature of having a fundamental frequency and harmonics is a property common to media of fixed composition, size and shape. Harmonics that are integer multiples of the fundamental

9 Ron Ferril SBCC Physics 101 Chapter ul06A Page 9 of 14 frequency are a typical property for linear media such as the spring I use to demonstrate standing waves. Designers of sound systems often encounter constructive and destructive interference for waves with the same frequency but being emitted from different areas of a room. Such waves constructively interfere in some places in space, but destructively interfere in other places. The resulting sound is loud at antinodes but may be quiet or inaudible at nodes. Audio technicians often walk around in an auditorium with music playing through the sound system, and the technicians listen to determine locations of nodes and antinodes. Tour guides in historic buildings sometimes demonstrate the audio effects in rooms where the guides can whisper so a volunteer on the opposite side of the room can hear the whisper but people between the guide and volunteer cannot hear the whisper. Sometimes these effects are due to interference. A similar effect can be achieved without constructive and destructive interference. Surfaces can reflect and focus sound at certain points in a room so sounds are loud there but sounds are relatively faint at other points. Resonance and Standing Waves An oscillator tends to oscillate at a frequency called the natural frequency of the oscillator. When an oscillator is disturbed for a moment or otherwise started but then allowed to oscillate on its own without being driven, the frequency of oscillation tends to be the natural frequency. Oscillating systems can be driven at different frequencies but tend to have maximum amplitudes for certain frequencies called resonant frequencies. This phenomenon is called resonance. The natural frequency of an oscillator is one of the oscillator's resonant frequencies. Resonance is caused by the existence of positive and negative work. Remember that work can be positive when the force and displacement have the same direction and negative when the force and displacement have opposite directions. In order to get a maximum amplitude, we can drive an oscillator at a frequency for which the work done on it will be mostly positive. When the oscillator is driven at the natural frequency, the work can be completely positive. A demonstration of resonance I like to do is with a pendulum which I drive by a small periodic force with a period equal to the natural frequency of the pendulum. Even though the force is small, the resulting amplitude of oscillation eventually becomes fairly large.

10 Ron Ferril SBCC Physics 101 Chapter ul06A Page 10 of 14 For a medium of fixed size and shape, the development of standing waves in or on the medium is an example of resonance. The fundamental frequency can be regarded as being like the natural frequency of an oscillator. For a linear medium such as the spring I use in class to demonstrate standing waves, the fundamental frequency and the harmonics are all resonances for that medium of fixed size and shape. Classical musical instruments use this type of resonance, in which standing waves are formed, to play musical notes. Standing waves form on the strings of stringed instruments, in the wind passages of wind instruments and on the surfaces of percussion instruments. Beats When two waves with the same shape and amplitude but slightly different frequencies are superposed, the result has two important features: a resulting frequency that is the average of the frequencies of the two waves superposed and a resulting amplitude that oscillates at a frequency equal to the difference of the frequencies of the superposed waves. The amplitude goes from zero to a maximum value that is twice the amplitude of the individual waves superposed, and then back to zero. These pulses of amplitude are called beats. When this is done with sound waves, the listeners hear the beats as a pulsating tone. If there is a separation of the sources of the two waves to be superposed to form beats, and the separation is larger than the wavelengths of the waves, then the interference of the two waves form a result similar to standing waves but with moving nodes and antinodes. In this case, these moving nodes and antinodes can be considered as the cause of the beats. If a listener is at rest relative to the two sources, the listener hears the beats. If the listener moves between the two sources in a direction opposite the direction the nodes and antinodes are moving, then the listener hears the beats at a higher pulse rate. If the listener moves in the direction the nodes and antinodes are moving, the beats are heard at a lower pulse rate. Typically the nodes and antinodes move fairly slowly so a person can walk along with an antinode keeping with it so the beats are not heard. This is a demonstration I like to do for people. The phenomenon of beats is used by instrument tuners. A tuning fork or other oscillator forms beats with the instrument being tuned and the tuner tries to adjust the instrument so the frequency of beats is reduced, ideally to zero. Also, pipe organs have celeste or "voix celeste" pipes that are deliberately mis-tuned so they form beats with the tones from other pipes. This produces a desirable pulsating effect.

11 Ron Ferril SBCC Physics 101 Chapter ul06A Page 11 of 14 Moving Sources of Waves Several effects result when the source of waves moves. If the source moves along the direction the waves travel, the source may start the next cycle of a wave with a crest near the crest of the previous cycle. The distance between the crests becomes smaller due to the motion of the source. This raises the frequency and lowers the wavelength of the wave. If the source moves the opposite way, the crest of the next cycle is further from the crest of the previous cycle. This lowers the frequency and raises the wavelength. This effect of changing frequency due to the motion of the source is called the Doppler effect. It occurs in water waves, sound and light. However, light is so fast that the source should move very fast for the human eye to notice any change in frequency. Suppose we receive and study waves from a moving source. If the source is moving toward us (relative to us), then the frequency is higher than it would have been if the source was not moving. Also the wavelength would be shorter if the source is moving toward us. If the source is moving away from us (relative to us), then the frequency would be lower, and the wavelength longer. Thus, the Doppler effect is useful for determining information about the movements of the source. Police have devices that bounce radar waves off of moving vehicles and use the reflected waves to determine the relative speed of vehicles. Astronomers determine whether a galaxy is moving toward Earth or away from Earth by determining whether the light from the galaxy is shifted to higher or lower frequencies. If a source moves as fast or faster then the speed of the waves, then the waves superimpose to create a wavefront with a large amplitude. For sound, this is the cause of sonic booms. A sonic boom can be caused even by a quiet aircraft because the pressure changes at the front and back of a jet are enough to cause the boom. Actually, a sonic boom should be a double boom because the pressure of air changes abruptly both when the nose of the jet arrives and when the tail arrives. Each of two sudden pressure changes produces a boom. The two booms often are too close in time for humans to notice that there are two. The separation of the two booms in time depends on the speed of sound and the length of the jet producing the booms. The speed of sound is lower at higher altitude because of the lower air pressure. Thus, sonic booms originating from high in the atmosphere from large vehicles (such as, for example, the Space Shuttle) may have double booms separated enough in time for humans to notice.

12 Ron Ferril SBCC Physics 101 Chapter ul06A Page 12 of 14 Reflection and Refraction of Waves When waves encounter a boundary between two media, some of the wave energy can be reflected at the boundary and some can be transmitted through the boundary into the next medium. The two media may be different materials in the case of sound or light. In the case of water waves, the two media may both be water but with two different depths of the water. Much of the energy of sound is reflected from hard surfaces but much is transmitted into soft materials that absorb the energy by converting it to heat. Waves crossing the boundary between media often undergo a change in speed. Refraction is the process of changing direction of travel caused by a change in speed. (People driving cars in snow encounter a type of refraction when they change from an icy lane to a snow-covered lane. The tires on the side of the car entering the snow-covered lane contact the snow and the snow tends to slow their motion while the tires on the ice are not slowed. The car turns its nose toward the snow-covered lane and car turns sharply into this lane. This causes many accidents.) Waves can undergo refraction when crossing from one medium into another. If the speed lowers as the waves enter the new medium, they turn more sharply into the medium. If the speed increases then the waves turn less sharply into the medium. To say this more precisely, consider a line perpendicular to the boundary between two media. We call this line the normal to the boundary. If the waves slow down on entering the new medium, the angle between their direction of travel and the normal becomes smaller in the new medium. If the waves speed up on entering the new medium, then the angle becomes larger. Since the frequency of the waves does not change in going from one medium to the next, the wavelength changes. Slowing at a boundary decreases the wavelength, and speeding up increases the wavelength. For example, when water waves from the deep ocean come to the shallower waters along a coast, the waves slow down and, thus, turn toward the coast and the distance between the waves decreases (which is fortunate for surfers). For another example, sound travels faster in warm air than cold air. When sound passes from warm air into colder air, the sound turns toward the normal to the boundary. Example Calculations Example 01. Suppose a wave has a wavelength of λ = 2 m and a frequency f = 20 Hz. We can calculate the speed of the wave. We can also calculate the period of the wave. v = f λ = 20 Hz x 2 m = 40 m/s

13 Ron Ferril SBCC Physics 101 Chapter ul06A Page 13 of 14 Thus, the period is one twentieth of a second. f = 1/T T = 1/f = 1 / [20 Hz] = 1 / [20 / s] = 0.05 s Example 02. Suppose two waves with the same shape and amplitude, but slightly different frequencies, f 1 = 19.9 Hz and f 2 = 20.1 Hz, are superposed. The result is beats. We can calculate the frequency of the resulting wave and the frequency of the beats. The frequency f of the resulting wave is the average of the two original frequencies. f = [f 1 + f 2 ] / 2 = [19.9 Hz Hz] / 2 = 20.0 Hz The frequency of the beats f B is the difference of the two original frequencies. Thus, the beat frequency is a fifth of a Hertz. f B = f 2 - f 1 = 20.1 Hz 19.9 Hz = 0.2 Hz Example 03. Suppose two identical waves are in phase with each other so they constructively interfere when they superpose. From the expression f(x,t) = f 1 (x,t) + f 2 (x,t) we can see that the amplitude of the resulting wave is double the original amplitude of either original wave. However, the expression can be misleading in other cases. For example, if the two waves are out of phase by a certain amount, they can destructively interfere and cancel each other so the amplitude drops to zero. Example 04. Suppose a string vibrates with fundamental frequency f 1 (the frequency of the first harmonic) and f 1 = 20 Hz. We can calculate the frequencies of the other harmonics. The nth harmonic has frequency f n = n f 1 = n x 20 Hz f 2 = 2 f 1 = 2 x 20 Hz = 40 Hz f 3 = 3 f 1 = 3 x 20 Hz = 60 Hz f 4 = 4 f 1 = 4 x 20 Hz = 80 Hz

14 Ron Ferril SBCC Physics 101 Chapter ul06A Page 14 of 14 Example 05. Consider a simple pendulum performing small oscillations. If the length of the pendulum, originally L, changes by a factor e so the new length is el, then the period changes from to so the period changed by a factor of e. T 1 = 1/(2π) [L/g] T 1 = 1/(2π) [el/g] = T 1 e