There are two major kinds of waves: transverse waves and longitudinal waves. The medium transmitting transverse waves oscillates in a direction perpendicular to the direction the wave is traveling. A good example is waves on water: the water oscillates up and down while transmitting a wave horizontally. Other common examples include a wave on a string and electromagnetic waves. By contrast, the medium transmitting longitudinal waves oscillates in a direction parallel to the direction the wave is traveling. The most commonly discussed form of longitudinal waves is sound. Transverse Waves: Waves on a String Imagine—or better yet, go grab some twine and set up—a length of string stretched between two posts so that it is taut. Each point on the string is just like a mass on a spring: its equilibrium position lies on the straight line between the two posts, and if it is plucked away from its resting position, the string will exert a force to restore its equilibrium position, causing periodic oscillations. A string is more complicated than a simple mass on a spring, however, since the oscillation of each point influences nearby points along the string. Plucking a string at one end causes periodic vibrations that eventually travel down the whole length of the string. Now imagine detaching one end of the string from the pole and connecting it to a mass on a spring, which oscillates up and down, as in the figure below. The oscillation at one end of the string creates waves that propagate, or travel, down the length of the string. These are called, appropriately, traveling waves. Don’t let this name confuse you: the string itself only moves up and down, returning to its starting point once per cycle. The wave travels, but the medium— the string, in this case—only oscillates up and down. The speed of a wave depends on the medium through which it is traveling. For a stretched string, the wave speed depends on the force of tension, , exerted by the pole on the string, and on the mass density of the string, : The formula for the wave speed is: EXAMPLE 276 A string is tied to a pole at one end and 100 g mass at the other, and wound over a pulley. The string’s mass is 100 g, and it is 2.5 m long. If the string is plucked, at what speed do the waves travel along the string? How could you make the waves travel faster? Assume the acceleration due to gravity is 10 m/s 2 . Since the formula for the speed of a wave on a string is expressed in terms of the mass density of the string, we’ll need to calculate the mass density before we can calculate the wave speed. The tension in the string is the force of gravity pulling down on the weight, The equation for calculating the speed of a wave on a string is: This equation suggests two ways to increase the speed of the waves: increase the tension by hanging a heavier mass from the end of the string, or replace the string with one that is less dense. Longitudinal Waves: Sound While waves on a string or in water are transverse, sound waves are longitudinal. The term longitudinal means that the medium transmitting the waves—air, in the case of sound waves—oscillates back and forth, parallel to the direction in which the wave is moving. This back-and-forth motion stands in contrast to the behavior of transverse waves, which oscillate up and down, perpendicular to the direction in which the wave is moving. Imagine a slinky. If you hold one end of the slinky in each of your outstretched arms and then jerk one arm slightly toward the other, you will send a pulse across the slinky toward the other arm. This pulse is transmitted by each coil of the slinky oscillating back and forth parallel to the direction of the pulse. 277 When the string on a violin, the surface of a bell, or the paper cone in a stereo speaker oscillates rapidly, it creates pulses of high air pressure, or compressions, with low pressure spaces in between, called rarefactions. These compressions and rarefactions are the equivalent of crests and troughs in transverse waves: the distance between two compressions or two rarefactions is a wavelength. Pulses of high pressure propagate through the air much like the pulses of the slinky illustrated above, and when they reach our ears we perceive them as sound. Air acts as the medium for sound waves, just as string is the medium for waves of displacement on a string. The figure below is an approximation of sound waves in a flute—each dark area below indicates compression and represents something in the order of 10 24 air molecules. Loudness, Frequency, Wavelength, and Wave Speed Many of the concepts describing waves are related to more familiar terms describing sound. For example, the square of the amplitude of a sound wave is called its loudness, or volume. Loudness is usually measured in decibels. The decibel is a peculiar unit measured on a logarithmic scale. You won’t need to know how to calculate decibels, but it may be useful to know what they are. The frequency of a sound wave is often called its pitch. Humans can hear sounds with frequencies as low as about 90 Hz and up to about 15,000 Hz, but many animals can hear sounds with much higher frequencies. The term wavelength remains the same for sound waves. Just as in a stretched string, sound waves in air travel at a certain speed. This speed is around 343 m/s under normal circumstances, but it varies with the temperature and pressure of the air. You don’t need to memorize this number: if a question involving the speed of sound comes up on the SAT II, that quantity will be given to you. Superposition Suppose that two experimenters, holding opposite ends of a stretched string, each shake their end of the string, sending wave crests toward each other. What will happen in the middle of the string, where the two waves meet? Mathematically, you can calculate the displacement in the center by simply adding up the displacements from each of the two waves. This is called the principle of superposition: two or more waves in the same place are superimposed upon one another, meaning that they are all added together. Because of superposition, the two experimenters can each send traveling waves down the string, and each wave will arrive at the opposite end of the string undistorted by the 278 other. The principle of superposition tells us that waves cannot affect one another: one wave cannot alter the direction, frequency, wavelength, or amplitude of another wave. Destructive Interference Suppose one of the experimenters yanks the string downward, while the other pulls up by exactly the same amount. In this case, the total displacement when the pulses meet will be zero: this is called destructive interference. Don’t be fooled by the name, though: neither wave is destroyed by this interference. After they pass by one another, they will continue just as they did before they met. Constructive Interference On the other hand, if both experimenters send upward pulses down the string, the total displacement when they meet will be a pulse that’s twice as big. This is called constructive interference. Beats You may have noticed the phenomenon of interference when hearing two musical notes of slightly different pitch played simultaneously. You will hear a sort of “wa-wa-wa” sound, which results from repeated cycles of constructive interference, followed by destructive interference between the two waves. Each “wa” sound is called a beat, and the number of beats per second is given by the difference in frequency between the two interfering sound waves: 279 EXAMPLE Modern orchestras generally tune their instruments so that the note “A” sounds at 440 Hz. If one violinist is slightly out of tune, so that his “A” sounds at 438 Hz, what will be the time between the beats perceived by someone sitting in the audience? The frequency of the beats is given by the difference in frequency between the out-of-tune violinist and the rest of the orchestra: Thus, there will be two beats per second, and the period for each beat will be T = 1/f = 0.5 s. Standing Waves and Resonance So far, our discussion has focused on traveling waves, where a wave travels a certain distance through its medium. It’s also possible for a wave not to travel anywhere, but simply to oscillate in place. Such waves are called, appropriately, standing waves. A great deal of the vocabulary and mathematics we’ve used to discuss traveling waves applies equally to standing waves, but there are a few peculiarities of which you should be aware. Reflection If a stretched string is tied to a pole at one end, waves traveling down the string will reflect from the pole and travel back toward their source. A reflected wave is the mirror image of its original—a pulse in the upward direction will reflect back in the downward direction—and it will interfere with any waves it encounters on its way back to the source. In particular, if one end of a stretched string is forced to oscillate—by tying it to a mass on a spring, for example—while the other end is tied to a pole, the waves traveling toward the pole will continuously interfere with their reflected copies. If the length of the string is a multiple of one-half of the wavelength, , then the superposition of the two waves will result in a standing wave that appears to be still. 280 Nodes The crests and troughs of a standing wave do not travel, or propagate, down the string. Instead, a standing wave has certain points, called nodes, that remain fixed at the equilibrium position. These are points where the original wave undergoes complete destructive interference with its reflection. In between the nodes, the points that oscillate with the greatest amplitude—where the interference is completely constructive—are called antinodes. The distance between successive nodes or antinodes is one-half of the wavelength, . Resonance and Harmonic Series The strings on musical instruments vibrate as standing waves. A string is tied down at both ends, so it can only support standing waves that have nodes at both ends, and thus can only vibrate at certain given frequencies. The longest such wave, called the fundamental, or resonance, has two nodes at the ends and one antinode at the center. Since the two nodes are separated by the length of the string, L, we see that the fundamental wavelength is . The string can also support standing waves with one, two, three, or any integral number of nodes in between the two ends. This series of standing waves is called the harmonic series for the string, and the wavelengths in the series satisfy the equation , or: In the figure above, the fundamental is at the bottom, the first member of the harmonic series, with n = 1. Each successive member has one more node and a correspondingly shorter wavelength. EXAMPLE 281 An empty bottle of height 0.2 m and a second empty bottle of height 0.4 m are placed next to each other. One person blows into the tall bottle and one blows into the shorter bottle. What is the difference in the pitch of the two sounds? What could you do to make them sound at the same pitch? Sound comes out of bottles when you blow on them because your breath creates a series of standing waves inside the bottle. The pitch of the sound is inversely proportional to the wavelength, according to the equation . We know that the wavelength is directly proportional to the length of the standing wave: the longer the standing wave, the greater the wavelength and the lower the frequency. The tall bottle is twice as long as the short bottle, so it vibrates at twice the wavelength and one-half the frequency of the shorter bottle. To make both bottles sound at the same pitch, you would have to alter the wavelength inside the bottles to produce the same frequency. If the tall bottle were half- filled with water, the wavelength of the standing wave would decrease to the same as the small bottle, producing the same pitch. Pitch of Stringed Instruments When violinists draw their bows across a string, they do not force the string to oscillate at any particular frequency, the way the mass on a spring does. The friction between the bow and the string simply draws the string out of its equilibrium position, and this causes standing waves at all the different wavelengths in the harmonic series. To determine what pitches a violin string of a given length can produce, we must find the frequencies corresponding to these standing waves. Recalling the two equations we know for the wave speed, and , we can solve for the frequency, , for any term, n, in the harmonic series. A higher frequency means a higher pitch. You won’t need to memorize this equation, but you should understand the gist of it. This equation tells you that a higher frequency is produced by (1) a taut string, (2) a string with low mass density, and (3) a string with a short wavelength. Anyone who plays a stringed instrument knows this instinctively. If you tighten a string, the pitch goes up (1); the strings that play higher pitches are much thinner than the fat strings for low notes (2); and by placing your finger on a string somewhere along the neck of the instrument, you shorten the wavelength and raise the pitch (3). 282 The Doppler Effect So far we have only discussed cases where the source of waves is at rest. Often, waves are emitted by a source that moves with respect to the medium that carries the waves, like when a speeding cop car blares its siren to alert onlookers to stand aside. The speed of the waves, v, depends only on the properties of the medium, like air temperature in the case of sound waves, and not on the motion of the source: the waves will travel at the speed of sound (343 m/s) no matter how fast the cop drives. However, the frequency and wavelength of the waves will depend on the motion of the wave’s source. This change in frequency is called a Doppler shift.Think of the cop car’s siren, traveling at speed , and emitting waves with frequency f and period T = 1/f. The wave crests travel outward from the car in perfect circles (spheres actually, but we’re only interested in the effects at ground level). At time T after the first wave crest is emitted, the next one leaves the siren. By this time, the first crest has advanced one wavelength, , but the car has also traveled a distance of . As a result, the two wave crests are closer together than if the cop car had been stationary. The shorter wavelength is called the Doppler-shifted wavelength, given by the formula . The Doppler-shifted frequency is given by the formula: Similarly, someone standing behind the speeding siren will hear a sound with a longer wavelength, , and a lower frequency, . You’ve probably noticed the Doppler effect with passing sirens. It’s even noticeable with normal cars: the swish of a passing car goes from a higher hissing sound to a lower hissing sound as it speeds by. The Doppler effect has also been put to valuable use in astronomy, measuring the speed with which different celestial objects are moving away from the Earth. EXAMPLE 283 A cop car drives at 30 m/s toward the scene of a crime, with its siren blaring at a frequency of 2000 Hz. At what frequency do people hear the siren as it approaches? At what frequency do they hear it as it passes? The speed of sound in the air is 343 m/s. As the car approaches, the sound waves will have shorter wavelengths and higher frequencies, and as it goes by, the sound waves will have longer wavelengths and lower frequencies. More precisely, the frequency as the cop car approaches is: The frequency as the cop car drives by is: Key Formulas Frequency of Periodic Oscillation Speed of Waves on a String Wave Speed Wavelengt h for the Harmonic Series Frequency for the Harmonic Series Beat Frequency Doppler Shift 284 Practice Questions 1. . Which of the following exhibit simple harmonic motion? I. A pendulum II. A mass attached to a spring III. A ball bouncing up and down, in the absence of friction (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III 2. . If a wave has frequency Hz and speed v = 100 m/s, what is its wavelength? (A) m (B) m (C) m (D) m (E) m 3. . Two strings of equal length are stretched out with equal tension. The second string is four times as massive as the first string. If a wave travels down the first string with velocity v, how fast does a wave travel down the second string? (A) v (B) v (C) v (D) 2v (E) 4v 285 [...].. .4 A piano tuner has a tuning fork that sounds with a frequency of 25 0 Hz The tuner strikes the fork and plays a key that sounds with a frequency of 20 0 Hz What is the frequency of the beats that the piano tuner hears? (A) 0 Hz (B) 0.8 Hz (C) 1 .25 Hz (D) 50 Hz (E) 45 0 Hz 5 How is the lowest resonant frequency, lowest resonant frequency,... proportional to the square root of the mass, waves on a string of quadrupled mass will be traveling half as fast 4 D 28 9 The frequency of the beats produced by two dissonant sounds is simply the difference between the two frequencies In this case, the piano tuner will hear beats with a frequency of 25 0 Hz – 20 0 Hz = 50 Hz 5 B A tube closed at one end can support a standing wave with a node at the closed end and... one-dimensional graph called the electromagnetic spectrum 29 2 A higher frequency—and thus a shorter wavelength—corresponds to a wave with more energy Though all waves travel at the same speed, those with a higher frequency oscillate faster, and a wave’s oscillations are associated with its energy Visible light is the part of the electromagnetic spectrum between roughly 40 0 and 700 nanometers (1 nm = m) When EM waves... mirror, and this small portion taken by itself is roughly flat As a result, we can still think of the normal as the line perpendicular to the tangent plane 29 8 The four basic kinds of optical instruments—the only instruments that will be tested on SAT II Physics are concave mirrors, convex mirrors, convex (or converging) lenses, and concave (or diverging) lenses If you have trouble remembering the difference... can calculate the ratio between and using Snell’s Law: 29 6 Since we know that the ratio of the value for Given / is equal to the ration of , we can now calculate the value for / , and since we know : m/s, we can also calculate that the index of refraction for the liquid substance is 2. 1, while the index of refraction for the gas substance is 1 .2 Total Internal Reflection The sine of an angle is always... through the center of the mirror or lens The vertex, represented by V in the diagram, is the point where the principal axis intersects the mirror or lens The only kind of curved mirrors that appear on SAT II Physics are spherical mirrors, meaning they look like someone sliced off a piece of a sphere Spherical mirrors have a center of curvature, represented by C in the diagram, which is the center of the... sound to be v 28 7 8 What is the frequency produced by the siren? (A) (B) (C) (D) (E) 9 What is the wavelength of the sound produced by the siren? (A) (B) (C) (D) (E) 10 An ambulance driving with velocity where is the speed of sound, emits a siren with a frequency of What is the frequency heard by a stationary observer toward whom the ambulance is driving? (A) (B) (C) (D) (E) Explanations 28 8 1 B Simple... adoption of the wave model of light in the nineteenth century In Newton’s time, light was studied as if it had only particle properties—it moves in a straight line, rebounds off objects it bumps into, and passes through objects that offer minimal resistance While this approximation of light as a particle can’t explain some of the phenomena we will look at later in this chapter, it’s perfectly adequate for... and reflected rays As a result, objects that we see in a different medium—a straw in a glass of water, for instance—appear distorted because the light bends when it passes from one medium to another 29 4 The phenomenon of refraction results from light traveling at different speeds in different media The “speed of light” constant c is really the speed of light in a vacuum: when light passes through matter,... that passes through the focal point will be reflected parallel to the principal axis The focal length, f, is defined as the distance between the 29 9 vertex and the focal point For spherical mirrors, the focal length is half the radius of curvature, f = R /2 Concave Mirrors Suppose a boy of height h stands at a distance d in front of a concave mirror By tracing the light rays that come from the top of . ball bouncing up and down, in the absence of friction (A) I only (B) II only (C) III only (D) I and II only (E) I, II, and III 2. . If a wave has frequency Hz and speed v = 100 m/s, what is its. string? (A) v (B) v (C) v (D) 2v (E) 4v 28 5 4. . A piano tuner has a tuning fork that sounds with a frequency of 25 0 Hz. The tuner strikes the fork and plays a key that sounds with a frequency of 20 0 Hz. What. Series Beat Frequency Doppler Shift 28 4 Practice Questions 1. . Which of the following exhibit simple harmonic motion? I. A pendulum II. A mass attached to a spring III. A ball bouncing up and down,