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1 This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: This content was downloaded on 16/09/2018 at 17:18 Please note that terms and conditions apply. You may also be interested in: The Everyday Physics of Hearing and Vision: Wave properties B Mayo An experiment with sound beats P J Fay Thermal conductivity and effective diffusion coefficient for vibrational energy : carbon dioxide ( K) P C Jain, S C Saxena and R Afshar Measurement of length using moire fringes W Noel Holland, John Mackenzie and Derek J Noble Teaching wave propagation and the emergence of Viete s formula J P Cullerne and M C Dunn Goekjian Inter-relations between classical cross sections for rotational transitions in sudden atom-molecule collisions S S Bhattacharyya and A S Dickinson Teaching wave theory by phasors S Everall Mapping of the five-parameter exponential-type potential model into trigonometric-type potentials Chun-Sheng Jia, Yong-Feng Diao, Min Li et al. Some solutions of linearized 5-d gravity with brane O I Vasilenko

2 IOP Concise Physics Musical Sound, Instruments, and Equipment Panos Photinos Chapter 1 Properties of waves 1.1 Introduction In everyday language the term wave has several uses; for example, a wave of s, a wave of enthusiasm, a wave of applause, a heat wave, and so on. The general idea is that something is suddenly going above or below normal. In more technical language, a wave indicates a repeating pattern of highs and lows in some quantity. In terms of sound, what is waving is the air pressure, going higher and lower than the ambient atmospheric pressure. This chapter will introduce the basic concepts that are commonly used to characterize waves, and sound waves in particular. 1.2 Periodic waves A most familiar wave is the pattern of circles generated by dropping a coin in a still pond, shown in figure 1.1. The pattern consists of highs and lows (crests and troughs, doi: / ch1 1-1 ª Morgan & Claypool Publishers 2017

3 Figure 1.1. A wave in a pond. respectively) traveling outwards from the center. At each point of the surface of the pond, the water level oscillates in cycles, above and below the undisturbed level of the pond. We will refer to the undisturbed level as the equilibrium level of the water surface. The pattern propagates away from the point of impact (the source of the wave) and at each point the propagation is along the line of sight to the source. In a shallow flat-bottomed pond, the crests (and troughs) travel at the same speed. As the speed is the same for all crests, the distance between successive crests remains the same as the wave travels. The distance between successive crests (or successive troughs) is defined as the wavelength. The time elapsed between the crossings of two successive crests through a given point is the period of the wave. In one period, the wave travels a distance of one wavelength; in other words, the speed (i.e. distance traveled divided by time of travel) is the ratio of the wavelength divided by the period. If we count the number of crests crossing through one point in a given time interval, say in one second, then we have a very important concept in the study of sound, namely the frequency. The frequency is the inverse of the period, i.e. frequency = 1/(period) and therefore period = 1/(frequency). If four successive crests cross a given point in 1 s (i.e. if the frequency is 4 crests per second) then the time elapsed between two successive crests (i.e. the period) is 1/4 of a second. As the speed of the wave equals the wavelength divided by the period, and since frequency is the inverse of the period, it follows that the speed of the wave is equal to the product of the wavelength times the frequency: Speed of wave = (frequency) (wavelength). The frequency is fundamental in characterizing sound tones, and is a measure of what we call the pitch. High pitch tones correspond to high frequencies, and low pitch tones correspond to low frequencies. The frequency is measured in units of Hertz (Hz for short). The difference in height between the top of a wave crest and the undisturbed water level is the amplitude of the wave. The amplitude of the wave depends on the weight and speed of the impacting object. A small coin will cause a smaller 1-2

4 amplitude than a huge rock. The amplitude relates to the intensity of the wave. Note that as the wave pattern expands, the amplitude diminishes, and eventually the wave dies out. With sound, this observation relates to everyday experience: the farther we are from the source, the weaker it sounds. The succession of crests and troughs in the water pond example occurs because of gravity. Water that happens to be above the equilibrium level of the surface is pulled down by gravity. The downward speed builds up, and that amount of water falls below the equilibrium level of the surface and becomes part of a trough. While moving down it pushes adjacent parts of the water upward, which become part of a crest, and so forth. The entire cycle can be viewed as an attempt of gravity to restore the water level back to the equilibrium level, as it was before the coin was dropped. In the process of restoring the equilibrium water level, it keeps overshooting the target. Thus, in our example, gravity is the restoring force. The overshoot occurs because of the energy imparted by the impacting coin. All waves require a restoring force. For example, in a vibrating string, the tension of the string acts as the restoring force. There is a relation between the frequency of the wave and the strength of the restoring force. A stronger restoring force makes the up-and-down oscillation faster, which means that the frequency will be higher. This relation will be discussed in more detail in connection with strings and string instruments. In the example of the water wave, the quantity that oscillates is the water level, as compared to the equilibrium level. In the case of sound waves in air, the oscillating quantity is the air pressure. The sound wave in air is a succession of layers of low and high pressures, the rarefactions and compressions, respectively. Low and high pressures are with reference to the undisturbed air pressure of the atmosphere. Note that, in the pond example, as the wave travels in the horizontal direction, the oscillation is up and down. In other words, the oscillation is at a right angle to the direction of travel. This is an example of a transverse wave: the oscillation is transverse to the direction of travel. In the case of a sound wave in air, the pressure oscillates back and forth, along the direction of travel. This is an example of a longitudinal wave. It is convenient to use graphs to represent waves. The simplest periodic waveform is the sinusoidal wave, i.e. described by the sin (sine) or cos (cosine) functions known from trigonometry. Three cycles of a sinusoidal are shown in figure 1.2. The cycle or Figure 1.2. Three cycles of a sinusoidal waveform. 1-3

5 Figure 1.3. Sinusoidal waveform. (a) Horizontal axis is distance. (b) Horizontal axis is time. periodicity of the repeating pattern is equal to the distance between two successive equivalent points; for example, two successive highs or two successive lows. The interval between adjacent highs and lows is half a cycle, and so on. The phase at any point of the graph is the fraction of the cycle elapsed from the starting point of the waveform. For example, in figure 1.2, the phase at the first high on the left is onequarter of a cycle; the phase at the first low is three-quarters of a cycle. When using graphs to represent waves, the vertical axis of the graph is used to show the value of the oscillating quantity (e.g. displacement or pressure). In the horizontal axis we usually have two options. We can choose the horizontal axis to show distance or time, as indicated in figure 1.3. With reference to the water wave in a pond, the vertical axis shows the height of the water level. From the graph, we see that the amplitude of the wave is 1 m. If we choose to show distance in the horizontal axis, then the graph will give the profile of the height at a given instant. If the horizontal axis is chosen to indicate time, then the graph will give the variation of the height at a given point. It is important to note that the two graphs provide different information about the wave. Recalling the definitions of the wavelength and period, we see that the distance between successive crests in figure 1.3(a) is equal to the wavelength. In this example, the first crest occurs at a distance of about 1.8 m, and the second crest occurs at a distance of about 7.8 m. The wavelength is found by taking the difference of the locations of the two crests, i.e = 6m.Infigure 1.3(b), the first crest occurs at time = 1 s, and the second crest at time = 5 s. The interval between successive crests is the period of the wave, and in this example it is 5 1 = 4 s. As the frequency is the inverse of the period, it follows that the frequency of this waveform is (1/4)= 0.25 Hz. 1-4

6 1.3 Addition of waveforms In this section we discuss simple ways in which waves can combine with each other. Comparison of the phase of the interacting waves is the key concept in understanding the outcome. Figure 1.4 shows two identical waveforms, i.e. they have the same amplitude and the same periodicity. The horizontal axis is not labeled, and can be either distance or time without affecting the conclusions. The graphs are offset vertically, and the horizontal lines represent zero displacement for each wave. To find the waveform resulting from combining waves A and B, we add the displacements at each point of the horizontal axis. The result is shown in the bottom graph, and is simply the sum of the two waves, i.e. the amplitude of the resulting wave is doubled, and the periodicity (which is the wavelength or the period depending on the choice of the horizontal axis) remains the same. In this case, we combined two waves that are in step, or in-phase. This means at each point of the horizontal axis, the two waves have the same phase, and the phase difference between them is zero. In figure 1.5, wave B is displaced to the left by about one quarter of a cycle. In this case the two waves are not in-phase, and there is a phase difference of one-quarter of a cycle. If the horizontal axis were distance, the two crests of the two waves would be separated by one quarter of a wavelength. In the same way, if the horizontal axis indicated time, then the crests of the two waves would be separated by one quarter of the period. To find the waveform resulting from combining waves A and B, we add the displacements at each point of the horizontal axis. The result is shown in the bottom of figure 1.5. We note that the sum of waves A and B has the same periodicity, but the amplitude is smaller than the sum of the amplitudes of A and B. The crest of the resulting wave occurs somewhere in between the crests of waves A and B. In figure 1.6, wave B is displaced to the left by half a cycle, and the result of adding these two waves is total cancellation. As the amplitudes of A and B are the same and the oscillations are in opposite directions, the cancellation is complete. If Figure 1.4. Adding two identical waveforms that are in-phase. 1-5

7 Figure 1.5. Adding two identical waveforms that are 1/4 of a cycle out of phase. Figure 1.6. Adding two identical waveforms that are 1/2 cycle out of phase. The result is complete cancellation. the amplitudes were not the same, the resulting wave would have amplitude equal to the difference between the amplitudes of A and B. For instance, if A had amplitude 3 and B had amplitude 1, the resulting wave would have amplitude equal to the difference 3 1 = 2. Using similar diagrams, we can add waves of the same periodicity. The general conclusions are: The resulting wave will always have the same wavelength and frequency. The amplitude will be equal to the sum of the amplitudes, if the phase difference is zero. The amplitude will be equal to the difference between amplitudes, if the phase difference is 1/2 cycle. 1-6

8 If the phase difference between waves A and B has value between 0 and 1/2 cycle, the resulting amplitude will have value between the amplitudes A + B and A B 1. Cancellation of waves as shown in figure 1.6 is the basic principle behind noise cancelling devices. For example, noise cancellation headphones use electronics to generate replicas of the sound before it reaches each ear. The replicas generated differ in phase from the incoming wave by half a cycle. Inside each ear, the incoming sound and the out-of-phase replica cancel each other. 1.4 Beats A very interesting situation occurs when adding two waves of slightly different frequencies. Figure 1.7 shows two waves, A and B, of equal amplitude. The horizontal axis shows time. By counting crests, we see that in the time it takes wave A to complete 10 cycles, wave B completes 8 cycles. In other words, wave A has a shorter period, therefore higher frequency than wave B. The result of adding waves A and B is shown at the bottom of figure 1.7. For our purposes, the important feature of the combined wave is that the amplitude has cycles of highs and lows, that occur approximately every 30 s. This change in the amplitude is referred to as beats. Thus, the amplitude of the resulting waveform A + Bismodulated, i.e. changes with time. The period of this modulation is about 30 s. A more detailed analysis 2 shows that the result of adding two waves of different frequencies is a wave of frequency equal to the average frequency of the two waves. The amplitude of the resulting wave is not constant, but modulated at a frequency equal to the difference between the frequencies of the two waves. Figure 1.7. Adding two waveforms of the same amplitude but different frequencies. 1 Or B A if the amplitude of B is larger. This is so because by convention, the amplitude is always a positive number. 2 Details are provided in appendix B

9 Figure 1.8. Sound produced by an out-of-tune piano. Note the four cycles of the beat pattern. So, if we add a wave of 400 Hz and a wave of 402 Hz, the result will be a wave of 401 Hz, with amplitude that is modulated with a frequency of = 2 Hz, which is the beat frequency. Therefore, the highs of the amplitude will repeat every 1/(2 Hz) = 0.5 s, i.e. the beats will occur every 0.5 s. The phenomenon of beats plays a significant role in music and musical instruments, particularly those with double or triple strings, such as the piano. Each string in a pair should produce sound of the same frequency. If there is a mismatch of 2 Hz between the frequencies of the two strings, then from our calculation above we find that the intensity of the sound will fluctuate up and down every 0.5 s, producing a tone of poor quality. Figure 1.8 shows the waveform of the sound produced by an out-of-tune piano. The entire waveform lasts about 2 s, from which we can estimate that the pattern of beats shown repeats about every 0.5 s. Therefore, the beat frequency is about 1/0.5 = 2 Hz, meaning that the two strings are off by 2 Hz, which, as will be discussed in section 5.6 is quite noticeable and unpleasant to the ear. Experienced tuners and string instrument players can use beats to tune their instruments very precisely. 1.5 Energy and intensity Waves carry energy. So, when we drop a coin in a pond, some of the energy of the falling coin (the so-called kinetic energy) is transferred to the water, and carried away by the wave generated by the impact. A heavier coin or faster moving coin will result in a wave of larger amplitude, because it imparts a larger amount of energy at the point of impact. In many cases, the important quantity is not the energy imparted at the point of impact, but how much energy is flowing at points away from the source of the wave. To understand the concept of energy flow, it would be helpful to make an analogy to collecting rainwater in a tub. The more rain is coming down, the more raindrops will be caught in the tub, but the amount of water collected will also depend on the size of the tub because a wider tub has larger collecting area. The amount of water collected will depend on time as well: the longer we wait, the more water will be collected. We can characterize the intensity of the rain by referring to a commonly agreed upon tub area (e.g. 1 m 2 ) in a specific time interval (e.g. the amount collected in 24 h). Also note that the amount collected will depend on the tilt of the tub. A vertical tub will not catch much water! The maximum amount of water will be collected if the tub is perpendicular to the direction of the falling drops. 1-8

10 For waves, we can define the intensity as the rate at which energy is flowing through an area of 1 m 2. The area must be oriented perpendicular to the direction of energy flow. The rate of energy flow (i.e. amount of energy flowing per unit time) is defined as the power, and it is measured in Watts (W). Therefore, the intensity is measured in Watts per square meter (W m 2 ). The intensity is related to the square of the amplitude of the wave. This means that if we double the amplitude of the wave the intensity is not doubled but quadrupled (following the square 2 2 ). If the amplitude is tripled, the intensity increases 3 2 = 9 times, and so on. As the wave propagates outward from the source, the energy that started at the source is spread over a larger and larger surface. Consequently, the intensity (and amplitude) will decrease more and more the further we move away from the source. In addition, some of the energy that started at the source is absorbed, usually converted to heat as a result of friction. We refer generically to the decrease in intensity as attenuation. Attenuation will be discussed in detail in section Further discussion Examples of waves Electromagnetic (EM) waves include light, radio waves, microwaves, ultraviolet radiation, x-rays, and more. The oscillating quantity in EM waves is an electric field combined with a magnetic field. In empty space, all EM waves travel at the speed of light. EM waves can be produced by the oscillation of electrical charges. WiFi is a wireless connection between devices based on transmitting and receiving EM waves of frequency 2.4 billion Hz (GHz for short) and 5 GHz. Similarly, an AM radio is a wireless connection, using EM waves of frequency in the range of about thousand Hz (khz). And an FM radio uses EM waves of frequency in the range from 87.5 to 108 million Hz (MHz). All these waves carry the sound information, but what we hear is not the EM wave itself. The EM wave is only the carrier, the so-called carrier wave. The sound information is imprinted on the carrier signal as a modulation of the amplitude (AM = Amplitude Modulation) or the frequency (FM = Frequency Modulation) of the carrier wave. The bottom graph in figure 1.7 is an amplitude-modulated signal. One of the most exciting discoveries in recent years was the detection of gravitational waves. In principle, gravitational waves can be produced by oscillations of huge amounts of mass (like stars) by analogy to EM waves that can result from oscillating electric charges. The oscillation quantity in the case of gravitational waves is the fabric (or the curvature) of space, as predicted by Einstein s general theory of relativity. 1.7 Equations Frequency, period, wavelength, and speed The following symbols are commonly used: c = speed λ = wavelength f = frequency 1-9

11 T = period f is defined as f = 1/T For sinusoidal waves we have: c = f λ and by the definition of f we have c = λ/t. 1.8 Questions 1. A person s heart rate is 120 beats per minute. (a) Find the period of the heart rate in seconds. (b) Find the frequency of the heart rate in Hz. 2. A wave has frequency 20 Hz and wavelength 200 meters. Find the speed of the wave. 3. Suppose we have two sound waves, A and B, traveling in air simultaneously. The amplitude of wave B is twice the amplitude of wave A. (a) Which of the two waves has higher intensity? (b) Which of the two waves travels faster? 4. Suppose we combine two sound waves of frequencies 100 and 106 Hz, respectively. Find the beat frequency resulting from the combination of the two waves. 1-10