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High School Physics Textbook

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The Free High School Science Texts: A Textbook for High School Students Studying Physics. 1 FHSST Authors December 9, 2005 1 See 1 Units 1.1 PGCE Comments ² Explain what is meant by `physical quantity'. ² Chapter is too full of tables and words; need ¯gures to make it more interesting. ² Make researching history of SI units a small project. ² Multiply by one technique: not positive Suggest using exponents instead (i.e. use the ¡1 table of pre¯xes). This also works better for changing complicated units (km=h to ¡1 m:s etc....). Opinion that this technique is limited in its application. ² Edit NASA story. ² The Temperature section should be cut-down. SW: I have edited the original section but perhaps a more aggressive edit is justi¯ed with the details de®ered until the section on gases. 1.2 `TO DO' LIST ² Write section on scienti¯c notation, signi¯cant ¯gures and rounding. ² Add to sanity test table of sensible values for things. ² Graph Celsius/Kelvin ladder. ² Address PGCE comments above. 1.3 Introduction Imagine you had to make curtains and needed to buy material. The shop assistant would need to know how much material was required. Telling her you need material 2 wide and 6 long would be insu±cient you have to specify the unit (i.e. 2 metres wide and 6 metres long). Without the unit the information is incomplete and the shop assistant would have to guess. If you were making curtains for a doll's house the dimensions might be 2 centimetres wide and 6 centimetres long 3Base quantity Name Symbol length metre m mass kilogram kg time second s electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd Table 1.1: SI Base Units It is not just lengths that have units, all physical quantities have units (e.g. time and tem- perature). 1.4 Unit Systems There are many unit systems in use today. Physicists, for example, use 4 main sets of units: SI units, c.g.s units, imperial units and natural units. Depending on where you are in the world or what area of physics you work in, the units will be di®erent. For example, in South Africa road distances are measured in kilometres (SI units), while in England they are measured in miles (imperial units). You could even make up your own system of units if you wished, but you would then have to teach people how to use it 1.4.1 SI Units (Systµ eme International d'Unit¶ es) These are the internationally agreed upon units and the ones we will use. Historically these units are based on the metric system which was developed in France at the time of the French Revolution. All physical quantities have units which can be built from the 7 base units listed in Table 1.1 (incidentally the choice of these seven was arbitrary). They are called base units because none of them can be expressed as combinations of the other six. This is similar to breaking a language down into a set of sounds from which all words are made. Another way of viewing the base units is like the three primary colours. All other colours can be made from the primary colours but no primary colour can be made by combining the other two primaries. Unit names are always written with lowercase initials (e.g. the metre). The symbols (or abbreviations) of units are also written with lowercase initials except if they are named after scientists (e.g. the kelvin (K) and the ampere (A)). To make life convenient, particular combinations of the base units are given special names. This makes working with them easier, but it is always correct to reduce everything to the base units. Table 1.2 lists some examples of combinations of SI base units assigned special names. Do not be concerned if the formulae look unfamiliar at this stage we will deal with each in detail in the chapters ahead (as well as many others) It is very important that you are able to say the units correctly. For instance, the newton is ¡2 another name for the kilogram metre per second squared (kg:m:s ), while the kilogram 2 ¡2 metre squared per second squared (kg:m :s ) is called the joule. Another important aspect of dealing with units is the pre¯xes that they sometimes have (pre¯xes are words or letters written in front that change the meaning). The kilogram (kg) is a 3 3 simple example. 1kg is 1000g or 1£ 10 g. Grouping the 10 and the g together we can replace 4Quantity Formula Unit Expressed in Name of Base Units Combination ¡2 Force ma kg:m:s N (newton) 1 ¡1 Frequency s Hz (hertz) T 2 ¡2 Work & Energy F:s kg:m :s J (joule) Table 1.2: Some Examples of Combinations of SI Base Units Assigned Special Names 3 3 the 10 with the pre¯x k (kilo). Therefore the k takes the place of the 10 . Incidentally the kilogram is unique in that it is the only SI base unit containing a pre¯x There are pre¯xes for many powers of 10 (Table 1.3 lists a large set of these pre¯xes). This is a larger set than you will need but it serves as a good reference. The case of the pre¯x symbol is very important. Where a letter features twice in the table, it is written in uppercase for exponents bigger than one and in lowercase for exponents less than one. Those pre¯xes listed in boldface should be learnt. Pre¯x Symbol Exponent Pre¯x Symbol Exponent 24 ¡24 yotta Y 10 yocto y 10 21 ¡21 zetta Z 10 zepto z 10 18 ¡18 exa E 10 atto a 10 15 ¡15 peta P 10 femto f 10 12 ¡12 tera T 10 pico p 10 9 ¡9 giga G 10 nano n 10 6 ¡6 mega M 10 micro ¹ 10 3 ¡3 kilo k 10 milli m 10 2 ¡2 hecto h 10 centi c 10 1 ¡1 deca da 10 deci d 10 Table 1.3: Unit Pre¯xes ¡3 As another example of the use of pre¯xes, 1£ 10 g can be written as 1mg (1 milligram). 1.4.2 The Other Systems of Units The remaining sets of units, although not used by us, are also internationally recognised and still in use by others. We will mention them brie°y for interest only. c.g.s Units In this system the metre is replaced by the centimetre and the kilogram is replaced by the gram. This is a simple change but it means that all units derived from these two are changed. For example, the units of force and work are di®erent. These units are used most often in astrophysics and atomic physics. Imperial Units These units (as their name suggests) stem from the days when monarchs decided measures. Here all the base units are di®erent, except the measure of time. This is the unit system you are most likely to encounter if SI units are not used. These units are used by the Americans and 5British. As you can imagine, having di®erent units in use from place to place makes scienti¯c communication very di±cult. This was the motivation for adopting a set of internationally agreed upon units. Natural Units This is the most sophisticated choice of units. Here the most fundamental discovered quantities (such as the speed of light) are set equal to 1. The argument for this choice is that all other quantities should be built from these fundamental units. This system of units is used in high energy physics and quantum mechanics. 1.5 The Importance of Units Without units much of our work as scientists would be meaningless. We need to express our thoughts clearly and units give meaning to the numbers we calculate. Depending on which units we use, the numbers are di®erent (e.g. 3.8 m and 3800 mm actually represent the same length). Units are an essential part of the language we use. Units must be speci¯ed when expressing physical quantities. In the case of the curtain example at the beginning of the chapter, the result of a misunderstanding would simply have been an incorrect amount of material cut. However, sometimes such misunderstandings have catastrophic results. Here is an extract from a story on CNN's website: (NOTE TO SELF: This quote may need to be removed as the licence we are using allows for all parts of the document to be copied and I am not sure if this being copied is legit in all ways?) NASA: Human error caused loss of Mars orbiter November 10, 1999 WASHINGTON (AP) Failure to convert English measures to metric values caused the loss of the Mars Climate Orbiter, a spacecraft that smashed into the planet instead of reaching a safe orbit, a NASA investigation concluded Wednesday. The Mars Climate Orbiter, a key craft in the space agency's exploration of the red planet, vanished after a rocket ¯ring September 23 that was supposed to put the spacecraft on orbit around Mars. An investigation board concluded that NASA engineers failed to convert English measures of rocket thrusts to newton, a metric system measuring rocket force. One English pound of force equals 4.45 newtons. A small di®erence between the two values caused the spacecraft to approach Mars at too low an altitude and the craft is thought to have smashed into the planet's atmosphere and was destroyed. The spacecraft was to be a key part of the exploration of the planet. From its station about the red planet, the Mars Climate Orbiter was to relay signals from the Mars Polar Lander, which is scheduled to touch down on Mars next month. \The root cause of the loss of the spacecraft was a failed translation of English units into metric units and a segment of ground-based, navigation-related mission software," said Arthus Stephenson, chairman of the investigation board. This story illustrates the importance of being aware that di®erent systems of units exist. Furthermore, we must be able to convert between systems of units 61.6 Choice of Units There are no wrong units to use, but a clever choice of units can make a problem look simpler. The vast range of problems makes it impossible to use a single set of units for everything without making some problems look much more complicated than they should. We can't easily compare the mass of the sun and the mass of an electron, for instance. This is why astrophysicists and atomic physicists use di®erent systems of units. We won't ask you to choose between di®erent unit systems. For your present purposes the SI system is perfectly su±cient. In some cases you may come across quantities expressed in units other than the standard SI units. You will then need to convert these quantities into the correct SI units. This is explained in the next section. 1.7 How to Change Units the \Multiply by 1" Technique Firstly you obviously need some relationship between the two units that you wish to convert between. Let us demonstrate with a simple example. We will consider the case of converting millimetres (mm) to metres (m) the SI unit of length. We know that there are 1000mm in 1m which we can write as 1000mm = 1m: 1 Now multiplying both sides by we get 1000mm 1 1 1000mm = 1m; 1000mm 1000mm which simply gives us 1m 1 = : 1000mm This is the conversion ratio from millimetres to metres. You can derive any conversion ratio in this way from a known relationship between two units. Let's use the conversion ratio we have just derived in an example: Question: Express 3800mm in metres. Answer: 3800mm = 3800mm£ 1 1m = 3800mm£ 1000mm = 3:8m Note that we wrote every unit in each step of the calculation. By writing them in and cancelling them properly, we can check that we have the right units when we are ¯nished. We m started with `mm' and multiplied by ` '. This cancelled the `mm' leaving us with just `m' mm the SI unit we wanted to end up with If we wished to do the reverse and convert metres to millimetres, then we would need a conversion ratio with millimetres on the top and metres on the bottom. 71.8 How Units Can Help You We conclude each section of this book with a discussion of the units most relevant to that particular section. It is important to try to understand what the units mean. That is why thinking about the examples and explanations of the units is essential. If we are careful with our units then the numbers we get in our calculations can be checked in a `sanity test'. 1.8.1 What is a `sanity test'? This isn't a special or secret test. All we do is stop, take a deep breath, and look at our answer. Sure we always look at our answers or do we? This time we mean stop and really look does our answer make sense? Imagine you were calculating the number of people in a classroom. If the answer you got was 1 000 000 people you would know it was wrong that's just an insane number of people to have in a classroom. That's all a sanity check is is your answer insane or not? But what units were we using? We were using people as our unit. This helped us to make sense of the answer. If we had used some other unit (or no unit) the number would have lacked meaning and a sanity test would have been much harder (or even impossible). It is useful to have an idea of some numbers before we start. For example, let's consider masses. An average person has mass 70kg, while the heaviest person in medical history had a mass of 635kg. If you ever have to calculate a person's mass and you get 7000kg, this should fail your sanity check your answer is insane and you must have made a mistake somewhere. In the same way an answer of 0:00001kg should fail your sanity test. The only problem with a sanity check is that you must know what typical values for things are. In the example of people in a classroom you need to know that there are usually 2050 people in a classroom. Only then do you know that your answer of 1 000 000 must be wrong. Here is a table of typical values of various things (big and small, fast and slow, light and heavy you get the idea): Category Quantity Minimum Maximum Mass People Height Table 1.4: Everyday examples to help with sanity checks (NOTE TO SELF: Add to this table as we go along with examples from each section.) Now you don't have to memorise this table but you should read it. The best thing to do is to refer to it every time you do a calculation. 1.9 Temperature We need to make a special mention of the units used to describe temperature. The unit of temperature listed in Table 1.1 is not the everyday unit we see and use. Normally the Celsius scale is used to describe temperature. As we all know, Celsius temper- atures can be negative. This might suggest that any number is a valid temperature. In fact, the temperature of a gas is a measure of the average kinetic energy of the particles that make up the gas. As we lower the temperature so the motion of the particles is reduced until a point is reached 8where all motion ceases. The temperature at which this occurs is called absolute zero. There is o no physically possible temperature colder than this. In Celsius, absolute zero is at¡273 C. Physicists have de¯ned a new temperature scale called the Kelvin scale. According to this scale absolute zero is at 0K and negative temperatures are not allowed. The size of one unit kelvin is exactly the same as that of one unit Celsius. This means that a change in temperature of 1 degree kelvin is equal to a change in temperature of 1 degree Celsius the scales just start in di®erent places. Think of two ladders with steps that are the same size but the bottom most step on the Celsius ladder is labelled -273, while the ¯rst step on the Kelvin ladder is labelled 0. There are still 100 steps between the points where water freezes and boils. 102 Celsius 375 Kelvin 101 Celsius 374 Kelvin water boils - 100 Celsius 373 Kelvin 99 Celsius 372 Kelvin 98 Celsius 371 Kelvin . . . 2 Celsius 275 Kelvin 1 Celsius 274 Kelvin ice melts - 0 Celsius 273 Kelvin -1 Celsius 272 Kelvin -2 Celsius 271 Kelvin . . . -269 Celsius 4 Kelvin -270 Celsius 3 Kelvin -271 Celsius 2 Kelvin -272 Celsius 1 Kelvin absolute zero - -273 Celsius 0 Kelvin (NOTE TO SELF: Come up with a decent picture of two ladders with the labels water boiling and freezingin the same place but with di®erent labelling on the steps) This makes the conversion from kelvin to Celsius and back very easy. To convert from Cel- sius to kelvin add 273. To convert from kelvin to Celsius subtract 273. Representing the Kelvin o temperature by T and the Celsius temperature by T , K C T = To + 273: (1.1) K C It is because this conversion is additive that a di®erence in temperature of 1 degree Celsius is equal to a di®erence of 1 kelvin. The majority of conversions between units are multiplicative. For example, to convert from metres to millimetres we multiply by 1000. Therefore a change of 1m is equal to a change of 1000mm. 1.10 Scienti¯c Notation, Signi¯cant Figures and Rounding (NOTE TO SELF: still to be written) 91.11 Conclusion In this chapter we have discussed the importance of units. We have discovered that there are many di®erent units to describe the same thing, although you should stick to SI units in your calculations. We have also discussed how to convert between di®erent units. This is a skill you must acquire. 10Chapter 2 Waves and Wavelike Motion Waves occur frequently in nature. The most obvious examples are waves in water, on a dam, in the ocean, or in a bucket. We are most interested in the properties that waves have. All waves have the same properties so if we study waves in water then we can transfer our knowledge to predict how other examples of waves will behave. 2.1 What are waves? 1 Waves are disturbances which propagate (move) through a medium . Waves can be viewed as a transfer energy rather than the movement of a particle. Particles form the medium through which waves propagate but they are not the wave. This will become clearer later. Lets consider one case of waves: water waves. Waves in water consist of moving peaks and troughs. A peak is a place where the water rises higher than when the water is still and a trough is a place where the water sinks lower than when the water is still. A single peak or trough we call a pulse. A wave consists of a train of pulses. So waves have peaks and troughs. This could be our ¯rst property for waves. The following diagram shows the peaks and troughs on a wave. Peaks Troughs In physics we try to be as quantitative as possible. If we look very carefully we notice that the height of the peaks above the level of the still water is the same as the depth of the troughs below the level of the still water. The size of the peaks and troughs is the same. 2.1.1 Characteristics of Waves : Amplitude The characteristic height of a peak and depth of a trough is called the amplitude of the wave. The vertical distance between the bottom of the trough and the top of the peak is twice the amplitude. We use symbols agreed upon by convention to label the characteristic quantities of 1 Light is a special case, it exhibits wave-like properties but does not require a medium through which to propagate. 11the waves. Normally the letter A is used for the amplitude of a wave. The units of amplitude are metres (m). Amplitude 2 x Amplitude Amplitude Worked Example 1 Question: (NOTE TO SELF: Make this a more exciting question) If the peak of a wave measures 2m above the still water mark in the harbour what is the amplitude of the wave? Answer: The de¯nition of the amplitude is the height that the water rises to above when it is still. This is exactly what we were told, so the answer is that the amplitude is 2m. 2.1.2 Characteristics of Waves : Wavelength Look a little closer at the peaks and the troughs. The distance between two adjacent (next to each other) peaks is the same no matter which two adjacent peaks you choose. So there is a ¯xed distance between the peaks. Looking closer you'll notice that the distance between two adjacent troughs is the same no matter which two troughs you look at. But, more importantly, its is the same as the distance between the peaks. This distance which is a characteristic of the wave is called the wavelength. Waves have a characteristic wavelength. The symbol for the wavelength is ¸. The units are metres (m). ¸ ¸ ¸ The wavelength is the distance between any two adjacent points which are in phase. Two points in phase are separate by an integer (0,1,2,3,...) number of complete wave cycles. They don't have to be peaks or trough but they must be separated by a complete number of waves. 2.1.3 Characteristics of Waves : Period Now imagine you are sitting next to a pond and you watch the waves going past you. First one peak, then a trough and then another peak. If you measure the time between two adjacent peaks you'll ¯nd that it is the same. Now if you measure the time between two adjacent troughs you'll 12¯nd that its always the same, no matter which two adjacent troughs you pick. The time you have been measuring is the time for one wavelength to pass by. We call this time the period and it is a characteristic of the wave. Waves have a characteristic time interval which we call the period of the wave and denote with the symbol T. It is the time it takes for any two adjacent points which are in phase to pass a ¯xed point. The units are seconds (s). 2.1.4 Characteristics of Waves : Frequency There is another way of characterising the time interval of a wave. We timed how long it takes for one wavelength to pass a ¯xed point to get the period. We could also turn this around and say how many waves go by in 1 second. We can easily determine this number, which we call the frequency and denote f. To determine the frequency, how many waves go by in 1s, we work out what fraction of a waves goes by in 1 1 second by dividing 1 second by the time it takes T. If a wave takes a second to go by then in 2 1 ¡1 1 second two waves must go by. = 2. The unit of frequency is the Hz or s . 1 2 Waves have a characteristic frequency. 1 f = T ¡1 f : frequency (Hz or s ) T : period (s) 2.1.5 Characteristics of Waves : Speed Now if you are watching a wave go by you will notice that they move at a constant velocity. The speed is the distance you travel divided by the time you take to travel that distance. This is excellent because we know that the waves travel a distance ¸ in a time T. This means that we can determine the speed. ¸ v = T ¡1 v : speed (m:s ) ¸ : wavelength (m) T : period (s) There are a number of relationships involving the various characteristic quantities of waves. A simple example of how this would be useful is how to determine the velocity when you have the frequency and the wavelength. We can take the above equation and substitute the relationship between frequency and period to produce an equation for speed of the form v = f¸ ¡1 v : speed (m:s ) ¸ : wavelength (m) ¡1 f : frequency (Hz or s ) Is this correct? Remember a simple ¯rst check is to check the units On the right hand ¡1 side we have velocity which has units ms . On the left hand side we have frequency which is 13¡1 measured in s multiplied by wavelength which is measure in m. On the left hand side we have ¡1 ms which is exactly what we want. 2.2 Two Types of Waves We agreed that a wave was a moving set of peaks and troughs and we used water as an example. Moving peaks and troughs, with all the characteristics we described, in any medium constitute a wave. It is possible to have waves where the peaks and troughs are perpendicular to the direction of motion, like in the case of water waves. These waves are called transverse waves. There is another type of wave. Called a longitudinal wave and it has the peaks and troughs in the same direction as the wave is moving. The question is how do we construct such a wave? An example of a longitudinal wave is a pressure wave moving through a gas. The peaks in this wave are places where the pressure reaches a peak and the troughs are places where the pressure is a minimum. In the picture below we show the random placement of the gas molecules in a tube. The piston at the end moves into the tube with a repetitive motion. Before the ¯rst piston stroke the pressure is the same throughout the tube. PiEC & q)k JLK4tu: 2U68 A + j;D (Xxd g ZoY - mcW, Qef MhOb’ vs3\ H%?. 5w /0G r_ Bn 7‘FaI" zSyTl R 9V1 N=p When the piston moves in it compresses the gas molecules together at the end of the tube. If the piston stopped moving the gas molecules would all bang into each other and the pressure would increase in the tube but if it moves out again fast enough then pressure waves can be set up. -Ɖ– AN†§ &0’œß K)”¨¸˘ O%S9flfi ł¡¢ J MT£‡› Q «Ł¿Œ "' +2H=—ˇ• UWVØ F´⁄ˆ ˝æ¯ … P:CB?‚ 8 »L¶ 7I(Dø „“˜ 465,/G˙` 1ER ˛·˚ƒı .¤ ;3º ¥‹ ª When the piston moves out again before the molecules have time to bang around then the increase in pressure moves down the tube like a pulse (single peak). The piston moves out so fast that a pressure trough is created behind the peak. ßj ªºƒ‚ lY p¶f a ¿£ n·⁄ ØŁ\ “´uy Zm ¡fl¢¨˚ owx tq‰˘‡ ˙fi łı« ˆ`v æ— ¯˝cd” ¥• bœ˛ X¤¸ Œk– „ sø_˜ Ƨˇ ›zg'‹hi» e†‘… r As this repeats we get waves of increased and decreased pressure moving down the tubes. We can describe these pulses of increased pressure (peaks in the pressure) and decreased pressure (troughs of pressure) by a sine or cosine graph. TÆO8U fl(¸.L3 ´˜ˆZ0 ˙G WC„'1˝K £¥⁄fi”–¯ †?y łlA) ›s:;/ H˛º6 JMØq ƒ¤§ˇ—ªE •opd `‰‘ıfe\= «˘k œ4Q ¶Vvt ,gh ‚‡i+ Xb5a S"% _Łu’mI2 ·z‹ Œj »æß&- øPD7YB ¿¢¡˚Fr ¨cwx9R …“nN 14Incident ray There are a number of examples of each type of wave. Not all can be seen with the naked eye but all can be detected. 2.3 Properties of Waves We have discussed some of the simple characteristics of waves that we need to know. Now we can progress onto some more interesting and, perhaps, less intuitive properties of waves. 2.3.1 Properties of Waves : Re°ection When waves strike a barrier they are re°ected. This means that waves bounce o® things. Sound waves bounce o® walls, light waves bounce o® mirrors, radar waves bounce o® planes and it can explain how bats can °y at night and avoid things as small as telephone wires. The property of re°ection is a very important and useful one. (NOTE TO SELF: Get an essay by an air tra±c controller on radar) (NOTE TO SELF: Get an essay by on sonar usage for ¯shing or for submarines) When waves are re°ected, the process of re°ection has certain properties. If a wave hits an obstacle at a right angle to the surface (NOTE TO SELF: diagrams needed) then the wave is re°ected directly backwards. If the wave strikes the obstacle at some other angle then it is not re°ected directly backwards. The angle that the waves arrives at is the same as the angle that the re°ected waves leaves at. The angle that waves arrives at or is incident at equals the angle the waves leaves at or is re°ected at. Angle of incidence equals angle of re°ection µ = µ (2.1) i r 15Incident ray µ = µ i r µ : angle of incidence i µ : angle of re°ection r µ µ i r In the optics chapter you will learn that light is a wave. This means that all the properties we have just learnt apply to light as well. Its very easy to demonstrate re°ection of light with a mirror. You can also easily show that angle of incidence equals angle of re°ection. If you look directly into a mirror your see yourself re°ected directly back but if you tilt the mirror slightly you can experiment with di®erent incident angles. Phase shift of re°ected wave When a wave is re°ected from a more dense medium it undergoes a phase shift. That means that the peaks and troughs are swapped around. The easiest way to demonstrate this is to tie a piece of string to something. Stretch the string out °at and then °ick the string once so a pulse moves down the string. When the pulse (a single peak in a wave) hits the barrier that the string is tide to it will be re°ected. The re°ected wave will look like a trough instead of a peak. This is because the pulse had undergone a phase change. The ¯xed end acts like an extremely dense medium. If the end of the string was not ¯xed, i.e. it could move up and down then the wave would still be re°ected but it would not undergo a phase shift. To draw a free end we draw it as a ring around a line. This signi¯es that the end is free to move. 162.3.2 Properties of Waves : Refraction Sometimes waves move from one medium to another. The medium is the substance that is carrying the waves. In our ¯rst example this was the water. When the medium properties change it can a®ect the wave. Let us start with the simple case of a water wave moving from one depth to another. The 2 speed of the wave depends on the depth . If the wave moves directly from the one medium to the other then we should look closely at the boundary. When a peak arrives at the boundary and moves across it must remain a peak on the other side of the boundary. This means that the peaks pass by at the same time intervals on either side of the boundary. The period and frequency remain the same But we said the speed of the wave changes, which means that the distance it travels in one time interval is di®erent i.e. the wavelength has changed. Going from one medium to another the period or frequency does not change only the wave- length can change. Now if we consider a water wave moving at an angle of incidence not 90 degrees towards a change in medium then we immediately know that not the whole wavefront will arrive at once. So if a part of the wave arrives and slows down while the rest is still moving faster before it arrives the angle of the wavefront is going to change. This is known as refraction. When a wave bends or changes its direction when it goes from one medium to the next. If it slows down it turns towards the perpendicular. 2 17Air Water If the wave speeds up in the new medium it turns away from the perpendicular to the medium surface. Air Water When you look at a stick 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 di®erent place. 18Air Water 2.3.3 Properties of Waves : Interference If two waves meet interesting things can happen. Waves are basically collective motion of parti- cles. So when two waves meet they both try to impose their collective motion on the particles. This can have quite di®erent results. If two identical (same wavelength, amplitude and frequency) waves are both trying to form a peak then they are able to achieve the sum of their e®orts. The resulting motion will be a peak which has a height which is the sum of the heights of the two waves. If two waves are both trying to form a trough in the same place then a deeper trough is formed, the depth of which is the sum of the depths of the two waves. Now in this case the two waves have been trying to do the same thing and so add together constructively. This is called constructive interference. A=0.5 B=1 A+B=1.5 If one wave is trying to form a peak and the other is trying to form a trough then they are competing to do di®erent things. In this case they can cancel out. The amplitude of the resulting 19wave will depend on the amplitudes of the two waves that are interfering. If the depth of the trough is the same as the height of the peak nothing will happen. If the height of the peak is bigger than the depth of the trough a smaller peak will appear and if the trough is deeper then a less deep trough will appear. This is destructive interference. A=0.5 B=1 B-A=0.5 2.3.4 Properties of Waves : Standing Waves When two waves move in opposite directions, through each other, interference takes place. If the two waves have the same frequency and wavelength then a speci¯c type of constructive interference can occur: standing waves can form. Standing waves are disturbances which don't appear to move, they look like they stay in the same place even though the waves that from them are moving. Lets demonstrate exactly how this comes about. Imagine a long string with waves being sent down it from either end. The waves from both ends have the same amplitude, wavelength and frequency as you can see in the picture below: 1 0 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5 To stop from getting confused between the two waves we'll draw the wave from the left with a dashed line and the one from the right with a solid line. As the waves move closer together when they touch both waves have an amplitude of zero: 201 0 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5 If we wait for a short time the ends of the two waves move past each other and the waves overlap. Now we know what happens when two waves overlap, we add them together to get the resulting wave. 1 0 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5 Now we know what happens when two waves overlap, we add them together to get the resulting wave. In this picture we show the two waves as dotted lines and the sum of the two in the overlap region is shown as a solid line: 1 0 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5 The important thing to note in this case is that there are some points where the two waves always destructively interfere to zero. If we let the two waves move a little further we get the picture below: 1 0 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5 Again we have to add the two waves together in the overlap region to see what the sum of the waves looks like. 1 0 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5 In this case the two waves have moved half a cycle past each other but because they are out of phase they cancel out completely. The point at 0 will always be zero as the two waves move past each other. 21