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The Free High School Science Texts:

A Textbook for High School Students Studying Physics.

About FHSST

[edit] Table of contents

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[edit] Introduction

<< Main Page -- First Chapter (Units) >>

[edit] Introduction

Physics is the study of the laws which govern space, structure and time. In a sense we are more qualified to do physics than any other science. From the day we are born we study the things around us in an effort to understand how they work and relate to each other. For example, learning how to catch or throw a ball is a physics undertaking.

In the field of study we refer to as physics we just try to make the things everyone has been studying more clear. We attempt to describe them through simple rules and mathematics. Mathematics is merely the language we use. The best approach to physics is to relate everything you learn to things you have already noticed in your everyday life. Sometimes when you look at things closely, you discover things you had initially overlooked.

It is the continued scrutiny of everything we know about the world around us that leads people to the lifelong study of physics. You can start with asking a simple question like "Why is the sky blue?", which could lead you to electromagnetic waves, which in turn could lead you to wave particle duality and energy levels of atoms. Before long you are studying quantum mechanics or the structure of the universe.

In the sections that follow notice that we will try to describe how we will communicate the things we are dealing with. This is our language. Once this is done we can begin the adventure of looking more closely at the world we live in.

[edit] Units

[edit] PGCE Comments

• Explain what is meant by `physical quantity'. -- clark: In the Introduction section, I replaced 'physical quantity' with 'measurement of a physical phenomenon'. Don't know if that's much clearer, really.
• Chapter is too full of tables and words need figures to make it more interesting.
• Make researching history of SI units a small project.
• Multiply by one technique: not positive! Suggest using exponents instead (i.e. use the table of prefixes). This also works better for changing complicated units (km·h^-1 to m·s⁻¹ etc....). Opinion that this technique is limited in its application.
• Edit NASA story.
• The Temperature section should be cut-down. SW: I have edited the original section but perhaps a more aggressive edit is justified with the details deffered until the section on gases.

[edit] Introduction

Science focuses on studying how things happen in the real world -- things you can see, touch or feel. In order to describe these things, it is necessary to carefully measure what is observed. Measurements must always be reported and discussed with appropriate units, which specify what type of quantity is being discussed.

As a simple example, 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 insufficient-- 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!

It is not just lengths that have units. Any measurement of any physical phenomenon--time, temperature, force, or voltage, just to name a few--has units.

Tip: Many physics problems ask you to determine a specific numeric quantity. When you solve the problem, do not forget to specify the units of your answer: even if you have the right number, your answer is not correct unless you include the correct units.

In the remainder of this class we will be using SI units, which are defined in the table below. These seven units are used to measure fundamental quantities, and are the basis of everything we will do, as will be discussed in more detail in the next section.

Table 1.1: SI Base Units

Base quantity	Name	Symbol
length	metre	m
mass	kilogram	kg
time	second	s
electric current	ampere	A
thermodynamic temperature	kelvin	K
countable amount of substance	mole	mol
luminous intensity	candela	cd

[edit] SI Units (Système International d'Unités)

These units are internationally agreed upon and form the system we will use. Historically these units are based on the metric system which was developed in France at the time of the French Revolution.

Table 1.1: SI Base Units

Base 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

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)). An exception to this rule is the litre, which is abbreviated as either L or l.

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 another name for the kilogram metre per second squared (kg·m·s⁻²), while the kilogram metre squared per second squared (kg·m²·s⁻²) is called the joule.

Table 1.2: Some Examples of Combinations of SI Base Units Assigned Special Names

Quantity	Formula	Unit Expressed in	Name of
		Base Units	Combination
Force	m·a	kg·m·s−2	N (newton)
Frequency		s−1	Hz (hertz)
Work & Energy	F·s	kg·m2·s−2	J (joule)
Electrical Potential	W/A	kg·m2·s−3·A−1	V (volt)

Another important aspect of dealing with units is the prefixes that they sometimes have (prefixes are words or letters written in front that change the meaning). The kilogram (kg) is a simple example: 1 kg is 1000 g or . Grouping the 10³ and the g together we can replace the 10³ with the prefix k (kilo). Therefore the k takes the place of the 10³. Incidentally the kilogram is unique in that it is the only SI base unit containing a prefix.

There are prefixes for many powers of 10 (Table 1.3 lists a large set of these prefixes). This is a larger set than you will need but it serves as a good reference. The case of the prefix 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 prefixes listed in boldface should be learnt.

Table 1.3: Unit Prefixes

Prefix	Symbol	Exponent	Prefix	Symbol	Exponent
yotta	Y	1024	yocto	y	10-24
zetta	Z	1021	zepto	z	10-21
exa	E	1018	atto	a	10-18
peta	P	1015	femto	f	10-15
tera	T	1012	pico	p	10-12
giga	G	109	nano	n	10-9
mega	M	106	micro	µ	10-6
kilo	k	103	milli	m	10-3
hecto	h	102	centi	c	10-2
deca	da	101	deci	d	10-1

As another example of the use of prefixes,

can be written as 1 mg (1 milligram).

[edit] 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 briefly for interest only.

[edit] 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 different. These units are used most often in astrophysics and atomic physics.

When electromagnetism comes into play, there are three CGS systems, adapted to the fundamental equations each theory views as basic: The electric CGS, the magnetic CGS, and the combined Gaussian. The latter has the advantage that corresponding electric and magnetic phenomena have the same units and related equations.

It has the additional advantage that there is only one natural constant in the equations, the speed of light, where the SI system has two. And experience, i.e. measurements, has shown that there is only one constant. So the Gaussion system is a bit more 'right'.

These unit systems also show that the choice of base units is arbitrary. In SI, there is a base unit for the current, the ampere [A], derived from it the unit of charge, coulomb [C]. The Gaussian system does without a dedicated unit for electricity. It simply defines the factor in the law of force between two charged particles as one - and lo, the unit C disappears; the esu (electrostatic unit) can be derived from g, cm, s - the C cannot, it is As, and A is basic.

[The same could be done with mass, leading to kg vanishing, just by setting the gravitational constant in Newton's law to one. kg would then be replaced by a combination of m and s.

[edit] Imperial Units

These units (as their name suggests) stem from the days when monarchs decided measures. As a result, different countries used different base units for each quantity (except for time). One version of the Imperial units is still in common use in the United States; however, the SI system is overwhelmingly predominant for scientific and technical use. The British once used another version of the imperial system, but now use SI units almost exclusively. Using different units in different places would make effective scientific communication very difficult. That is why the scientific community has adopted SI units as its internationally agreed upon standard.

[edit] 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.

[edit] 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 different (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 specified 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 firing 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 difference 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 different systems of units exist. Furthermore, we must be able to convert between systems of units!

[edit] 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 different systems of units.

We won't ask you to choose between different unit systems. For your present purposes the SI system is perfectly sufficient. 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.

[edit] How to Change Units-- the "Multiply by 1" Technique

Also known as fractional dimensional analysis, the technique involves multiplying a labeled quantity by a conversion ratio, or knowledge of conversion factors. First, a relationship between the two units that you wish to convert between must be found. Here's a simple example: converting millimetres (mm) to metres (m)-- the SI unit of length. We know that there are 1000 mm in 1 m which we can write as

Now multiplying both sides by

we get

which simply gives us

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 3800 mm in metres.

Answer:

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 finished. We started with mm and multiplied by

This cancelled the mm leaving us with just m, which is the SI unit we wanted! 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.

It is helpful to understand that units cancel when one is in the numerator and the other is in the denominator. If the unit you are trying to cancel is on the top, then the conversion factor that you multiply it with must be on the bottom.

This same technique can be used to not just to convert units, but can also be used as a way to solve for an unknown quantity. For example: If I was driving at 65 miles per hour, then I could find how far I would go in 5 hours by using as a conversion factor.

This would look like

This would yield a result of 325 miles because the hours would cancel leaving miles as the only unit.

[edit] Practice Problem

Problem: Convert 3 millennia into seconds.

Most people don't know how many seconds are in a millennium, but they do know enough to solve this problem. Since we know 1000 years = 1 millennium, 1 year = about 365.2425 days, 1 day = 24 hours, and 1 hour = 3600 seconds we can solve this problem by multiplying by one many times.

[edit] 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'.

[edit] 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 70 kg, while the heaviest person in medical history had a mass of 635 kg. If you ever have to calculate a person's mass and you get 7000 kg, 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 0.00001 kg 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 20-50 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):

Table 1.4: Everyday examples to help with sanity checks Category Quantity Minimum Maximum People Mass Height

(NOTE TO SELF: Add to this table as we go along with examples from each section.)

Note that you do not have to memorize this table. However, read it so that you can refer to it when you do a calculation.

[edit] Temperature

In everyday life, many people measure temperatures in Celsius. But in physics, we prefer to use kelvin.

As we all know, Celsius temperatures 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 where all motion ceases. The temperature at which this occurs is called absolute zero. There is no physically possible temperature colder than this. In Celsius, absolute zero is at -273 °C.

Physicists have defined a new temperature scale called the Kelvin scale. According to this scale absolute zero is at 0 K and negative temperatures are not allowed. The size of one unit kelvin is exactly the same as that of one unit degree Celsius. This means that a change in temperature of 1 kelvin is equal to a change in temperature of 1 degree Celsius—the scales just start in different 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 first step on the Kelvin ladder is labelled 0. There are still 100 steps between the points where water freezes and boils (at 1.0 atmosphere of pressure).

                         |----|   102 degrees Celsius    |----|  375 kelvin
                         |----|   101 degrees Celsius    |----|  374 kelvin
 water boils  --->       |----|   100 degrees Celsius    |----|  373 kelvin
                         |----|   99  degrees Celsius    |----|  372 kelvin
                         |----|   98  degrees Celsius    |----|  371 kelvin
                                             .
                                             .
                                             .
                         |----|   2   degrees Celsius    |----|  275 kelvin
                         |----|   1   degree Celsius     |----|  274 kelvin
 ice melts    --->       |----|   0   degrees Celsius    |----|  273 kelvin
                         |----|   -1  degree Celsius     |----|  272 kelvin
                         |----|   -2  degrees Celsius    |----|  271 kelvin
                                             .
                                             .
                                             .
                         |----|  -269 degrees Celsius    |----|  4 kelvin
                         |----|  -270 degrees Celsius    |----|  3 kelvin
                         |----|  -271 degrees Celsius    |----|  2 kelvin
                         |----|  -272 degrees Celsius    |----|  1 kelvin
 absolute zero --->      |----|  -273 degrees Celsius    |----|  0 kelvin
 

(NOTE TO SELF: Come up with a decent picture of two ladders with the labels --water boiling and freezing--in the same place but with different labelling on the steps!)

This makes the conversion from kelvin to degree Celsius and back very easy. To convert from degrees Celsius to kelvins add 273. To convert from kelvins to degrees Celsius subtract 273. Representing the Kelvin temperature by T_K and the Celsius temperature by T_°C,

or

Converting between kelvin and Celsius is additive -- so a difference in temperature of 1 degree Celsius is equal to a difference 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 1 m is equal to a change of 1000 mm.

Although it seems as though there is not much reason for the scientific community to use the Kelvin scale over the Celsius scale, there is actually a very remarkable difference in using the Kelvin scale other than the reminding effect stated above. It happens that the idealised form of some materials varies proportionately according to the Kelvin scale, such that some required values can be simply found by multiplication and division. In such calculations, it is just a chore to use the Celsius scale.

[edit] Scientific Notation, Significant Figures and Rounding

If you are only sure of say, both digits of a two-digit number, and put it in a formula and get a long series of numbers to the right of the decimal place, then those digits are probably not very accurate. This is the idea of significant figures.

Take 10 and divide by 3. If you are not sure that the number 10 is perfectly accurate, then you do not need to write down 3.333... and can get away with something like 3.3 or 3.33

(NOTE TO SELF: still to be written)

The accuracy of a measurement using significant figures is represented by the number of digits that it contains. A number is said to have the number of significant figures equal to the number of digits in the number not including leading 0s or trailing 0s unless there is a decimal point. The table below contains a list of numbers and how many significant digits each contains.

Table ?: Significant Digits Number Significant Digits 1000 1 1000. 4 10.0 3 010 2 232 3 23.2 3 1 3 As you may have noted, some numbers cannot be shown in proper significant figure notation without the use of scientific notation. For example, the number 1000 can only be shown to have 1 or 4 or more significant digits by the inclusion of a decimal point. However, by rewriting 1000 as any number of significant digits may be added by simply add additional 0s after the decimal point. The Free High School Science Texts: A Textbook for High School Students Studying Physics. Main Page - Next Chapter (Waves and Wavelike Motion) >> Units Units - PGCE Comments - TO DO LIST - Introduction - Unit Systems - The Importance of Units - Choice of Units - How to Change Units-- the Multiply by 1 Technique - How Units Can Help You - Temperature - Scientific Notation, Significant Figures and Rounding - Conclusion [edit] Conclusion In this chapter we have discussed the importance of units. We have discovered that there are many different units to describe the same thing, although you should stick to SI units in your calculations. We have also discussed how to convert between different units. This is a skill you must acquire. [edit] Waves and wavelike motion The Free High School Science Texts: A Textbook for High School Students Studying Physics. Main Page - << Previous Chapter (Units) - Next Chapter (Vectors) >> Waves and Wavelike Motion Waves and Wavelike Motion - What are waves? - Two Types of Waves - Properties of Waves - Practical Applications of Waves- Sound Waves - Practical Applications of Waves: Electromagnetic Waves - Important Equations and Quantities [edit] 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, but sound waves and electromagnetic waves are other, less visible examples. We are most interested in the properties that waves have. All waves have the same basic properties, so by studying waves in water then we can transfer our knowledge and predict how other types of waves will behave. Waves are associated with energy. As the waves move, they carry energy from one to another point in space. It is true for water waves as well. You can see the wave energy working while a ship drifts along the wave in rough sea. The most spectacular example is the enormous amount of energy we receive from sun in the form of light and heat, which are transmitted as electromagentic wave - not even requiring a medium to propagate. [edit] Simple Harmonic Motion Simple Harmonic motion is a wavelike motion. It is considered wavelike because the graph of time vs. displacement from the equilibrium position is a sine curve. An example of simple harmonic motion is a mass oscillating on a spring. It will be hard to understand the forces involved this early in the course that cause the motion to simple harmonic, but it is still possible to look at a mass oscillating on a spring and understand that it is indeed simple harmonic. When a mass is oscillating on a spring, the further the string stretches, the slower the mass will be moving. Then the mass reaches a point where the string won't stretch any further, so it quits moving and then it reverses direction. As it moves closer to the equilibrium position is moves faster. The Free High School Science Texts: A Textbook for High School Students Studying Physics. Main Page - << Previous Chapter (Units) - Next Chapter (Vectors) >> Waves and Wavelike Motion Waves and Wavelike Motion - What are waves? - Two Types of Waves - Properties of Waves - Practical Applications of Waves- Sound Waves - Practical Applications of Waves: Electromagnetic Waves - Important Equations and Quantities [edit] What are waves? Waves are phenomena that everyone experiences constantly; water waves, sound waves, light waves, human waves when the home team scores... the list goes on. When asked what makes a wave a wave, the most common responses would probably be that a wave is something that moves, or propagates, or perhaps that it is something that repeats over and over again. These properties do capture the essential qualities of waves. Now we must determine these properties quantitatively, and discover what governs their behavior. Generally, a wave is defined as any phenomenon which can be modeled by a function of the form where the r-vector represents a position in space, and t represents a time, and the k-vector and omega are both constants. Don't be intimidated by the vectors in the argument - most of our time at first will be spent on one-dimensional waves. If the wave is in only one spatial dimension 'x', for instance a wave travelling on a taut string, it can be written simply as f(kx − wt). Any function of this form "propagates" along the direction over time. As time increases, the argument of the function increases; over time the form of the function effectively advances through space. Try coming up with functions of this form, and plot them at time t = 0, then plot them again at a later time. This progressive property will become obvious. Try to figure out the velocity with which your function advances! (we will study this later) The negative sign in front of the time term causes the wave to propagate in the direction defined as positive (if that seems confusing, try plotting more functions over time, and examine the results). If you replace the negative with a positive (or instead consider a negative value of omega), the wave will propagate in the negative direction. A very special and important case of a wave is the mathematical function , or in one dimension, f(kx − ωt) = sin(kx − ωt). This is a sinusoidal wave - it oscillates up and down infinitely in both directions, and moves as time progresses. I mentioned that waves have the quality of repeating over and over, the quality of periodicity. However, many functions of the form mentioned above do not seem to repeat. However, you will find that ALL waves can be decomposed into a sum of many of these simple, infinitely repeating waves when you learn about Fourier transformations. More than any other concept, physicists are finding that waves characterize the structure of the universe at every scale imaginable. As you learn about the physics of waves in everyday life, keep an open mind towards finding waves and wave behavior everywhere you turn. Lets consider a very well-known case of a wave phenomenon: 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. So waves have peaks and troughs. This could be our first property for waves. The following diagram shows the peaks and troughs on a wave. 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. --Sure kr06 13:04, 8 January 2006 (UTC): Waves are repetitions of physical quantity in a periodic manner, carrying energy in the process. The water waves, for example, can be visualized to repeat any of the physical quantities like "peaks", "troughs", "potential energy" or "kinetic energy". Even, we can visualize water waves as the motion of disturbance (energy). It is the energy aspect of waves that is central to the understanding of different types of waves, many of which are not visible. Looking closely at the water wave, we can recognize that crests and troughs basically represent of extreme potential and kinetic energies in addition to representing rise and fall of water from the still level. At the peak, energy is only potential, whereas energy is only kinetic at the trough. Similarly, propagation of electromagnetic wave is associated with repetitions of magnetic and electric field in space with certain periodicity. As existence of electrical or magnetic fields does not require any medium, electromagnetic waves can move even in the absence of any medium. [edit] Characteristics of Waves : Amplitude We use symbols agreed upon by convention to label the characteristic quantities of the waves. 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. 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 definition 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. [edit] 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 fixed distance between the peaks. Similarly, 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 greek letter lambda, λ. 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. [edit] 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 find that it is the same. Now if you measure the time between two adjacent troughs you'll find 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. The period of the wave is denoted with the symbol T. [edit] 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 go past. 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 second by dividing 1 second by the time it takes T. If a wave takes 1/2 a second to go by then in 1 second two waves must go by. . The unit of frequency is the Hz or s-1. Waves have a characteristic frequency.     f  : frequency (Hz or s-1) T  : period (s) generally, the frequency of a wave is the number of crests that pass by per unit time. [edit] Characteristics of Waves : Speed Now if you are watching a wave go by you will notice that they move at a constant velocity. Thinking back to rectilinear motion you will be able to remember that we know how to work out how fast something moves. 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 λ in a time T. This means that we can determine the speed.     v  : speed (m.s-1) λ  : 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 = fλ     v  : speed (m.s-1) λ  : wavelength (m) f  : frequency (Hz or s-1) Is this correct? Remember a simple first check is to check the units! On the right hand side we have speed which has units ms-1. On the left hand side we have frequency which is measured in s-1 multiplied by wavelength which is measure in m. On the left hand side we have ms-1 which is exactly what we want. [edit] Speed of a wave through strings The speed of a wave traveling along a vibrating string (v) is directly proportional to the square root of the tension (T) over the linear density (μ): μ is equal to the mass of the string divided by the length of the string. The Free High School Science Texts: A Textbook for High School Students Studying Physics. Main Page - << Previous Chapter (Units) - Next Chapter (Vectors) >> Waves and Wavelike Motion Waves and Wavelike Motion - What are waves? - Two Types of Waves - Properties of Waves - Practical Applications of Waves- Sound Waves - Practical Applications of Waves: Electromagnetic Waves - Important Equations and Quantities [edit] 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 are two additional types of waves. The first is called longitudinal waves and have 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 pressure waves 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 first piston stroke the pressure is the same throughout the tube. 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. 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 and trough, a single wave cycle). 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. The second additional type of wave is the torsional wave. The peaks and troughs rotate around the direction of motion. In simpler terms, a "twisting motion" is transmitted through the medium. Of the three wave types, this is the hardest one to describe and visualize. 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. The Free High School Science Texts: A Textbook for High School Students Studying Physics. Main Page - << Previous Chapter (Units) - Next Chapter (Vectors) >> Waves and Wavelike Motion Waves and Wavelike Motion - What are waves? - Two Types of Waves - Properties of Waves - Practical Applications of Waves- Sound Waves - Practical Applications of Waves: Electromagnetic Waves - Important Equations and Quantities [edit] Properties of Waves We have discussed some of the simple properties of waves that we need to know. These have just been describing the characteristics that waves have. Now we can progress onto some more interesting and, perhaps, less intuitive properties of waves. [edit] Properties of Waves : Reflection When waves strike a barrier they are reflected. This means that waves bounce off things. Sound waves bounce off walls, light waves bounce off mirrors, radar waves bounce off planes and how bats can fly at night and avoid things as small as telephone wires. etc. The property of reflection is a very important and useful one. (NOTE TO SELF: Get an essay by an air traffic controller on radar) (NOTE TO SELF: Get an essay by on sonar usage for fishing or for submarines) When waves are reflected, the process of reflection 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 reflected directly backwards. If the wave strikes the obstacle at some other angle then it is not reflected directly backwards. The angle that the waves arrives at is the same as the angle that the reflected waves leaves at. The angle that waves arrives at or is incident at equals the angle the waves leaves at or is reflected at. Angle of incidence equals angle of reflection (2.1)     θi  : angle of incidence θr  : angle of reflection 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 reflection of light with a mirror. You can also easily show that angle of incidence equals angle of reflection. If you look directly into you see yourself .... Need to mention that the incident wave, normal to the surface and the reflected wave all lie in the same plane. The same also holds for refraction at a surface. [edit] Phase shift of reflected wave When a wave is reflected 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 flat and then flick 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 tied to, it will be reflected. The reflected wave will look like a trough instead of a peak. This is because the pulse had undergone a phase change. The fixed end is like reflection off a more dense medium. If the end of the string was not fixed, i.e. it could move up and down then the wave would still be reflected but it would not undergo a phase shift. [edit] Properties of Waves : Refraction Sometimes waves move from one medium to another. The medium is the substance that is carrying the waves. In our first example this was the water. When the medium properties change it can affect the wave. Let us start with the simple case of a water wave moving from one depth to another. The 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 different i.e. the wavelength has changed. Going from one medium to another the period or frequency does not change only the wavelength 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 wave 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. If the wave speeds up in the new medium it turns away from the perpendicular to the medium surface. 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 different place. [edit] Properties of Waves : Interference If two waves meet interesting things can happen. Waves are basically collective motion of particles. So when two waves meet they both try to impose their collective motion on the particles. This can have quite different 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 efforts. 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. If one wave is trying to form a peak and the other is trying to form a trough then they are competing to do different things. In this case they can cancel out. The height of the peak less the depth of the trough will be the resulting effect. 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. Image:Fhsst waves23.png [edit] Properties of Waves : Standing Waves When two waves move in opposite directions, through each other, constructive interference happens. If the two waves have the same frequency and wavelength then a specific type of constructive interference can occur: standing waves can form. Standing waves are disturbances which don't appear to move, they stand in the same place. 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: 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: 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. 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: 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: Again we have to add the two waves together in the overlap region to see what the sum of the waves looks like. 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. When the waves have moved past each other so that they are overlapping for a large region the situation looks like a wave oscillating in place. If we focus on the range -4, 4 once the waves have moved over the whole region. To make it clearer the arrows at the top of the picture show peaks where maximum positive constructive interference is taking place. The arrows at the bottom of the picture show places where maximum negative interference is taking place. As time goes by the peaks become smaller and the troughs become shallower but they do not move. Image:Fhsst waves33.png For an instant the entire region will look completely flat. Image:Fhsst waves34.png The various points continue their motion in the same manner. Image:Fhsst waves35.png Eventually the picture looks like the complete reflection through the x-axis of what we started with: Image:Fhsst waves36.png Then all the points begin to move back. Each point on the line is oscillating up and down with a different amplitude. If we superimpose the two cases where the peaks where at a maximum and the case where the same waves where at a minimum we can see the lines that the points oscillate between. We call this the envelope of the standing wave as it contains all the oscillations of the individual points. A node is a place where the two waves cancel out completely as two waves destructively interfere in the same place. An anti-node is a place where the two waves constructively interfere. To make the concept of the envelope clearer let us draw arrows describing the motion of points along the line. Every point in the medium containing a standing wave oscillates up and down and the amplitude of the oscillations depends on the location of the point. It is convenient to draw the envelope for the oscillations to describe the motion. We cannot draw the up and down arrows for every single point! [edit] Reflection from a fixed end If waves are reflected from a fixed end, for example tieing the end of a rope to a pole and then sending waves down it. The fixed end will always be a node. Remember: Waves reflected from a fixed end undergo a phase shift. The wavelength, amplitude and speed of the wave cannot affect this, the fixed end is always a node. [edit] Reflection from an open end If waves are reflected from end, which is free to move, it is an anti-node. For example tieing the end of a rope to a ring, which can move up and down, around the pole. Remember: The waves sent down the string are reflected but do not suffer a phase shift. [edit] Wavelengths of standing waves with fixed and open ends [edit] Beats If the waves that are interfering are not identical then the waves form a modulated pattern with a changing amplitude. The peaks in amplitude are called beats. If you consider two sound waves interfering then you hear sudden beats in loudness or intensity of the sound. The simplest illustration is to draw two different waves and then add them together. You can do this mathematically and draw 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 (drawn in coloured dashed lines) you can see the resulting wave pattern: To make things clearer the resulting wave without the dashed lines is drawn below. Notice that the peaks are the same distance apart but the amplitude changes. If you look at the peaks they are modulated i.e. the peak amplitudes seem to oscillate with another wave pattern. This is what we mean by modulation. The maximum amplitude that the new wave gets to is the sum of the two waves just like for constructive interference. Where the waves reach a maximum it is constructive interference. The smallest amplitude is just the difference between the amplitudes of the two waves, exactly like in destructive interference. The beats have a frequency which is the difference between the frequency of the two waves that were added. This means that the beat frequency is given by fB = | f1 − f2 | (2.2) fB = | f1 − f2 |     fB  : beat frequency (Hz or s-1) f1  : frequency of wave 1 (Hz or s-1) f2  : frequency of wave 2 (Hz or s-1) [edit] Properties of Waves : Diffraction One of the most interesting, and also very useful, properties of waves is diffraction. When a wave strikes a barrier with a hole, only part of the wave can move through the hole. If the hole is similar in size to the wavelength of the wave diffractions occurs. 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. This bending around the edges of the hole is called diffraction. To illustrate this behaviour we start with Huygen's principle. [edit] Huygen's Principle Huygen's principle states that each point on a wavefront acts like a point source or circular waves. The waves emitted from each point interfere to form another wavefront on which each point forms a point source. A long straight line of points emitting waves of the same frequency leads to a straight wave front moving away. To understand what this means lets think about a whole lot of peaks moving in the same direction. Each line represents a peak of a wave. If we choose three points on the next wave front in the direction of motion and make each of them emit waves isotropically (i.e. the same in all directions) we will get the sketch below: What we have drawn is the situation if those three points on the wave front were to emit waves of the same frequency as the moving wave fronts. Huygens principle says that every point on the wave front emits waves isotropically and that these waves interfere to form the next wave front. To see if this is possible we make more points emit waves isotropically to get the sketch below: You can see that the lines from the circles (the peaks) start to overlap in straight lines. To make this clear we redraw the sketch with dashed lines showing the wavefronts which would form. Our wavefronts are not perfectly straight lines because we didn't draw circles from every point. If we had it would be hard to see clearly what is going on. Huygen's principle is a method of analysis applied to problems of wave propagation. It recognizes that each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves and that the advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already traversed. This view of wave propagation helps better understand a variety of wave phenomena, such as diffraction. [edit] Wavefronts Moving Through an Opening Now if allow the wavefront to impinge on a barrier with a hole in it, then only the points on the wavefront that move into the hole can continue emitting forward moving waves - but because a lot of the wavefront have been removed the points on the edges of the hole emit waves that bend round the edges. The wave front that impinges (strikes) the wall cannot continue moving forward. Only the points moving into the gap can. If you employ Huygens' principle you can see the effect is that the wavefronts are no longer straight lines. Image:Fhsst waves51.png Riaan Note: still cant find this image, have to get it from the pdf For example, if two rooms are connected by an open doorway and a sound is produced in a remote corner of one of them, a person in the other room will hear the sound as if it originated at the doorway. As far as the second room is concerned, the vibrating air in the doorway is the source of the sound. The same is true of light passing the edge of an obstacle, but this is not as easily observed because of the short wavelength of visible light. This means that when waves move through small holes they appear to bend around the sides because there aren't enough points on the wavefront to form another straight wavefront. This is bending round the sides we call diffraction. [edit] Properties of Waves : Dispersion Dispersion is a property of waves where the speed of the wave through a medium depends on the wavelength. So if two waves enter the same dispersive medium and have different wavelengths they will have different speeds in that medium even if they both entered with the same speed. The Free High School Science Texts: A Textbook for High School Students Studying Physics. Main Page - << Previous Chapter (Units) - Next Chapter (Vectors) >> Waves and Wavelike Motion Waves and Wavelike Motion - What are waves? - Two Types of Waves - Properties of Waves - Practical Applications of Waves- Sound Waves - Practical Applications of Waves: Electromagnetic Waves - Important Equations and Quantities [edit] Practical Applications of Waves: Sound Waves [edit]