Grade 11 Quarter 3 Physics Lesson Notes PDF
Document Details
Uploaded by LionheartedKineticArt
Isaac A. David Sr. Memorial School
Tags
Summary
These are lesson notes for Grade 11 physics, covering Chapter 5 on waves. The notes detail wave characteristics, properties, and types, along with examples and experiments.
Full Transcript
CHAPTER P11CH05 5 WAVES Chapter Contents 5.1 Nature, Characteristics and Properties of Waves and Types of Wa...
CHAPTER P11CH05 5 WAVES Chapter Contents 5.1 Nature, Characteristics and Properties of Waves and Types of Waves y Summary y Review Exercises Waves Chapter Outcome Learners will able to: y recognize and appreciate the importance of the nature of wave’s characteristics and components in daily activities. Chapter Objectives Upon completion of this chapter, learners will: y analyze characteristics, concept and components of waves. y elaborate on the properties and the categories of waves. y design methods of production and transmission of sound wave and its application. y compute the speed of sound relative to its temperature. y analyze the Doppler Effect. y distinguish between y Loudness and intensity of sound; y Intensity and intensity level; y Music and noise: y Stringed and non-stringed music instruments. y analyze vibrations in strings and tubes (pipes). 213 Physics — Grade 11 Textbook Introduction The idea of energy has to be explored as much as possible because life on earth much depends on the vital use of energy. It is the transfer of energy that gets things done. In this unit we will investigate an important way of transferring energy. One way of transferring energy is by wave motion. All waves transfer energy. The idea of waves is important in understanding sound. Sound plays so important role in our lives that we communicate each other using it. Periodic vibrations can cause disturbances that move through a medium in the form of waves. Many kinds of waves occur in nature, such as sound waves, water waves and electromagnetic waves. As a wave propagates, it carries energy. This chapter deals about the types, characteristics and properties of waves. 5.1 Nature Characteristics and Properties of Waves Types of Waves The characteristics of waves (wave speed, Key terms amplitude, period, frequency, and wavelength) y Wave: A disturbance in a and the properties of waves (reflection, refraction, medium without carrying the diffraction, interference and polarization) will be particles of the medium. discussed in this section. y Medium: A material through Definition of waves which a wave travels. Waves are disturbances which originate from some vibrating source and propagate through a medium and vacuum. A medium is the substance through which a wave can propagate. Water is the medium of water waves. Air is the medium through which we hear sound waves. The electric and magnetic fields are the medium of light. The most important property of wave is it transfer energy from one point to another. Waves transfer energy without transferring the particles of medium. When a finger is dipped, or a stone is thrown into the water, the water particles are pushed away (disturbed) from their rest position. The surface is now disturbed. It is higher than normal in some places and lower than normal in others. The disturbed water at the point of impact disturbs the water next to it, which in turn disturbs the water next to it, and so on. If you place a cork or anything that floats on water at the middle of the disturbed water it bobs up and down and stays in its position as the ripples pass beneath it. It doesn’t move along with the disturbance. Thus when the 214 Waves disturbance moves forward the particles of the medium vibrate up and down about their mean position of rest but they do not move forward along with the wave. Key terms When the water surface is disturbed only once, we y Pulse: A single disturbance create a pulse through the water. A single non- y Crest: The highest point of a repeating disturbance traveling in a medium is transverse wave. called a pulse. But when the water is disturbed y Trough: The lowest point of repeatedly at regular intervals, we create a wave in a transverse wave. the water. Thus a pulse is a single disturbance while a wave is a disturbance that is repeated at regular intervals. Examples of common waves that we encounter in our daily life are sound and light. All waves can be characterized by the following characteristics: amplitude, wavelength, period, frequency, and speed. D id you know? Ocean waves are caused by wind. When the wind blows over the surface of the ocean, it creates friction, transferring energy from the wind and into the water. The actual wave height is dictated by the wind speed and the size of the area hit by the wind. Characteristics of Waves The most basic wave characteristics that are used to describe a wave are amplitude, wavelength, frequency, period and speed. 1. Crest and Trough Waves in water consist of moving crests and troughs. A crest is a place where the water rises higher than when the water is still. A trough is a place where the water sinks lower than when the water is still. The crest is the highest point of the wave and the trough is the lowest point of the wave. Figure 1 shows the crests and troughs on a wave. Figure 1. Crest and trough in a water wave. Amplitude The maximum displacement of the wave from the equilibrium (or rest position) is called the amplitude 215 Physics — Grade 11 Textbook of the wave. It is the maximum height from the center line to the crest or the trough. Amplitude is directly related to the amount of energy carried by a wave. The greater the force that produces a wave, the greater the amplitude of the wave and the greater the energy carried by the wave. Sounds with greater amplitude will be louder; light with greater amplitude will be brighter. The symbol A is used for the amplitude of a wave. The unit of amplitude is meter (m). D id you know? The tallest wave ever measured is at Lituya Bay, just off the coast of Alaska. The breaking waves from this tsunami created a mountain of water that stretched to a massive 1,720 feet or 524 meters and devastated everything in its path. Wavelength The wavelength (λ) is the distance between any two adjacent points which are in phase. It could be the distance between two adjacent crests or troughs. It is the distance the wave travels in one complete cycle. The wavelength is usually represented by the Greek letter lambda (λ). Figure 2. Wavelength of a wave Period The period (T) is the time taken for a wave to make one complete vibration or it is the time taken for a wave to move a distance of one wavelength. The period is usually represented by the upper case “T.” As the period is time, it is measured in units of time such as seconds or minutes. 216 Waves Frequency Key terms The frequencyof a wave, f, is a measure of how y Amplitude: The maximum many waves pass by, or how many complete displacement from the equilibrium position. oscillations occur, in one second. The frequency y Wavelength: The distance a of a wave is the same as the frequency of the wave travels after one full vibrating body which produces the wave. The cycle. higher the frequency, the shorter wavelength and y Period: The time taken for greater energy. Frequency is often represented by one full cycle. the lower case “f.” The unit of frequency is hertz y Frequency: The number of (Hz) which is equal to one wave per second. complete oscillations made per unit time. We can easily see that period and frequency are y Speed: The distance the inversely related. The period is the reciprocal of wave travels per unit time. the frequency and vice versa. 1 1 T= or f = f T Speed The speed of a wave is how fast the disturbance of the wave is moving. The speed of a wave refers to the distance traveled by a given point on the wave in a given interval of time. Knowing wavelength and wave period, the wave speed (v) can be calculated by dividing wavelength (λ) by period (T). v T 1 Since, = f , it implies that the speed (v) of a wave is equal to the product of T frequency and wavelength. v f T The above equation is known as the wave equation. Note that the speed of a wave depends on the properties of the medium through which the wave propagates and not on the mechanism that is generating the wave. 217 Physics — Grade 11 Textbook For example, the speed of sound waves depends on the pressure, density, and temperature of the air through which they propagate, and not on what is making the sound. Experiment: Calculating the speed of a water wave using a ripple tank. Aim of the experiment: To measure the frequency, wavelength and speed of waves in a ripple tank. Materials required: 1. Ripple tank plus accessories 2. Suitable low voltage power supply 3. Meter ruler Light Source Support Wooden Bar Supported by Elastic Bands Water Ruler Screen Wavefronts Read these instructions carefully before you start work. Set up the ripple tank. A large sheet of white card or paper needs to be on the floor under the tank. Method: 1. Set up the ripple tank as shown in the diagram with about 5 cm depth of water. 2. Adjust the height of the wooden rod so that it just touches the surface of the water. 218 Waves 3. Switch on the lamp and motor and adjust the speed of the motor until low frequency waves can be clearly observed. 4. Place a meter ruler at right angles to the waves shown in the pattern on the card. 5. Measure across as many waves as possible. Then divide that length by the number of waves. This gives the wavelength of the waves. 6. Count the number of waves passing a point in ten seconds then divide by ten to record frequency (the frequency of waves is also determined by frequency of vibration of the dipper). 7. Calculate the speed of the waves using the equation: Wave speed = frequency × wavelength v=fλ Examples If a wave has a wavelength of 20 meters and a period of 4 seconds, what is its speed? Solution l 20 m v= = = 5.0m/s T 4s Examples A wave has a distance of 1.6m between consecutive crests. If it takes 4 sec for one complete vibration, what are (a) the frequency and (b) the speed of the wave? Solution λ = 1.6m; T = 4s 1 1 (a) f = = = 0.25 Hz T 4s (b) v = λ f = 1.6m × 0.25 Hz = 0.4 m/s Examples The FM radio station broadcasting for Addis Ababa and the surrounding has a frequency of 97.1MHz. What is the wavelength of this radio wave? Solution f = 97.1MHZ = 97.1 × 106HZ, C = 3 × 108m/s v 3 ´ 108 m/s l= = = 3.1m f 97.1 ´ 106 Hz 219 Physics — Grade 11 Textbook Exercises 1. Define a wave motion. 2. Does a wave motion involve the motion of the medium as a whole, as the wave progresses in the medium? Explain. 3. Define (a) pulse, (b) wave. 4. Define the following terms: (a) crest, (b) trough, (c) amplitude, (d) frequency, (e) period, and (f) wavelength 5. The distance between a crests and adjacent trough of a wave motion is 1m. A person counted 11 crests passing a given point in the medium in one second. What is the speed of the wave? Wave fronts and Rays Wave front is defined as the imaginary surface constructed by the locus of all points of a wave that have the same phase. This could be where all the crests are, where all the troughs are or any phase in between. Wave fronts are useful for showing how waves move in two dimensions. The length between two lines on a wave front is exactly one wavelength. Ray Wavefront Source Straight Wavefront Circular Wavefront Figure 3. (a) Plane wave front and (b) spherical wave front Key terms The direction of travel of the wave fronts is shown y Wave front: A wave front is by a straight line with an arrow. It is called a ray. the set of all locations in a A ray is a line extending outward from the source medium where the wave is at and representing the direction of propagation of the the same phase. wave at any point along it. Wave fronts and rays are y Ray: Rays are lines that show always perpendicular to each other. the direction of propagation of a wave. 220 Waves Properties of Waves Key terms The properties of a wave include: reflection, y Reflection of waves: The refraction, diffraction, interference and polari- turning back of a wave in zation. reaching to a boundary through which they cannot Reflection of waves pass. When a wave is incident on the boundary separating the two media it is reflected (turned) back into the first medium without any change in the speed or wavelength of the wave. This Normal Incident Relfected Ray phenomenon is known as the reflection of Ray waves. Sound waves bounce off walls, light Normal waves bounce off mirrors, and radar waves bounce off planes. Bats use reflection while they fly at night and avoid things as small as Incident telephone wires. Figure 4 Shows the incident Waves and reflected wave fronts represented by Barrier their crests. Figure 4. Incident and Reflected Ray. Activity 1 Compare the wavelength, frequency and velocity of the incident and reflected wave. Refraction of Waves Key terms The speed and wavelength of a wave depends on the y Refraction: The bending medium through which the wave travels. Therefore, of a wave in moving from when a wave approaches a boundary obliquely (at one medium to another of an angle), their direction is changed, because of different density. the change in the speed of the wave. This change in the direction of waves at the boundary between two different media is known as refraction. During refraction, the speed and wavelength of the wave changes, but the period and frequency of the wave remain the same. When you look at a pencil that emerges from water it looks like it is bent. This is because the light from below the surface of the water bends when it leaves the water. Your eyes project the light back in a straight line and so the object looks like it is a different place. 221 Physics — Grade 11 Textbook New Medium Refraction of pencil Original Wave Refracted Wave Figure 5. Refraction of waves Refraction of Water Waves A deep water and a shallow water acts like a different media. It is observed that as the waves pass into the shallow water from the deep water the wavelength of the wave decreases (see Figure 6). Deep Shallow Water Water Figure 6. Refraction of water Wave. Activity 2 From Figure 6 we see that the wavelength of a water wave shortens when it enters the shallow water. What does this tells you about the speed of water wave in deep and shallow water? Key terms There are two types of refraction: Refraction y When a wave slows down towards the normal and refraction away from in the new medium it bends the normal. When the speed of a wave decreases towards the normal. on entering the second medium, its wavelength y When a wave speeds up in also decreases. Therefore, the angle of refraction the new medium it bends will be less than the angle of incidence (r < i). As away from the normal. a result the wave refracts towards the normal, see Figure 7 (a). But if the speed of a wave increases on entering the second medium, its speed and hence it wavelength increases. Thus, the angle of refraction will be greater than the angle of incidence (r > i). As a result it refracts away from the normal, see Figure 7 (b). 222 Waves Incident Normal Incident Normal Ray Ray I I Air Glass Glass Air R R Refracted Refracted Ray Ray A) B) Figure 7. a) Refraction towards the normal b) Refraction away from the normal Activity 3 1. How will (a) mechanical waves, and (b) electromagnetic waves refract towards the normal and away from the normal? 2. A wave that moves from medium A to B is refracted as shown in the figure below. In which medium is the speed of the wave greater and in which medium is smaller? Incident Ray Normal Mediuum A Mediuum B Refracted Ray Examples Water wave propagates with a speed of 0.8m/s and wavelength 4cm in the shallow section of the ripple tank. When the wave enters into the deep section its wavelength becomes 6cm. What is the speed of the wave in the deep section? Given: V1 = 0.8 m/s, λ1 = 4cm = 0.04m, λ2 = 6cm = 0.06m. 223 Physics — Grade 11 Textbook Solution Since the frequency of the wave is the same in both media. f1 = f2 V1 V = 2 l1 l2 V1l 2 0.8m/s ´ 0.06m Þ V2 = = = 1.2m/s. l1 0.04m Diffraction of Waves Diffraction happens when waves move through Key terms a gap or around an obstacle. The wave fronts y Diffraction of waves: is the change their shape when they pass through the spreading out of a wave gap. They spread out behind the gap, and change after passing through a small their direction. When the gap is very narrow as opening or bouncing of a in Figure 8 (a), the gap seems to act as a source wave around an obstacle. of circular waves. When the gap is wider as in Figure 8 (b) the effect are not very noticeable. This change of direction of the wave fronts when they pass through a gap is called diffraction. Diffraction effects are greater when the opening between the objects is about the same size as or smaller than the wavelength of the waves. If you place a solid obstacle in the straight waves, a) Diffraction of b) Diffraction of you will see that the wave bend behind the waves through waves through small opening large opening obstacle as in Figure 8 (c). They are able to pass into the region behind the obstacle, even though they are not passing through it. Thus we can also define diffraction as the bending of a wave front into the region behind the obstacle. It is because of diffraction that sounds can sometimes be heard around corners and in the shadow of buildings. It is also why radio signals, particularly those c) Diffraction around an obstacle with a long wavelength, can be received in the Figure 8. shadow of hills. 224 Waves The frequency of the wave depends on the frequency of the source generating the wave, which is constant in this case. Therefore, during diffraction the speed and wavelength do not change, but the direction of the wave changes as they pass the edge of the obstacle. Note that waves that have a longer wavelength are diffracted more strongly than waves of smaller wavelength. Activity 4 1. Suppose your friend is calling you being behind a house through which there is no gap for sound to pass through. Then how do you hear his call? 2. Why is the diffraction of sound more easily observable than the diffraction of light? Interference of Waves When two or more identical waves (same wavelength, amplitude and frequency) travel in the same medium at the same time, they interfere each other. Such a mixing up of waves is called interference. When the two waves are in phase (the crest of one meets the crest of the other or the trough of one meets the trough of the other) they reinforce each other. In such cases the interference is said to be constructive. When the interfering waves are out of phase (the crest of one coincides with the trough of the other) they tend to cancel each other and the interference is said to be destructive. The amplitude of the resulting wave will depend on the amplitudes of the two waves that are interfering. During constructive interference the amplitude of the resulting wave is larger than the amplitude of either of the interfering waves, see Figure. 9 (a). During destructive interference the amplitude of the resultant wave is less than the amplitude of the interfering waves, see Figure. 9 (b). If they have the same amplitude, the resultant wave will have zero amplitude while they cross over each other. Such kind of interference is known as complete destructive interference. The Superposition Principle The amplitude of the resultant wave is determined by use of the superposition principle stated below. “When two or more waves travelling in the same medium at the same time interfere, the resulting displacement of the new wave is the algebraic sum of the displacement caused by the individual waves.” 225 Physics — Grade 11 Textbook Figure 9. Constructive and destructive interference. Interference of Water Waves Key terms A vibrating dipper on a water surface sends out y Interference of waves: when circular waves. Two sets of circular waves are two or more identical waves produced in this way in a ripple tank. These two travel in the same medium at sets of waves pass through each other continuously the same time they mix up and produce an interference pattern (see Figure 10). together Demonstration: To study the properties of water waves with a ripple tank: A ripple tank is a device used in demonstrating wave properties such as reflection and refraction. It consists of a shallow transparent tray of water with a point light source above it and a white screen on the floor below (Figure 11). Before adding water, the tray is leveled with a spirit-level to ensure a uniform water depth of rather less than 1 cm. Straight parallel waves may be produced by a horizontal Figure 10. Interference of metal strip, or circular waves by a vertical ball-ended rod. water waves When either of these is dipped into the water, a pulse of ripples is sent across the surface. The bar is moved up and down by the vibrations of a small electric motor having an eccentric metal disc on its rotating spindle. A rheostat in the motor circuit controls the speed and hence the frequency of the waves sent out. Owing to the lens effect of the wave crests and troughs, the light source produces a bright and dark wave pattern on the white screen. 226 Waves Figure 11. The ripple tank Reflection Place in a barrier (metal bar) somewhere in the tank. Then send the ripples towards the barrier. The ripples reflect from the bar. Reflection of straight wave front by a straight barrier is shown in the figure Figure 12. Figure 12. Reflection in a ripple tank Refraction Refraction of water waves can be observed in a ripple tank if the tank is partitioned into a deep and a shallow section. A piece of glass slab is placed in the ripple tank 227 Physics — Grade 11 Textbook so that two regions with different depth are created. Waves traveling from the deep end to the shallow end can be seen to refract. Region B Region A Region B Glass Block Region A Figure 13. Refraction in a ripple tank Diffraction Put two solid barriers in a ripple tank one pair with a narrow gap and the other with a wider gap between them. Then let a straight wave to pass through the gaps. The waves that comes through the hole no longer looks like a straight wave front. It bends around the edges of the hole. If the hole is small enough it acts like a point source of circular waves. Figure 14. (a) Barrier with a small gap (b) Barrier with a wider Interference Plane waves in a ripple tank strike two narrow gaps. Each gap produces circular waves beyond the barriers, and the result is an interference pattern. The pattern is the same as would be produced by two dippers vibrating in phase at the gaps, but fainter. 228 Waves Figure 15. (a) Interference of water waves with (b) two dippers Polarization A wave that is vibrating in more than one plane Key terms is referred to as unpolarized. Polarized waves are y Polarization is the process waves in which the vibrations occur in a single of transforming unpolarized plane. The process of transforming unpolarized wave into polarized wave wave into polarized wave is known as polarization. Light waves are often polarized using a polarizing filter. Only transverse waves can be polarized. Longitudinal waves, such as sound, cannot be polarized because they always travel in the same direction of the wave. Figure 16. Polarization of wave Exercises 1. Define the terms: (a) reflection; (b) refraction; (c) diffraction; (d) interference; and (e) polarization. 2. Explain how refraction towards the normal and refraction away from the normal takes place. 229 Physics — Grade 11 Textbook 3. What is the difference between diffraction through a small opening and a large opening? 4. When are the waves interfering (a) constructively? (b) destructively? Types of Waves Not all waves are the same type. Some can travel Key terms through vacuum but others cannot. Some waves y Mechanical waves: Waves that require a material travel parallel to the direction of propagation of medium to transmit the the particles of the medium while others move disturbance. perpendicular to the direction of propagation of the y Electromagnetic waves: medium. These and other types of waves will be Waves that do not require discussed in this section. material medium (that can 1. Based on their ability or inability to travel propagate through vacuum) through a vacuum, waves are classified into two: mechanical and electromagnetic. A mechanical wave is a wave that is not capable of transmitting its energy through a vacuum. Mechanical waves require a medium in order to transport their energy from one location to another. Sound waves, water waves and waves that travel along a spring (slinky) or a string are all examples of mechanical waves. Mechanical waves cause oscillations of particles in a solid, liquid or gas and must have a medium to travel through. All mechanical moves can not travel through vacuum. Electromagnetic waves are waves that do not require a material medium to transmit the disturbance. They can propagate through transparent materials and can also propagate easily through vacuum. Light wave, radio and TV waves, microwaves, infrared, UV – rays, x – rays and gamma rays are all electromagnetic waves. All electromagnetic waves have the same velocity in moving through vacuum (or air). This speed is what we commonly refer as speed of light= 3.0 ×108 m s. Electromagnetic waves are produced by the periodic changes that take place in magnetic and electric fields and therefore known as Electromagnetic Wave. 2. Based on the basis of the direction Key terms of movement of the individual y Transverse waves are particles of the medium relative to the waves in which the particle direction that the waves travel waves of the medium oscillate can be classified as transversal or perpendicular to the direction longitudinal. of propagation of the wave. 230 Waves A transverse wave is a wave in which particles of the medium move in a direction perpendicular to the direction that the wave moves. For example, if you attach a horizontal spring to a wall and move the other end up and down vertically while the wave travels horizontally along the spring (Figure 17). In this case, the particles of the medium move perpendicular to the direction that the pulse moves. This type of wave is a transverse wave. Figure 17. Transverse wave in a) spring b) string You will also get the same kind of wave if you disturb one end of a string up and down, the other end fixed to something. Light waves, waves that travel along ropes and waves across the surface of water, and all electromagnetic waves are transverse. Parts of transversal waves For transversal waves the wavelength is the distance between two consecutive crests or troughs or the distance between any two points which are in the same phase (having the same position and direction of propagation, see Figure 18. Figure 18. Characteristics of a transverse wave. Longitudinal waves A longitudinal wave is a wave in which particles of the medium move in a direction parallel to the direction that the wave moves. In the production of sound waves by clapping your hands, the air molecules oscillate about an equilibrium position in the same direction as the wave propagates, Figure 19(a). 231 Physics — Grade 11 Textbook Figure 19. A longitudinal wave in a) air b) spring Longitudinal waves are also produced by a spring. Fix one end of a long spring (slinky) to a rigid body. Then move the free end back and forth along its length in a regular pattern. The wave will travel along the length of the spring parallel to the disturbance see Figure. 5.19 (b). Key terms In longitudinal waves no crests and troughs are y Longitudinal waves: Waves produced. But in course of their movements, the in which the particle of the particles come closer together in some places than medium oscillate parallel to their normal separation in the medium. In this the direction of propagation region a compression is formed. In other places of the wave. the particles move farther apart than their normal separation and a rarefaction is formed. The distance from a compression to a compression, or from rarefaction to rarefaction gives the wavelength, l, of the wave, see Figure 20. Figure 20. Compression and Rarefaction in a longitudinal wave. 1. Based on their appearance waves are classified as traveling wave and standing wave. A traveling wave is a disturbance that travels through a medium. Consider the ripples (waves) made by a rock dropped in a pond (Figure 5.21). The ripples travel outwards from where the rock was dropped. The individual water molecules will move in small circles about an equilibrium Figure 21. A water wave position, but they do not move along with the waves. 232 Waves Standing waves Standing waves are waves that do not appear to be propagating. They are also called stationary waves. These waves arises when a wave meets its own reflection under the right circumstances. Standing waves do not transport energy through the medium. For example, a vibrating string on a violin is a standing wave. Figure 22. A standing wave by a string Examples The speed of sound in water is 1500 m/s. If a tuning fork vibrating at 600Hz is immersed in water, what is the wavelength of the resulting wave? V = 1500m s, f = 600Hz V 1500m/s l= = = 2.5m f 600Hg Activity 5 Longitudinal waves can be set up through solid, liquid or gases, but transverse waves are produced only in a solids and liquids but not in gases. Explain why. Exercises 1. What are the requirements for the production of mechanical waves? 2. Distinguish between: (a) mechanical and electromagnetic waves (b) Longitudinal and transversal waves (c) Travelling and standing waves. 3. Give at least three examples of: (a) mechanical waves. (b) Electromagnetic waves. (c) Transversal waves. (d) Longitudinal waves. 4. How is energy transferred by mechanical waves? 5. Are water waves longitudinal waves or transverse waves? Explain. 233 Physics — Grade 11 Textbook 6. A wave traveling in the + x direction is shown in the figure. What is (a) the amplitude (b) the wavelength of the wave? 7. If the distance between the points A and B in the figure is 60cm, what is (a) the period? (b) the frequency? (c) the wave length? (d) the speed of the wave? y (cm) 10 x (cm) 2 4 6 8 10 12 14 16 18 -10 60 cm t(s) 0.2 0.6 1.0 1.4 1.8 2.2 B Sound Waves Key terms Sound waves are the most important example of y Source of sound: A vibrating longitudinal waves. In this section we discuss body that could produce the characteristics of sound waves, how they are sound. produced, what they are, and how they travel y Compressions: the particles through matter. We then investigate what happens are close together (high when sound waves interfere with each other. pressure region). Sound y Rarefactions: the particles are spread apart (low Sound is a form of energy. From the source of sound pressure region). this energy is propagated through the surrounding medium in the form of waves. When the vibration of the source has a frequency between 20Hz and 20,000Hz these compressional waves are able to cause a sensation of hearing and are referred to as sound waves. In the previous section, we have seen that sound is a mechanical longitudinal wave. Thus when the source of sound vibrates, it makes compressions and rarefactions 234 Waves which spread out through medium, which may be a solid, liquid or gas. Sound cannot travel through a vacuum because it needs particles to vibrate. D id you know? With the help of sounds, most animals can detect dangers and hazards before they affect them. Dogs and cats can hear much higher pitched sounds than we can. Dolphins, for example, can’t hear sounds as low as we can, but can hear high sounds of over 100,000 Hz. Animals that have large ears can hear better as compared to animals with small ears. Flies cannot hear any kind of sound. Not even their own buzzing. The production and Transmission of Sound Any source of sound (like a tuning fork, a drum, a guitar,...) is always in a state of vibration. When the source of sound vibrates it causes successive compressions and rarefactions. When the molecules are pushed together by a vibrating source a compression area (a region of higher pressure) will be formed. These compressed molecules of the air will pass the compression to the adjacent molecules of the air. These molecules in turn compress the next adjacent molecules and so on. In this way this compressed wave travels away from the source into the surrounding air. Next to the compression we have rarefactions where the air molecules are far apart. This expansion of few air molecules into a free space causes rarefaction (a region of low pressure), see Figure 23. Rarefaction Compression Tuning Frok Figure 23. When a source of sound vibrates it forms successive compressions and rarefactions. Sound is produced by vibrating bodies. The objects that produce sound are called sources of sound. Vibrating strings such as violin and human vocal cords; and vibrating plates and membrane such as drum and loudspeakers are some of the sources of sound. 235 Physics — Grade 11 Textbook D id you know? Sound energy can be used in medicine as well as for therapeutic purposes. For example, sound vibrations are utilized in ultrasounds. Activity 6 Take a drum. Touch it when not in use. Again touch it when producing sound. What do your hands feel when drum is beaten and produce sound? Can you feel the skin of the drum vibrating? Place small pieces of papers on the drum and play it. Observe what happens to the pieces of paper when the drum is beaten. What did you conclude from your observation Activity 7 Observing sound propagation in solids (a) Hold one end of a meter stick against your ear. Let your friend slightly scratch at the other end. You may hear the scratching quite clearly although it is hardly audible through the air medium. (b) Your may have played with a tin-can-telephone during your child hood. It could be a surprising experience to those students who use the telephone for the first time. Figure 24. Tin-can-telephone Activity 8 Propagation of sound through liquids Take a glass and fill it with water. Take a bell and ring it inside the water. Ask your friend to listen to the sound by keeping his / her ears touching walls of the glass. The conclusion is that sound propagates through matter in all the three states – solid, liquid and gas. Figure 25. Sound throught liquid 236 Waves Activity 9 Sound des not travel through vacuum To show Sound needs a medium to travel. Take an electric bell and airtight jar. The electric bell is suspended inside the airtight bell jar. With air still in jar ring the bell. Now take out air slowly by using vacuum pump. Ring the bell again. What difference did you observe? Observation: Sound of bell can be heard when air is inside the jar. As more and more air is removed from the glass jar, the sound of ringing bell becomes fainter and fainter. And when all the air is removed from the glass jar, no sound can be heard at all. This shows that sound can’t travel through vacuum. Figure 26. Sound can’t travel through vacuum Characteristics of Sound Like any kind of waves, sound also have five main characteristics: wavelength, amplitude, frequency, time period and velocity. Wavelength: In transverse waves we have measured the wavelength of the wave from crest to crest or trough to trough. But in longitudinal waves like sound, we measure the wavelength as the distance between two consecutive compressions or rarefaction which are in same state of vibration, center to center or end to end, see Figure 27. 237 Physics — Grade 11 Textbook Figure 27. A traveling wave traveling through a spring Amplitude: The amplitude is the maximum displacement of the particles the sound wave disturbed. Frequency: The frequency refers to the number of sound waves a source produces per second. Sound frequency is not dependent upon the medium the sound is passing through. The S.I unit of frequency is hertz or Hz. A vibrating body emitting 1 wave per second is said to have a frequency of 1 hertz. Sometimes a bigger unit of frequency is known as kilohertz (kHz) that is 1 kHz = 1000 Hz. Time Period – The time period is the time required to produce a single complete cycle. Each vibration of the source producing the sound is equal to a cycle. The time period is the reciprocal of the frequency. Speed of Sound in Different Media The speed of sound depends on the medium through which it is propagating. We have already seen that sound travels best through media that are dense. Table 1 shows this fact. Table 1 Speed of sound in different media. S.No. Nature of the Name of the medium Speed of sound (in medium m s-1) 1 Copper 5010 2 Solid Iron 5950 3 Aluminium 6420 238 Waves 4 Kerosene 1324 5 Liquid Water 1493 6 Sea water 1533 7 Air (at 0 C) o 331 Gas 8 Air (at 20 C) o 343 As you see from the table the speed of sound tends to increase in a denser medium. Solids are denser than liquids and liquids are denser than gases. Therefore, the speed of sound is greater in solids than in liquids and greater in liquids than in gases. Solids, liquids and gases transmit sound, but sound can’t travel through vacuum. Activity 10 Discuss in group why Sound waves travel faster in liquids than gases, and fastest of all through solids. Speed of Sound in Air The speed of sound in air depends on the temperature of the air. When the temperature of a medium increases, the kinetic energy of all the molecules in the medium increases and hence the speed of sound in moving through this medium also increases. For sound traveling through air, the relationship between wave speed and medium temperature is: Tc v 331.5 1 m/s 273.15 o C where Tc is the temperature in °C and 331.5m/s is the speed of sound at 0°C. This equation is approximated as: v (331.5 0.6Tc ) m / s From this equation we see that, at a temperature of 00C, sound waves travel with a speed of 331.5m/s. The speed of sound increases by 0.6 meters per second (m/s) for every degree-Celsius increase in temperature. Examples What is the speed of sound in air at room temperature (200C)? Solution Using the above equation v(331.50.620)m/ s343.5m/ s 239 Physics — Grade 11 Textbook Speed of Sound in Solids and Fluids Sound is a mechanical wave. Thus its speed depends on how close the particles of the medium are through which it is moving. The speed, υ,of a sound wave in various materials is related to the elasticity and density, ρ, of the material by the equation Y For solids: υ = ρ Where Y= young’s modulus of the solid, and ρ is the density of the solid. B For fluids: Where B = the bulk modulus of the fluid (liquid or gas), and ρ is the density of the fluid. Examples Determine the speed of sound in water. Solution Water is a fluid, and its bulk modulus is Bw = 2.1 × 109 N/m2, and its density is ρw = 103 kg.m3. Thus, the speed of sound in water is Bw 2.1 ´ 109 N/m 2 vw = = 1449.14 m/s rw 103 kg/m3 Examples Find the speed of a sound pulse in an aluminum bar struck at one end with a hammer. Solution Aluminum is a solid and its Young modulus is YAl = 7 × 1010 N/m2, and its density is ρAl = 2.7 × 103 kg.m3. Thus, the speed of sound in Aluminum is YAl 7 ´ 1010 N/m 2 v Al = = 5091.75 m/s rAl 2.7 ´ 103 kg/m3 Reflection of Sound – Echo Because sound is a wave it will be reflected back from a tall building, a mountain even from the wall of an auditorium. If sound is reflected back from an obstacle to the observer in such a way that he/she hears it distinctly as a repetition of the original sound an echo is said to be produced. 240 Waves Note that sound waves are reflected off hard surfaces like glass and wood. Sound does not reflect off from soft surfaces like cloth and rubber. These surfaces absorb sound. That is why the walls of auditoriums and cinema halls are covered by a soft material like curtain, to decrease the reverberation time (the time taken for sound to die away in a building). If the echo time (t), that is the time taken for the sound to move from the source to the obstacle, then back to the source is measured, the speed of sound at that temperature is determined by the equation v = total distance travelled time taken The distance the sound travels will be 2s, because the sound wave will travel to the obstacle and reflected back to the source., where is the distance between the soruce of sound and the obstacle. Thus, v = 2s t Applications of Reflection of sound – Echo Echoes have many uses some of them are mentioned below: 1. Determination of the velocity of sound in air. As discussed earlier if the distance between the source and obstacle, and echo time is measured, the velocity of sound at that temperature can be determined by using the equation. 2s v= t 2. Determination of the depth of an ocean (sea). The depth of oceans or seas are determined using a device called sonar. A sonar makes underwater sound waves, which travels to the bottom of the sea. This wave is then reflected off the seabed and returns to the ship. The time for the sound wave to return back to the ship (echo time) is measured. By use of the speed v of sound in sea water, we can then determine the depth of the sea or ocean using the equation. vt s= 2 3. The working of a stethoscope is also based on the reflection of sound. In a stethoscope, the sound of the patient’s heartbeat reaches the doctor’s ear by multiple reflections of sound. 4. The soundboard is based on the reflection of sound. Sound board is a big concave board and is set in such a fashion behind the stage that speaker is at the 241 Physics — Grade 11 Textbook focus. Sound coming from speaker falls over sound board and gets reflected towards the audience. As a result, the audience sitting in the hall even at far distance from the speaker can clearly hear what the speaker is saying. Examples A depth measuring device produces sound signals at a frequency of 3,000Hz and having a wavelength of 0.5m in ocean water. These waves are reflected from the bottom of an ocean and received by the device at the top after 2sec. What is the depth of the ocean? Solution f = 3000Hz; λ = 0.5 m; t = 2 sec vt flt s= = 2 2 3000Hz ´ 0.5m ´ 2s s= = 1500m 2 Noise and music Key terms We hear different types of sounds around us. Some y Noise: Unpleasant sound sounds are pleasant to the ear, whereas some are y Musical note: sounds which are pleasant to hear not. Such unpleasant sounds are called noise. The sounds which are pleasant to hear are called music. Some sources of sound produce pleasant sounds or musical notes while others produce unpleasant sound or noises. Musical notes are sounds that are easy to listen to because they are rhythmic. The waves that carry these notes change smoothly and their wave patterns occur at regular interval of time. Figure 28. Noise and music D id you know? The majority of cows that listen to music end up producing more milk than those who do not. 242 Waves Activity 11 Discuss in group why two astronauts talking on the surface of the moon cannot be heard each other. Loudness and pitch The Loudness of a sound is the magnitude of the auditory sensation produced by the sound. It depends on the amplitude of the sound wave. The greater the amplitude of the sound the louder the sound will be. Actually, loudness is a difficult quantity to measure because it depends on the Judgment of the listener rather than any physical measurement. The pitch of a sound is the subjective feeling experienced by a listener due to the frequency of the sound. It is associated with the physical Lower Pitch Higher Pitch characteristics of frequency of vibration. The higher the frequency of the sound waves the higher Figure 29. Pitch and frequency their pitch. Audible and Inaudible Sounds All sound waves cannot produce sensation of hearing. Sound waves that are able to produce sensation of hearing are called Audible, and those waves that cannot produce sensation are Inaudible. Audibility depends on frequency. The frequency of audible sound lies between 20Hz to 20,000Hz. The lower limit of audibility is 20Hz and the upper limit of audibility is 20,000Hz. A sound wave whose frequency is less than the lower limit of audibility is called Infrasonic Wave. A sound wave whose frequency is greater than the upper limit of audibility is called Ultrasonic. D id you know? Bats produce ultrasound wave signals. When these signals bounce off objects, they return echoes, which helps them to know whether or not an obstacle is in the way. Key terms Intensity and Intensity Level y Intensity is a measure of the rate of sound energy (power) Any vibrating source can set up sound waves in air. per unit area. The greater the energy of vibration the greater the 243 Physics — Grade 11 Textbook amount of energy carried by the sound wave. So, we can see that the amount of energy carried by a sound wave depends on the amount of energy with which the source vibrates. The amount of energy (E) transferred per unit time (t) per unit Area (A) perpendicular to the direction of motion is called the intensity (I) of the wave. Since energy per unit time is power, intensity can also be defined as power per unit area. Energy / time Power I= = Area Area P I= A The SI unit of intensity is watt per meter square (W/m2). Because sound waves spread out at the same speed in all directions from a vibrating source, they spread out as a spherical wave front. The spherical volume around a vibrating source in a medium will be filled with the energy from the vibrating source. The area of a sphere is given by the equation, A = 4πr2. Thus, for a point source that sends out spherical sound waves, the intensity (I) of the sound at a distance, r, from the source is I= P = P A 4pr 2 The above equation shows that the intensity of a sound wave is inversely proportional to the square of the distance r between the source and listener, see Figure 30. Intensity at Sphere Area Surface of Sphere I 2 4πr P 9 I =I Source Power 4pr 2 4 P I r The energy twice as far from 2r the source is spread over four 3r times the area, hence one-fourth the intensity. Figure 30. The intensity of a sound is inversely proportional to the square of the distance 244 Waves Inverse square law P From the above equation of intensity I , you can understand that the 4r 2 intensity of a sound wave is inversely proportional to the square of the distance from the sound source. This principle is known as the inverse square law. 1 I 2 r Ir 2 cons tan t The intensity of a sound wave can also be determined using the equation 1 I 2 A2 2 Where ρ is the density, υ the speed, ω the angular velocity, and A the amplitude of vibration of the wave. From this equation we see that the intensity of a sound wave is directly proportional to the square of the amplitude of vibration of the source, IαA2. Examples The total effective area of the auricle of a listener ear is about 40cm2. (a) How much power enters the ear of a person from a sound of intensity 10-6w/m2. (b) How long will it take for one micro joule of energy to enter the listener’s ear? Solution A = 40cm2 = 40 × 10-4m2 = 4 × 10-3m2; I = 10-6 w/m2; E = 1µJ = 10-6J (a) P = I × A = 10-6 w /m2 × 4 ×10-3 m2 = 4 × 10-9 W W E (b) P= = t t E 10-6 J t= = = 0.25 ´ 103s = 250s P 4 ´ 10 W -9 Examples A sound wave with an intensity of 10-8 w/cm2 is incident on an eardrum of area 4 × 10-4 m2. How much energy is absorbed by the ear drum in 5 min? 245 Physics — Grade 11 Textbook Solution I = 10-8w/cm2 = 10-8 W/(10-2 m) 2 =10-4W/m2, A = 4 × 10-4m2, t = 5min = 300s E/t I= A E = IAt = 10-4 w/m2 × 4 × 10-4m2 ×300sec = 1.2 × 10-5J Examples The sound from an organ pipe has an intensity of 100 W/m2 at a point 1m from the open end of the pipe. What is the intensity of the sound at a distance of 20m? Solution I1r12 = I 2 r22 I1r12 100w/m 2 ´ (1m) 2 I2 = 2 = 2 = 0.25 w/m 2 r2 (20m) Intensity Level The loudness of a sound depends upon the listener. It is quite subjective. So we can’t measure it. But it is easy to accept that as the intensity of a sound increases the sound becomes more louder. The relationship between intensity and loudness is logarithmic. If we take the intensity of the least audible sound as a reference level, (the threshold of hearing (I0), the intensity level of a sound, having intensity I, is given by I Intensity level()10 log Io Where the reference intensity (Io = 10-12 W/m2) is the intensity of the threshold of hearing: Io = 10-12 W / m 2 = 10-16 W / cm 2. We express the loudness of the sound coming from various sources in decibels (dB). If a person is being exposed to the sound of 80dB continuously it may lead to hearing problems. A whisper is about 30 dB, normal conversation is about 60 dB, and a motorcycle engine running is about 95 dB. Loud noise above 120 dB can cause immediate harm to your ears. 246 Waves D id you know? The cry of a human baby, is about 115 decibels, it is louder than a car horn. Examples What is the intensity level of a sound of intensity 10−10 W/m2? Solution I = 10-10 w/m2, Io= 10-12 w/m2 æ Iö æ 10-10 w/m 2 ö 2 b = 10log ç ÷ = 10log10 ç -12 2÷ = 10log1010 è Io ø è 10 w/m ø = 10 × 2 log 1010 = 20dB Examples What is the intensity of a sound wave whose intensity level is 70 dB? Solution β = 70dB, Io = 10-12w/m2 æ Iö 70 = 10 log10 ç ÷ è Io ø æ Iö 7 = log10 ç ÷ è Io ø I 107 = Io I = 107× Io =107×10-12W/m2 =10-5W/m2 The ear and hearing The loudness of a sound depends on intensity and frequency. For a given frequency an increase in intensity produces an increase in loudness, but the sensitivity of the ear is so different in the various frequency ranges. Figure 5. shows a diagram giving the relation between frequency, intensity and hearing. The range of frequencies and intensities to which the ear is sensitive is conveniently represented by a hearing curve shown in Figure 31. 247 Physics — Grade 11 Textbook Figure 31. The Hearing curve From Figure 31 we see that the ear is most sensitive to frequencies between 2000Hz and 5000Hz. Intensities below the line indicating the threshold of hearing are insufficient to produce any sensation of hearing. The curve indicates that the normal ear is sensitive in the frequency range between 20Hz and 20,000Hz. The lower curve called the threshold of hearing indicates the minimum intensity level at which sound waves of different frequencies can be heard. The upper limit called the threshold of pain indicates the upper intensity level for audible sounds. The upper curve represents the intensity level of the loudest sound that can be tolerated. This upper limit of audibility is called the threshold of pain. It is about 130dB. The lower limit of audibility corresponds to 5dB. This valve varies from person to person. It depends on age, the shape and health of the ear. Exercises 1. Define sound. 2. How are sound waves propagated? 3. Why is sound not transmitted through vacuum? 4. Distinguish between (a) Intensity and loudness. (b) Frequency and pitch. (c) A noise and a musical note. (d) Threshold of pain and threshold of hearing. 5. What is the range of audio frequencies? 6. What name is given to sound vibrations 248 Waves (a) Below the audio range? (b) Above the audio range? 7. What is the speed of sound at 25oC? 8. How long does it take sound to travel 3.5 km in air at a temperature of 30oC? 9. What is the wavelength of sound in air at 30oC if its period is 0.4s? 10. Suppose a man stands at a distance from a cliff and claps his hands. He receives an echo from the cliff after 4 second. Calculate the distance between the man and the cliff. Assume the speed of sound to be 343 m/s. 11. The human ear can just detect sound waves of intensity 10−2W/m2. Calculate the sound energy incident each second on eardrum of area 20mm2 at this intensity. 12. What is the wavelength of a 10KHz sound wave in a bar of iron? (Y= 1.9 ×1011Pa, ρ =7.8 g/cm3). 13. What is the difference in their intensity level of two sound waves with intensities: 1012W/ cm2 and 108W / cm2? 14. By how much will the intensity of sound decreases if the distance from a point source is tripled? 15. A loudspeaker radiates sound uniformly in all directions. If the intensity of the sound at a distance of 40m from the source is 4 × 10-6 w/m2, what will be the intensity at a distance of 80m? 16. A beetle vibrates its wings to give out a sound wave of intensity 4 × 10-3 w/m2, when you are 5m away from it. (a) How much power is transmitted by this sound wave? (b) What is its intensity at 2m? The Doppler Effect Key terms In 1842 Austrian scientist Christian Doppler, y Doppler’s Effect: The discovered that when a source of sound and a variation in the pitch of a listener are in motion relative to each other, the sound due to the relative frequency of the sound heard by the listener is motion between a source of not the same as the source frequency. This effect sound and a listener. works not only for sound but also for light and radio waves. This is the Doppler Effect, which has important applications in medicine and technology. So far, we have only considered stationary sources of sound and stationary listeners (or observers). In this case, the frequency of sound reaching a listener’s ear will be the same as the frequency of the source when the source of sound, the medium and 249 Physics — Grade 11 Textbook the listener (observer) are all at rest. If however, the source and the observer are in relative motion, the frequency of the wave noted by the observer appears to be different from the true frequency. Suppose you are standing on a sidewalk and an ambulance is approaching you. As the sound waves move towards you, they compress which increases the frequency resulting in a higher pitch. But, as the ambulance is moving away from you, the sound waves spread further apart so the frequency lowers resulting in a lower pitch (Figure 32). The sound the ambulance is producing is not changing, but the frequency of the sound perceived by our ear changes. This phenomena of variation of the frequency of the sound heard by the relative motion of the source or the listener is known as Doppler effect. It is illustrated in the Figure 32. Figure 32. Doppler Effect Different Cases of Doppler Effect The apparent frequency due to Doppler Effect for different cases can be deduced as follows. Case I: When the source and observer are relatively at rest with respect to each other, then the frequency heard by the observer is equal to the actual frequency produced by the source. fL= fS Case II: When a listener moving towards the stationary source. V = f L f s 1 + L V Where VL is the speed of the listener, and V is the speed of sound. 250 Waves Case III: A listener moving away from the stationary source. V f L f s 1 L V Case IV: When the source of sound is moving toward a stationary listener, the frequency of the sound produced by the source ( fs ) and the frequency of the sound received by the listener ( fL ) are related by: V 1 =f L f= s fs V V − Vs 1− s V Where V is the speed of sound, and Vs is the speed of the source. Case V: When a source of sound is moving away from the stationary listener, they are related by V 1 =f L f= s fs V V + Vs 1+ s V Case VI: When a source and a listener moving towards each other. V + Vo fL = fs V − Vs Case VII: A source and a listener moving away from each other. V − Vo fL = fs V + Vs Applications of Doppler Effect Here we will take a look at some common applications of the Doppler Effect in real life. 1. To measure the speed of an automobile Here is how a police officer uses a radar to know the speed of a vehicle. 251 Physics — Grade 11 Textbook y A police officer takes position on the side of the road. y The officer aims his radar gun at a approaching vehicle. y The gun sends out a burst of radio waves at a particular frequency. y The radio waves strike the vehicle and bounces back towards the radar gun. y The radar gun measures the frequency of the returning waves. Because the car is moving towards the gun, the frequency of the returned waves will be higher than the frequency of the waves initially transmitted by the gun. The faster the car’s speed, the higher the frequency of the returning wave. y The difference between the emitted frequency and the reflected frequency is used to determine the speed of the vehicle. The computer inside the gun performs the calculation instantly and display a speed to the officer. 2. Radar Radar is a device, which transmits and receives radio waves. Radar sends high frequency radio waves towards an airplane. The reflected waves are detected by the receiver of the radar station. The difference in frequency is used to determine the elevation and speed of an airplane. 3. Sonar (sound navigation and ranging) Sound waves generated from a ship fitted with SONAR are transmitted in water towards an approaching submarine. The frequency of the reflected waves is measured and hence the speed of the submarine is calculated. 4. Doppler echocardiogram Doppler echocardiography is a procedure that uses Doppler ultrasonography to examine the heart. An echocardiogram uses high frequency sound waves to create an image of the heart while the use of Doppler technology allows determination of the speed and direction of blood flow by utilizing the Doppler effect. Examples (a) A train moving toward a detector at 30m/s blows a 305Hz horn. What frequency is detected by the detector? (b) repeat question (a) if the train were moving away from the detector? Take velocity of sound as 340 m/s. Solution Vs= 30m/s, fs =305Hz V=340m/s 252 Waves 1 1 f L = fS = 350H z = 334.5H z 1 Vs/v 1 30/340 1 1 f L = fS = 350H z = 280.3H z 1 + Vs/V 1 + 30/340 Demonstration: Demonstrate Doppler Effect Using a tuning fork In this demonstration, Doppler effect is demonstrated in the classroom using a tuning fork. Method I y Attach a string to the stem of a tuning fork. y Hit the tuning fork with the rubber hammer and rotate it on a horizontal plane above your head. y The class should be able to hear the changes in pitch. Method II y Let one student hold a tuning fork and hit it with the rubber hammer. y At the same time another student approach and then recede from the fork. y Observe the difference in the pitch your ears will intercept as you approach and recede from the fork. y What do you observe, if you move rapidly towards and away from the tuning fork? Exercises 1. State Doppler’s Effect. 2. Write at least two applications of Doppler Effect in real life. 3. A boy in a train moving at 25m/s hears a sound wave of frequency 300Hz from a stationary source. Find the frequency of sound heard by the listener if he is moving (a) towards the source, and (b) away from the source. (Take speed of sound at that temperature to be 334m/s). 4. A car moving at a speed of 10m/s produces a sound wave of frequency 400Hz. Find the observed frequency of a listener standing at the side of the road, if: (a) the car is moving towards the listener. (b) the car is moving away from the listener. (Speed of sound in air at that temperature is 330m/s.) 253 Physics — Grade 11 Textbook Vibrations in strings and tubes (pipes) Key terms Waves on a string and pipes play an important y The property of light role in music. String instruments produce sound travelling in a straight line by vibration of the string and wind instruments is called as the rectilinear produce sound by vibration of sound waves in propagation of light. the pipe. In this section we will discuss, how the different modes of vibrations are produced in these instruments. Almost all the musical instruments that produce music can be grouped in three major classes. 1. String instruments: - produce musical sounds by vibrating strings. Examples: piano, guitar, violin, flute, cello. Figure 33. String musical instruments 2. Wind instruments: - Produce musical sounds when wind is blown into or through a tube. Examples: clarinet, flute, trumpets and saxophones. Figure 34. Wind musical instruments 254 Waves 3. Percussion instruments:- Produce musical sounds when struck by something. Figure 35. Percussion musical instruments Standing Wave on Strings Let’s now consider a string of a definite length L, clamped at both ends. Such strings are found in many musical instruments, including pianos, violins, and guitars. When a guitar string is plucked, a wave will be sent in both sides, to the left and to the right. These waves are reflected and re-reflected from the ends of the string, making a standing wave. This standing wave on the string in turn produces a sound wave in the air, with a frequency determined by the properties of the string. Figure 36. The standing wave pattern of a string The standing waves on strings cannot be made at any frequency. The special frequencies at which the standing waves are made are called the resonance 255 Physics — Grade 11 Textbook frequencies. At these frequencies nodes and antinodes are formed in the string, and the wave seems to be standing still. As the frequency increases different pattern of mode of vibration where the standing waves are formed will appear. These resonance frequencies are named as the fundamental frequency of the string or overtones of the fundamental frequency. The fundamental frequency is the lowest frequency which have two nodes and an antinodes between them, see Figure 36 (a). All other standing wave patterns that are set up are multiples of this fundamental frequency. These frequencies are known as harmonics of the fundamental. The fundamental is the first harmonic. Each of higher harmonics are named by the integer by which the frequency of the fundamental must be multiplied to give its frequency. Thus the modes of vibration with the frequency 2f, is the 2nd harmonic or first overtone, that with a frequency 3f, is the third harmonic or second overtone, and so on. In all modes of vibration there must be a node at each end of the string. In general 1. L = n l n 2 2. λn = 2L n v 3. fn = n = nf1 2L Where n is the number of harmonic or number of segments. Note that: y The distance between adjacent antinodes is equal to ½λ. y The distance between adjacent nodes is equal to ½λ. y The distance between a node and an adjacent antinode is λ/4. Examples A standing wave is formed in a stretched string that is 3m long. What is the wavelength of the first three harmonics? Solution 2L L = 3m, n = n 256 Waves 2L 1 = = 2 (3m) = 6m 1 2L 2 (3m) 2 = = = 3m 2 2 2L 2 (3m) 3 = = = 2m 3 3 Examples A standing wave is formed in a 4m long string that transmits waves with a speed of 12m/s. What are the frequencies of the first three harmonics? Solution L = 4m; υ = 12m/s 12m/s f1 = = = 1.5Hz 2L 2 (4m) f2 = 2f1 = 2 (1.5Hz) = 3Hz f3 = 3f1 = 3 (1.5Hz) = 4.5Hz Examples A string 100cm long is adjusted so that it vibrates in 8 segments when subjected to a frequency of 120Hz. What is the speed of the wave in the string? Solution L = 100cm = 1m , n = 8 ; f = 120Hz v f = n 2L 2Lf 2 (1m) (120Hz) v= = = 30m/s n 8 The speed of a wave in a Vibrating string The wave speed is the speed at which the disturbance propagates through the medium. It is not the speed of the individual particles making up the medium. The speed, of transverse waves on a stretched string depends on the properties of the string that affect its elasticity and its inertial properties. For a mechanical wave travelling along 257 Physics — Grade 11 Textbook a string the speed depends on the tension in the string and the mass per unit length. (µ=m/l) known as the linear mass density (µ) cof the string. Mathematically, T Where m = mass per unit length, and T is the tension in the sting. It is crucial to control the speed of sound in stringed instruments like the guitar, which is why they have tuning knobs at one end (to control T) and the strings are of different mass (to control μ). The fundamental frequency of a vibrating string can then be determined by the equation V 1 T f1 2L 2L In general for the nth harmonic n I f n nf1 2L Examples If a string 1.25m long and a mass of 20g vibrates under a tension of 64ON, what is (a) the liner mass density of the string? (b) the speed of the wave and (c) the fundamental frequency produced by this string? Solution m 0.02kg L = 1.25m, m = 20g = 0.02kg, T = 640N, = = = 1.6 10-2 kg/m 1.25m T 640N = = = 200m/s 1.6 10-2 kg/m 200m/s f1 = = = 80Hz 2L 2(1.25m) Examples A Stretched string vibrates with a frequency of 50Hz. The string is 1.0m long, has of 10g and is stretched with a tension of 400N. What is the wavelength of the wave produced? 258 Waves Solution f = 50Hz, l=1.0m, m= 10g = 0.01kg,T = 400N T T/ 400N 1.0m V= = = = 200m/s m/ m 0.01kg 200m/s = = = 4m f 500Hz Standing Wave in Pipes A longitudinal standing wave is set up in wind instruments which produce sound by a vibrating air column in a pipe. The pipe is called closed when it is open at one end and closed at the other, or open when it is open at both ends. Standing Wave in Closed Pipe If any source of sound is held over an open end of a closed pipe a longitudinal wave is generated which travel through the air column to the closed end of the pipe. On reaching there the wave is reflected back towards the open end. These two identical waves traveling in opposite directions give rise to a longitudinal standing (stationary) wave in the pipe. Since the closed end acts like a rigid barrier a node (N) is formed there, but at the open end an antinode (A) is formed. Note that because it is difficult to represent longitudinal stationary waves in diagram form, they are usually drawn as transverse waves instead. The different modes of vibration of air in a closed pipe The standing wave pattern (or mode of vibration) with the lowest frequency has a node at the closed end and an antinode at the open end, see Figure. 37. We can easily find that the length L of the pipe (the distance between a node and the next 1 antinode) is one – fourth of the wavelength of the wave produced, L =. 4 We then increase the length of the pipe to form the next mode of vibration. This is usually made when the player closes the holes on the pipes of some of the instruments, like the flute or Clarinet. When the length of the pipe is increased by one segment we will have two nodes and two antinodes. Thus length of the pipe will be, L = 3 ( 1 4 ). The next mode of vibration will have 3 – nodes and 3 – antinodes. Thus, L = 5 ( 1 4 ) etc. 259 Physics — Grade 11 Textbook L A First hramonic = fundamental 1 L = 1 4 B υ f1 = 4L Third harmonic A 3 L = 3 4 B 3υ f3 = =3 f1 4L Overtones A Fifth harmonic 5 L = 5 4 B 5υ f5 = =5 f1 4L Figure 37. Standing in closed pipe Now we come up with a general formula for the length of the pipe and the frequency of that mode of vibration. 1. L = n n 4 v 2. fn = n 4L Where n is the number of harmonic and n = 1, 3, 5, 7... The different standing waves made in a closed pipe have specific names. The standing wave with only one node is called the first harmonic or fundamental note. All other standing waves made in a closed pipe are called overtones. We can also call the standing wave with two nodes having a frequency 3 – times the fundamental as the third harmonic, and the standing wave with three nodes having a frequency 5–times the fundamental the fifth harmonic etc. v f1 = – fundamental or first harmonic 4L v f3 = 3f1 = 3 – third harmonic or first overtone 4L v f5 = 5f1 = 5 – fifth harmonic or second overtone. 4L 260 Waves This leads to the fact that there are only odd harmonics in a closed pipe, no even harmonic. Examples A closed organ pipe has a length of 0.8m at room temperature (20oc). What are the (a) frequencies and (b) wavelengths of the first three harmonics? Solution L = 0.8m; T = 20oc 0.6 (20°c) v = 331.5 + = 343.5m/s 1°c v 343.5m/s f1 = = = 107.3Hz, f3 = 3f1 = 322Hz, f5 = 5f1 = 536.7Hz 4L 4(0.8m) 4L n = n 4 (0.8m) 4 (0.8m) 4 (0.8m) 1 = = 3.2m, 3 = = 1.07m, 5 = = 0.64m 1 3 5 Examples What is the length of a pipe, closed at one end, which gives a fundamental note of 250Hz? (The speed of sound is 340m/s.) Solution f = 250Hz; v = 340 m/s v = , since the pipe is closed L = f 4 v 4L = f v 340m/s 340 L= = = = m = 0.34m = 34cm 4f 4 (250Hz) 1000 Standing Waves in Open Pipes Standing waves can also occur in open pipes. In open pipes antinodes will always occur at the two open ends. Therefore, the fundamental note will have one node in the center of the pipe and two antinodes at the two ends, see Figure 38. Thus 261 Physics — Grade 11 Textbook L n=1 1 = 2L f1 = f 0 Fundamental standing wave, first harmonic n=2 2 = L f 2 = 2f 0 Second harmonic n=3 2 3 = L 3 f3 = 3f 0 Third harmonic nth harmonic: 2 n = L fn = nf 0 n Figure 38. standing wave in open pipes Examples What is the frequency of (a) the fundamental (b) the first overtone of an organ pipe open at both ends and of length 125cm? Assume the velocity of sound in air to be 350m/s. Solution L = 125cm = 1.25m, V = 350 m/s v 350m/s (a) f1 = = = 140Hz 2L 2 (1.25m) (b) The first overfore is the second harmonic (n = 2). Thus f2 = 2f1 f2 = 2 (140Hz) = 280Hz 262 Waves Examples What is the shortest length of a tube open at both ends that will resound to a fork of frequency 480Hz, if the velocity of sound in air is 340m/s? Solution f1 = 480Hz, v = 340 m/s v f1 = 2L v 340m/s L= = = 0.354m 2f1 2 (480Hz) Exercises 1. Explain why it is not possible to produce a standing wave pattern for all frequencies in every stretched string. 2. Define the terms (a) fundamental frequency, and (b) harmonics 3. An organ pipe closed at one end has a length of 125cm. If the speed of sound is 350m/s, calculate the frequency of (a) the fundamental (b) the first overtone (c) the fifth harmonic. 4. The mode of vibration for a certain organ pipe is shown in the figure. If the length of the pipe is 6m, what is (a) the number of harmonic? (b) The wavelength of the wave? N A N A N A 5. A string fixed at both ends resonates in 3 segments at a frequency of 165Hz. At what frequency would the string resonate in 2 segments? 6. For the modes of vibration shown in the figure determine (a) the wavelength of the wave (b) the fundamental frequency if the speed of sound is 330 m/s. A N A N A N A 1.2m 1.2m Beats When two sound waves having slightly different frequencies are sounded together, the combined sound grows and falls in loudness. The principle of superposition 263 Physics — Grade 11 Textbook tells us that the waves reinforce one another when Key terms they arrive in phase and cancel out when they arrive y Beats are the periodic and out of phase. When reinforcement occurs a louder repeating fluctuations heard in sound is heard and when cancellation occurs a soft the intensity of a sound when sound or no sound at all is heard. This variation in two sound waves of very loudness of a sound heard is called beat. similar frequencies interfere with one another. The simplest illustration of beat is to draw two different waves and then add them together. You can do this by drawing them yourself to see the pattern that occurs. Here is wave 1: Now we add this to another wave, wave 2: When the two waves are added the resulting wave is drawn below. Notice that the peaks are the same distance apart but the amplitude changes. If you look at the peaks, the peak amplitudes seem to oscillate with another wave pattern. Figure 39. Beat The beats have a frequency which is the difference between the frequency of the two waves that were added. If two sound waves have frequencies f1 and f2, then the number of beats per second, called the beat frequency (fb) is given by: fb= | f1- f2 | 264 Waves Examples One tuning fork has as frequency of 445Hz. When a second fork is struck, beat notes occur with a frequency of 3Hz. What are the two possible frequencies of the second fork? Solution f1 = 445Hz, fb = 3Hz First possibility fb = f1- f2 ⇒ f2 = f1- fb=445Hz-3Hz=442Hz Second possibility fb = f2- f1 ⇒ f2 = f1+ fb=445Hz+3Hz=448Hz Exercises 1. If one tuning fork vibrates at 440 Hz and a second one vibrates at 445 Hz, find the beat frequency. 2. A tuning fork produces 8 beats per second with another fork of frequency 340 Hz. What are the two possible frequencies of the former fork? Experiment Standing Waves in a Closed Tube Theory: A tuning fork is held by hand just above the open end of the tube. When the tuning fork is struck by a rubber hammer, it vibrates and longitudinal waves are sent down the air column. These waves are reflected at the water surface and thus produce standing waves. Here, the surface of water will act as the closed end. The sound waves reflected from the water surface change their phase by 180° and therefore are completely out of phase with the incident sound waves. In other words, the amplitude of the standing waves must be zero at the water’s surface. Nodes are produced at the water surface and antinodes are produced at the open end. The length of the water column may be changed by raising or lowering the water level while the tuning fork is held over the open end of the tube. When the frequency of waves in the air column becomes equal to the natural frequency of tuning fork, a loud sound is produced in the air column, indicating the formation of resonance. 265 Physics — Grade 11 Textbook Since the distance from a node to antinode is one fourth of a wavelength, the resonance conditions can only be satisfied when the column length (L) is: l L = n , n = 1, 2, 3, 4, 5 4 where L is the length of the tube and λ is the wavelength. Objectives: y To observe the resonance phenomenon in a closed cylindrical tube. y Using tuning forks of known frequency, the velocity of sound in air at room temperature is determined. Materials: y Tuning forks of different frequencies. y A tube open at both ends (here a graduated cylinder with one end sawed off was used). y A large jar partially filled with water. Procedure: y Fill the tube 2/3 full of water. y Submerge the tube in the jar of water. y Strike a tuning fork and hold it over the open end of the tube. y Adjust the height of the tube until you hear the resonance frequency. When a resonance is found, a pronounced reinforcement of the sound will be heard. Move the water surface up and down several times to locate the point of maximum sound intensity and mark that point with a rubber band on the outside of the tube. y Lower the water further to find the next resonant length. Continue in this manner as far as the length of the tube will permit. Obtain the lengths λ/4, 3λ/4, etc. in meters from your measurements. You will need to check to see if your column lengths follow the progression 1, 3, 5, 7, -- since you may have missed a resonance or counted one of the fainter spurious resonances which sometimes occur. Calculate the wavelength and velocity of sound. y Repeat the procedure for the other tuning forks supplied. Please record the room temperature for reference since the velocity of sound increases with increasing air temperature. 266 Waves f=? λ/4 3λ/4 5λ/4 First resonance Second resonance (third Water harmonic) Third resonance (fifth harmonic) Data: The first resonance occurs only when the length of air column is proportional to one-fourth of the wavelength of sound waves having frequency equal to frequency of tuning fork. i.e.; For first resonance, λ=4L Thus, using the above equation we can find the wavelength of the sound from the length of the tube that forms resonance. Finally using the frequency of the tuning fork we can easily determine the speed of sound at that temperature. v=fλ Conclusion: Compare your resu
