Strand E. Waves Unit 1. Measuring Waves Contents Page Types of Wave 2 Measuring Waves 6 Phase 10
E.1.1 Types of Wave Ripples on a pond, sunlight, musical sounds and earthquakes are all wave phenomena. Waves can occur wherever there is a disturbance of a system, and when that disturbance can travel from one region to another. Waves that travel through a substance, like waves on a string or water waves, are called mechanical waves. The substance that the mechanical wave travels through is known as the medium. Mechanical waves need the medium to travel through, and without it, the wave cannot exist. For example, sound is a mechanical wave that can travel through air. Sound progresses through air by moving the air molecules back and forth, which then moves the adjacent air molecules and so on, passing the vibration along the line of progression like a chain. In the vacuum of space there is no air, or any other molecules to pass along the vibration, and therefore the sound wave cannot exist. Some waves can travel through a vacuum since they do not require a medium to travel through. These are electromagnetic waves like light, radio waves, and infrared waves (heat). If electromagnetic waves couldn t travel across a vacuum, the heat and light from our sun, and the light from all the other stars in the night sky, couldn t reach us. A waves purpose is to transfer energy from one region to another without transferring the particles within the medium. Waves only transfer energy in the direction in which they travel. For instance, a storm hundreds of miles out to sea transfers its energy to the sea by creating waves where the high winds of the storm are in contact with the water. These waves then travel hundreds of miles, giving up the energy only when they reach the shore and break. A stationary boat floating on the sea does not travel with the wave. Instead, the boat stays in its position and moves up and down as the wave passes underneath it. The same is true for the water molecules; they oscillate about equilibrium, without going anywhere overall. Longitudinal and transverse waves. Waves can be classified according to the direction of vibration relative to energy transfer into two distinct types, longitudinal and transverse. Longitudinal waves are those where the displacement of the particles about their equilibrium position is in the same direction (parallel) to the direction of wave travel. The direction of wave travel is the propagation of the wave, and as such energy transfer occurs in the direction of propagation. As a longitudinal wave propagates through a medium, the particles collectively oscillate about an equilibrium position and this collective oscillation causes regions of rarefaction (stretching out) and compression, as shown by the oscillating spring in Figure 1.1.1. The loops of the spring can be considered as the particles of the medium. The wave is created by pushing on one end of the spring, oscillating the particles. Rarefaction and compression occurs due to the fact that the particles are oscillating along the chain in the direction of wave travel slightly out of time with each other. 2
Oscillation Wave propagation Figure 1.1.1 Rarefaction Compression The leading edge of the region of compression is known as the wave front, and the particles of the medium (in this case the loops of the spring) do not travel anywhere overall. Instead they oscillate about an equilibrium position with an amplitude that is parallel to the wave propagation, a defining characteristic of longitudinal waves. A longitudinal wave is a mechanical wave for which the displacement of the medium is parallel to the propagation of the wave, or energy transfer. Longitudinal waves cannot travel across a vacuum. Transverse waves can be mechanical, such as waves on a string, or electromagnetic, and also cause the particles of the medium to oscillate about an equilibrium position. For a transverse wave the amplitude of oscillation is perpendicular to wave propagation as shown by Figure 1.1.2. Wave Amplitude Oscillation Wave Propagation Figure 1.1.2 3
A transverse wave is one for which the displacement is perpendicular to the propagation of the wave, or energy transfer. A transverse wave can be either mechanical or electromagnetic Electromagnetic waves all travel through a vacuum at a constant speed c = 3 10 8 m/s or 300,000km per second. All electromagnetic waves are transverse, and whereas longitudinal waves are polarised by definition, the displacement of a transverse wave can either occur in a single plane (plane-polarised) or the vibrations are free to occur in all planes, continually switching from one plane to another (unpolarised). Light from the sun, a light bulb, or a candle is unpolarised, but can be polarised. Figure 1.3 shows an unpolarised transverse wave being polarised into a single plane by a slit in an opaque material. The material blocks the parts of the wave that is not oscillating in the plane parallel to the slit. Unpolarised wave Polarised wave Figure 1.1.3 4
Exercise E.1.1 1. Define both longitudinal and transverse waves in terms of particle displacement. 2. Complete the following table Wave Type Longitudinal Transverse Mechanical Electromagnetic Sound Radio Light Waves on a String Pressure Waves on a Spring 3. Draw a transverse wave travelling on a string, showing the direction of energy transfer and the direction of particle displacement at each peak and trough. 4. What is rarefaction? State which type of wave exhibits rarefaction and suggest an example. Challenge Question 5. Jim has sat on and broken his polarised sunglasses. He holds the two lenses up one in front of the other so that they are initially aligned parallel to each other. While holding one fixed, he rotates the other about 180. Describe what Jim observes. 5
E.1.2 Measuring Waves In order to gain information regarding the energy being conveyed by a wave, or the waves size and speed, we need to be able to measure it. The wave properties that provide important information on the wave are shown schematically for a transverse wave in Figure 1.1.4, and listed below. These wave properties are equally valid for longitudinal waves. Wave Amplitude (A) Wavelength (λ) Peak Oscillation Wave Propagation Figure 1.1.4 One cycle Trough The displacement x is a measurement of the distance and direction of the vibrating particles from their equilibrium position. For transverse and longitudinal waves displacement is perpendicular and parallel to the energy transfer respectively. Displacement is measured in metres. The amplitude A is the maximum displacement of a wave element from equilibrium measured from the center of the wave to the wave peak or trough. Amplitude is therefore the maximum value of x, can be either positive or negative, and is measured in metres. For a sound wave, a larger amplitude means a louder sound. The wavelength λ of the wave is the shortest distance in metres between two identical points on the wave between adjacent wave pulses. The wavelength can be measured from peak to peak, trough to trough, or any other two identical points. One wavelength could be copied and translated to form the entire wave and as such, a wavelength contains all the information required to categories a wave. One cycle of the wave is the distance between a peak and the next adjacent peak of the wave. The period T of the wave is the time in seconds for a vibrating particle to complete 1 full cycle, or the time in seconds between the arrival of two adjacent crests at a given point in space. The frequency f is the number of cycles of vibration of a particle per second, or the number of crests, troughs, or any other point on the wave that passes a given 6
point in one second. Frequency is measured in Hertz (Hz), where 1Hz = 1 cycle per second. The higher the frequency of the wave, the higher the energy. For a wave of frequency f, the period T is given by ff = 1 TT Worked example BBC Radio 2 broadcasts at a frequency of 90.2MHz. Calculate the time period between successive peaks of the radio waves. How many wave peaks will a radio antenna detect in a 5 second period? Answer The time interval between successive peaks is the time period T. Therefore TT = 1 ff = 1 90.2 10 6 = 1.11 108 ss Since a frequency of 1Hz is 1 cycle (one wave peak) arriving at the antenna per second, in 5 seconds; 90.2 10 6 5 = 4.51 10 8 wave peaks arrive at the antenna in a 5 second period. To create transverse waves on a slinky spring or on a rope, one must wiggle or oscillate the end of the rope. The faster the oscillation, the greater the frequency, because more wave peaks are created every second. Since it takes more energy to oscillate the end of the rope faster, the higher the frequency the higher the energy of the wave. If the waves created travel at a constant speed, then many wave peaks per second are produced, with each peak being closer together. Therefore the higher the frequency, the shorter the wavelength. If a wave is created with a frequency of 3Hz, then 3 wave crests pass a fixed point in the medium every second. If the wavelength of the wave is 9m, then the third wavecrest needs to travel a distance of 3 9 = 27m in this time. Since speed = distance / time, We can see that ssssssssss = 27mm 1ss = 27mmss 1 wwwwwwww ssssssssss vv = ddddssssssssssss tttttttttttttttttt iiii 1 cccccccccc tttttttt tttttttttt ffffff 1 cccccccccc 7
since the wavelength of the wave is the distance travelled in 1 cycle and the frequency is 1/T, wwwwwwwwwwwwwwwwwwh λλ wwwwwwww ssssssssss vv = tttttttt pppppppppppp TT = λλ = ffff 1 ffffffffffffffffff ff Hence vv = ffff where v is the wave velocity in m/s, f is the frequency in Hz, and λ is the wavelength in m of the wave. Worked example A ship sends a sound pulse from its sonar vertically downward toward the bottom of the ocean. The sound pulse bounces off a submarine and gets detected by the sonar operator 1.2 seconds after the initial pulse was sent. If the frequency of the sonar pulse is 15kHz with a wavelength of 0.1m, how deep is the submarine? Answer The frequency f = 15kHz and wavelength λ = 0.1m. Therefore, the velocity of the sound wave is; vv = ffff = 15000HHHH 0.1mm = 1500mmss 1 This seems fast for sound which travels at approximately 340ms -1 in air, but in water (a denser medium) sound travels at least four times faster. The time t for the pulse to travel down to the submarine and back to the ship (twice the depth) is 1.2s. Therefore, it takes only 0.6s to reach the sub; ssssssssss ss = dddddddddddddddd dd tttttttt tt dd = ss tt = 1500mmss 1 0.6ss = 900mm 8
Exercise E.1.2 1. Consider the following transverse wave. 4 3 displacement (m) 2 1 0-1 time (s) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16-2 -3-4 (a) Determine the following, stating appropriate units; Amplitude Time period Frequency (b) From part (a), if the wave is a water wave travelling through the ocean at 6 km/h, calculate the wavelength of the wave. 2. A sound wave travels in air at 340m/s. Calculate the wavelength of the sound wave if it has a time period of 50µs. 3. Give two similarities and two differences between a longitudinal and a transverse wave. 9
4. All electromagnetic waves travel at the speed of light c = 3 10 8 m/s. Complete the following table; EM Wave Frequency (Hz) Wavelength (m) Microwave 2 10 9 Visible Light 500 10-9 UV 200 10-9 Radio 90 10 6 Challenge Question 5. A towed sonar array is used to map the sea floor of an ancient sunken city. In order for the archeologists to achieve a good resolution, sound with a wavelength of 2.5mm must be used. If sound travels at 1400m/s in seawater, what is the minimum frequency sound wave that can be used in khz? E.1.3 Phase The phase of a wave is an extremely useful concept, which allows a comparison between waves, and also allows us to add and subtract waves when they meet. The phase of the wave describes the position at a point on the wave at an instant of time within one cycle. Consider points A and B of Figure 1.3.1 (a). Here we see to identical points on the wave. The particles A and B may be on different cycles of the wave, but their vertical displacement is the same. To be identical points on a wave, two points must be separated by a whole number of wavelengths. If this is the case, these points are said to be in phase. As such, all points shown by Figure 1.3.1 (b) are in phase. Any two points on a wave are in phase if separated by a whole number of wavelengths. A (a) (b) C (c) D Figure 1.3.1 λ E F B G In order to describe the position of a point on the wave at an instant in time, each cycle or wavelength of the wave is divided up into 360 points, referred to as degrees of phase. Thus two adjacent points of phase on a wave are separated by a phase difference of 1 degree. Therefore point C of Figure 1.3.1 (c) is 180 out of 10
phase (one half of a cycle) with point E, 90 out of phase with point D and point G, and 270 out of phase with point F. The difference in phase between any two points on a wave or between any two waves is known as the phase difference θ, as shown by Figure 1.3.2. Displacement x (m) Time t (s) Figure 1.3.2 There are 360 degrees of phase in any one cycle of a wave. The phase difference θ between two points is the fraction of the cycle between the vibrations of the two particles. If two waves have a phase difference of 0 they are in phase. If two waves have a phase difference of 180 they are in anti-phase Worked Example x(m) Consider the following transverse wave; 4 3 2 1 0-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16-2 -3-4 t (s) (a) Mark two points on the wave that are in phase, and two points which are in anti-phase. Mark the direction of travel of a particle at these points. (b) Sketch a second wave on the diagram, which possesses the same wavelength and amplitude, but is ahead of the original wave by 90 degrees of phase. Answer (a) There are a multitude of pairs of points on the wave that are in phase and in antiphase. We therefore choose two which are easy to mark, because they are 11
either at a wave peak or trough, or occur on the horizontal axis, but note that these points could have been chosen anywhere on the wave form as long as they were separated by 0 of phase (red points) or 180 of phase (blue points). The direction of travel of the particles at these points must be the same for particles that are in phase and opposite for particles that are in anti-phase. x(m) 4 B b D 3 2 1 A a c 0 C 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16-1 -2-3 d -4 t (s) (b). Now we must sketch a second wave (red line) which is behind the original by 90 degrees (1 quarter of a cycle since there are 360 degrees in a whole cycle). This means that when the original wave is at zero amplitude, the lagging wave is at maximum amplitude, and when the original is at maximum amplitude the lagging wave is at zero displacement. 4 3 2 1 x(m) 0-1 -2-3 -4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 t (s) 12
Exercise 1.3 Consider the following diagram. 2.5 2 1.5 1 0.5 0-0.5-1 -1.5-2 -2.5 a c g e f b 0 1 2 3 4 5 6 7 8 9 10 d 1. Which pairs of points are in phase? Which pairs of points are in anti-phase? 2. State the phase difference between the following points; i) a and b ii) a and d iii) e and f iv) c and g v) b and d 3. Sketch a transverse wave with an amplitude of 3cm and a wavelength of 5cm. Your transverse wave should have zero amplitude at the origin. 4. On your sketch mark two points with arrows showing particle direction for the following: (a) two points that are out of phase by 90 (b) two points that are out of phase by 360 (c) two points that are in phase (d) two points that are out of phase by 270 What do you notice about the two points that are 360 out of phase? Challenge Question 5. On your sketch, add a wave form with the same amplitude and wavelength as the original wave form but is out of phase by 180. 13