WAVES PRINCIPLE EXPLAINED BY MR. AKINS

I WISH TO INTRODUCE YOU TO ANOTHER IMPORTANT TOPIC IN PHYSICS. FOLLOW ME AS WE EXPLORE  THE PRINCIPLE OF WAVES. What is a Wave?

A mechanical wave is a disturbance in a medium which moves through the medium thus transferring energy from one place to another.       Examples include, waves on the surface of a liquid, waves on stretched wires/strings, sounds waves       Electro-magnetic radiations like light, x-rays, radio waves etc, have wave-like properties but do not need a medium to travel through. See here for more detail.       All waves can be put into one of two categories, transverse or longitudinal.   This distinction is illustrated below, considering two different ways of sending a wave along a long stretched spring, for example, a "slinky" spring.       Transverse Waves     The medium through which the wave is moving is the spring.   The direction of motion of the wave (direction of propagation) is at 90° to the direction of the disturbance (or displacement) of the medium.       Longitudinal Waves     c - compression e - expansion (or rarefaction)   Again, the medium through which the wave is moving is the spring.   However, in this case, the direction of propagation of the wave is parallel to the direction of the disturbance .   Sound waves are longitudinal waves.       The velocity of propagation of a wave depends on the properties of the medium through which it moves.   For example, the tension in the spring and the mass per unit length of the spring determine how fast waves travel along it. Mr. Akins, how is mechanical waves different from particle waves? Hmmmmm....mechanical waves is different from particle waves in that only ENERGY is transfered in mechanical waves while both the ENERGY and PARTICLES traveled together in particle waves. hope you get the idea...... Refraction: Snell's Law Huygens’ principle has been used to make predictions about the position of a wave-front after waves have crossed a boundary between two different media.       The diagram below results from the application of the principle to waves moving from a region where the velocity is v1 to a different medium where the velocity is v2 (in this case, v2<v1)         Angles ABC and ADC are 90°.       Therefore we can write     which gives         which means that, for a given pair of media, the ratio, sinθ1 to sinθ2 is a constant, for a given pair of media (equal to the ratio of the velocities of light in the two media).   This relation is called Snell’s law after the Dutch astronomer Willebrord Snellius (or the Snell-Descartes law) of refraction.   It is especially easily verified by experiment using light passing from, for example, air to a glass prism.       The constant is called the refractive index (symbol, n12) for waves passing from medium 1 to medium 2   and, as the path of the waves is reversible, we can write            The angle between a normal line and the direction of propagation in medium one is called the angle of incidence (θ1 in the diagram above).   The angle between a normal line and the direction of propagation in medium two is called the angle of refraction (θ2 in the diagram above).         Snell's law:    The sine of the angle of incidence divided by the sine of the angle of refraction is equal to a constant called the refractive index, n, of the two media.      When considering light,, if medium 1 is a vacuum (or air) then the ratio v1/v2 is called the absolute refractive index of medium 2 Now let us consider another important aspect of WAVE which is the graphical representation of waves. Graphs Representing Waves The following graphs represent continuous waves (as opposed to the single pulses shown here).   These graphs have the same shape as graphs of sine of angle against angle. For this reason the waves they describe are often called sine waves (or sinusoidal waves, if you want to sound more intelligent).   This is the type of wave which results when the disturbance of the medium is produced by a body oscillating with simple harmonic motion, s.h.m.     Graph of Displacement of the Medium against Distance along a Typical Wave             The maximum displacement of the medium from the equilibrium position, r, is called the amplitude of the wave.       λ  is the wavelength.   This is the distance moved by the disturbance during one time period (see below).       For a transverse sine wave, this graph can be considered to be a picture of the wave at a given instant in time.   However, it should be remembered that the graph could also represent a longitudinal wave.       Graph of Displacement of a Particular Point in the Medium against Time for a Typical Wave           T, is the time for one "cycle" of the wave.   This is called the time period.       The frequency, f, of the wave is the number of cycles per second.   This is determined solely by the source of the waves (the frequency of a wave is equal to the frequency of the source).   As with any cyclical phenomenon, the relation between frequency and time period is         Relation between f, λ and Velocity of Propagation, v   As stated above, λ is the distance moved by the disturbance during one time period.       From the definition of velocity we have     where s is distance moved and t time taken.   So, in the particular case of a disturbance forming a wave we can write this as     therefore     This relation shows us that, for a given velocity, the wavelength is inversely proportional to the frequency.   In other words, as frequency increases wavelength decreases. Reflection of Waves This is the principle of the first law wave that explains the property of wave when a wave strike a plane. All waves can be reflected.       A wave is reflected at the boundary of the medium through which it is traveling or at any point where there is a change in the velocity of propagation of the wave.     Some aspects of reflection of waves can be studied by observing waves on springs.   Hold a long spring vertically, as shown below, and give it a short sharp shake, horizontally. A transverse pulse will travel down the spring and be reflected from the other end.             What happens during the process of reflection depends on the nature of the boundary.   Notice that, if the end of the spring is fixed in place, the pulse "flips over" to the other side.   We say that there has been a change of phase.   The reflected pulse is said to be in anti-phase with the incident pulse or, by comparison with sine waves, 180° or pradians out of phase.       When the end is free to move, the pulse goes back up on the same side, so we say no phase change has occurred.     Reflection at a Point where the Wave Velocity Changes   In the above examples, the change is rather dramatic... we go abruptly from wave to no wave (by definition, you can't have a wave on a spring where there's no spring!)   A slightly less dramatic, though equally important case, is one in which some property of the medium through which the wave is traveling changes, resulting in a change of the velocity of propagation of the wave.   For example, if a wave goes from a spring having a certain mass per unit length to another of different mass per unit length, we find that a certain proportion of the energy of the wave is reflected back from the point where the velocity of propagation of the wave changed.   Some of the energy continues past the change in medium: this forms what we call the transmitted wave.       In the examples below, imagine that the blue spring has a greater mass per unit length than the red spring, so the wave velocity is lowerin the blue spring.       Wave slows down at the change of medium, v2 < v1       Before Reflection         After Reflection     Reflected wave, phase change Transmitted wave, no phase change   The phase change of the reflected wave should be no surprise, having seen that it occurs when the end of the spring is fixed.   Imagine an extreme example with a (blue) spring of ginormous mass per unit length (I mean, like, really really massive...)   In this case the end of the red spring would be more or less fixed in place by the much more massive spring so the two situations would be very similar.       Wave moves faster after the change of medium, v2 > v1       Before Reflection         After Reflection     Reflected wave, no phase change. Transmitted wave, no phase change.   Again, this should be no surprise after the non-phase-change from a free end.   Again, to convince yourself, consider a very very low mass spring (I mean, like, really really low mass), you get the idea... Now, let us use the Huygens principle to explain the law of reflection. Huygens’ Principle The Dutch scientist Christiaan Huygens suggested a graphical method of predicting the future position of a wave-front, knowing its current position.   His principle is stated as follows.   Each point on the existing wave-front can be considered to act as a source of waves (sometimes referred to as secondary wavelets).   For physical evidence to support this suggestion, just consider the phenomenon of diffraction of waves by a small aperture.   When a small part of a plane wave-front is isolated, it does behave like a point source... I rest my case (well, Huygens' case)...       In the following examples it is perhaps easiest to imagine waves on the surface of water (as observed in a ripple tank), but the results can be applied to any two (or three) dimensional waves.       First, a rather trivial example.   Consider the set of plane (straight) waves shown below.   The velocity of propagation of the waves is v.   To apply the principle, we must pretend that we cannot guess where the wave-front will be, say, t seconds later!       First chose a point (any point), A, on the wave-front and draw an arc of radius vt.   This is the distance the secondary wavelets will have moved in t seconds.             Now chose another point, B, at random, on the wave-front and repeat the process.             The new wave-front is the tangent to the two curves…well what a surprise, it's just where we expected!   However, we will now consider two situations where the principle can help make useful predictions.       Reflection of Waves Using Huygens’ Principle   Consider a set of plane waves moving towards a reflecting surface, indicated by the line x-x’.   At time t = 0, point A on the wave-front reaches the reflecting surface.   The red arrow is a "ray" showing the direction of motion of the waves. A ray is always at 90° to the wave-front.             We will try to find the position of the wave-front at time t, the instant when point B reaches the reflecting surface.     First, draw an arc of radius equal to the distance B C (see next diagram).   The secondary wavelets from point A will have travelled this far by the time the waves at point B reach point C.   The new wave-front is the tangent to this arc which passes through point C.         Now, using the observed fact that the direction of propagation of a wave is always at 90° to the wave front, we can predict the direction of motion of the waves after reflection.         The angle of incidence is the angle between the direction of propagation of the waves and a normal to the reflecting surface before reflection.       The angle of reflection is the angle between the direction of propagation of the waves and a normal to the reflecting surface after reflection.       We therefore wee that Huygens' method predicts that waves obey the familiar law of reflection, easily observed using a light beam and a mirror.         Refraction of Waves Using Huygens’ Principle   When waves travel across a boundary between two different media, the speed of propagation changes.   For example, the speed of light in a vacuum is 3×108ms-1, whereas in glass its speed is about 2×108ms-1.   The change in speed can result in a change in direction of propagation of the waves.   This change in direction is called refraction.       The diagrams below show how to use Huygens’ principle to predict the position of the wave-front when waves move from a medium in which they have speed v1 to a medium in which they have speed v2.   In this case, v2 < v1   In these diagrams the line x x’ represents the boundary between the two media.             At time t = 0, point A on the wave-front reaches the boundary.   Consider secondary wavelets emitted from A at time t = 0.   At time t seconds later, point B reaches the boundary.   At time t, the secondary wavelets emitted from A have moved a distance v2t.     The position of the new wave-front is shown by line C D.   The situation at a later time is shown in the next diagram.             Notice that the change in speed of the waves inevitably produces a change in the wavelength, as explained here. Phase and Phase Difference If two oscillations reach their maximum displacement at the same time, they are said to be in phase at that time (if they also have exactly the same frequency, they will remain in phase).       If one oscillation is at its maximum displacement when another is at its equilibrium position, the two oscillations are said to have a phase difference of one quarter of a time period (T/4).     For many oscillations (a pendulum, a mass on a spring etc) a graph of displacement, x against time, t is very similar to a graph of sine of angle against angle, θ.   For this reason, phase differences are usually expressed in terms of angles rather than times.       For example, the two oscillations shown below are T/4 or π/2rads (or 90°) out of phase with each other.               Similarly, the next diagram shows two oscillations which are T/2 or πrads (or 180°) out of phase with each other (in this case we also use the term “in anti-phase”).               As a continuous wave is a series of oscillations, having varying phase relationships, the above ideas can also be applied to waves. Refraction: Snell's Law Huygens’ principle has been used to make predictions about the position of a wave-front after waves have crossed a boundary between two different media.       The diagram below results from the application of the principle to waves moving from a region where the velocity is v1 to a different medium where the velocity is v2 (in this case, v2<v1)         Angles ABC and ADC are 90°.       Therefore we can write     which gives         which means that, for a given pair of media, the ratio, sinθ1 to sinθ2 is a constant, for a given pair of media (equal to the ratio of the velocities of light in the two media).   This relation is called Snell’s law after the Dutch astronomer Willebrord Snellius (or the Snell-Descartes law) of refraction.   It is especially easily verified by experiment using light passing from, for example, air to a glass prism.       The constant is called the refractive index (symbol, n12) for waves passing from medium 1 to medium 2   and, as the path of the waves is reversible, we can write            The angle between a normal line and the direction of propagation in medium one is called the angle of incidence (θ1 in the diagram above).   The angle between a normal line and the direction of propagation in medium two is called the angle of refraction (θ2 in the diagram above).         Snell's law:    The sine of the angle of incidence divided by the sine of the angle of refraction is equal to a constant called the refractive index, n, of the two media.      When considering light,, if medium 1 is a vacuum (or air) then the ratio v1/v2 is called the absolute refractive index of medium 2. Interference Consider a transverse wave moving through a medium, for example, a wave moving through a stretched spring.   Each point in the medium is oscillating in a direction perpendicular to the direction of propagation of the wave.   (If the wave is a sine wave, the oscillation of each point is the simplest type of oscillation, called simple harmonic motion.)   The points are all oscillating but different points reach their maximum displacement at different times.   To describe this situation we say that a wave consists of many oscillations which are out of phase with each other.   The actual phase difference between two oscillations depends on the distance between them (measured along the direction of propagation of the wave).       In many situations, we need to consider two sets of wave, of the same frequency, travelling through the same medium.   The combined effect of the two waves at a given point in the medium will depend on the phase difference between their oscillations at the point considered.       We will consider the two extreme cases, that is,   1. the two waves arrive at the point in phase with each other and   2. they arrive in anti-phase (p radians or 180° out of phase)       1. Two waves arriving at a point in phase result in constructive interference.             The resultant of these two oscillations will be an oscillation of amplitude equal to sum of the amplitudes of the two original oscillations.         2. Two waves arriving at a point in anti-phase result is destructive interference.                   In this case the algebraic sum of the oscillations results in an oscillation of smaller amplitude.   If the amplitudes of the two oscillations are equal the result is no oscillation.       The fact that the resultant amplitude can be found by adding together the individual amplitudes is sometimes referred to as the principle of superposition of waves. Diffraction The phenomena observed when waves are obstructed by obstacles or pass through apertures are called diffraction effects. All (two or three dimensional) waves exhibit diffraction effects but the magnitude of many of these effects depends on the wavelength of the waves.   Diffraction effects are most obvious when the object or aperture causing the diffraction is similar in size to the wavelength of the waves.       As with interference, these effects can be seen quite clearly using a ripple tank to observe small waves on the surface of water.           If plane waves (the "straight" waves on the left of this diagram) are sent towards a barrier with a well defined edge, they will "bend round" the object.   The edge of the barrier appears to behave like a point source of waves.                     Plane waves sent towards a large aperture show the same effect at each edge (duh, of course...)           However...             ...if the aperture is small enough (of the same order of size as the wavelength, or smaller), it will behave like a point source of waves.                     Plane waves incident on a small object (again, the same order of size as the wavelength, or smaller), it will also behave like a point source of waves but in this case sending the waves back too in what can be considered to be a reflection.   This phenomenon is useful to bats which used echo location to find insects.     An arrow drawn on a diagram to show the direction of propagation of a set of waves is called a ray.   A ray is always at 90° to the wavefront. Diffraction Patterns Diffraction patterns can be seen using various light sources but are most easily observed using a laser.   If we direct a laser beam towards a narrow aperture (say, 10-4m wide or less) and observe the transmitted light on a screen a couple of metres away, we find a central bright patch of light (much wider than the aperture) and bright and dark fringes on the sides.       The diagram below represents such a diffraction pattern.   The red patches show approximately what is seen on the screen.   The curve above the patches is a graph of relative intensity of light against position on the screen.         Although we are referring to this as a diffraction pattern, it should bring to mind interference effects (see Interference and Interference Patterns)       The next diagram shows the same situation but with an even narrower aperture.         Two changes are noticeable: the central maximum (the bright patch in the middle) is wider and the maximum intensity is lower.   Neither of these changes is surprising, having seen diffraction effects in, for example, water waves.       However, we might ask; how do we obtain interference effects when there is only one source of light (one aperture)?       One way to consider the situation is to imagine the narrow aperture (the slit, to save typing) to be made up of two slits each half the width... ok, I know, sounds like a trick but just accept it for now!       Calculation of the Fringe Spacing for a Single Slit Diffraction pattern   We imagine secondary sources (see Huygens' principle) of light at points A, B and C in the slit.   Consider light leaving these points at angle, θ to the normal line, as shown below.   The slit width is b and B is the mid-point.     We will assume that the rays drawn here meet at p, a point on a screen, a few metres away.   As the slit width, b is very small compared with the distance to the screen, we can consider the rays to be very nearly parallel.       Suppose that θ is such that the difference between C-p and A-p is equal to one wavelength of the light (this is called the path difference).   This means that the path difference for light from A and B to p will be λ/2.   Therefore the light from A and B will interfere destructively on the screen.   From the diagram it is clear that     In practice, the angle θ is very small, so we can write     (see here for explanation) where θ is in radians.       Now consider the slit to be made up of pairs of point sources, A and B, A’ and B’, etc, as shown here.   Light from all these pairs of points will also interfere destructively at point p.   Therefore we will have a minimum intensity (a dark fringe) of the diffraction pattern at an angleθ to the normal..               If the distance between the slit and the screen is D, then the width of the central maximum (central bright patch) of a single slit diffraction pattern is given by         The above reasoning should not be considered as an explanation of the diffraction pattern.   It does, however, give a method of predicting where the minima (dark fringes) of the pattern will occur.       Even if it's not really an explanation...  these predictions agree with experiment, and that's what counts!       Double Slit Diffraction/Interference Pattern   If we now consider sending light through two slits placed near each other (for example as in the Young's famous double slit experiment) we obtain an intensity distribution as shown below.     Here we are assuming that the distance between the slits is much greater than their width.       Note that the single slit diffraction pattern is still visible in the "envelope" (shown by the broken line).     The fringes due to the double slits are much closer together than in the single slit case because the distance between the slits is greater then their width. Interference Patterns for Red and Blue Light The diagrams below represent the patterns observed in experiments like Young's double slit experiment.       We imagine that the experiment is performed first with a red filter on the light source, then with a blue filter.   We expect the patterns to differ because red and blue light have different wavelengths. Diagrams not to scale (although the last one is about what you see when all works well...)       Red light interference pattern (λ = about 650nm)         Blue light interference pattern (λ = about 450nm)         The fringe spacing is smaller for blue light than for red, as expected.   Beats The term beats is used to describe an effect due to the interference of two waves (or oscillations) of very nearly (but not quite) equal frequencies.   Musicians often use this phenomenon to tune instruments.   If two strings of, for example, a guitar, which are very close in pitch (frequency) are sounded together, the resulting sound has periodic variations in volume (loudness).   These variation in volume, beats, are not heard if the two strings vibrate at exactly the same frequency.   Thus, by changing the tension in one string until beats are not heard, you can tune the instrument.       The diagrams below represent graphs of displacement against time for waves of slightly different frequencies, f1 and f2 (f2 > f1)         At t = 0, the two oscillations are in phase with each other.   At t = tA, they are in anti-phase and at t = tP they are again in phase.   So, at time t = 0 and t = tp the sum of the amplitudes of the two oscillations will produce a large amplitude oscillation.   At t = tA a low amplitude oscillation will result (going to zero if the two amplitudes are equal, as shown here)   The next diagram represents the sum of the two waves.     Beats can occur in all kinds of waves but if the graphs represent sound waves, then we would hear a loud sound at t = 0 and t = tP but a quiet sound near t = tA.       This phenomenon can be observed in any situation where we have two periodic variations of different frequencies.   For example, two masses on two different springs, as shown in this animation.       Relation between f1  f2 and F the Beat Frequency   Recall that the relation between frequency and time period is     T1 is the time period of wave 1, T2 is the time period of wave 2 and T is the time period of the beats.       If there are N time periods of wave 1 between t = 0 and t = tP, then there will be (N+1) time periods of wave 2.   Therefore, we can write   and   Eliminating N from these two equations gives     so     which means that         Thus the beat frequency is simply equal to the difference between the two frequencies. Stationary (or Standing) Waves If two sets of identical waves travel in opposite sense though the same medium, interference occurs between them. The interference can be constructive or destructive.   Places where destructive interference occurs are called nodes or nodal points (or lines).   Places where constructive interference occurs are called anti-nodes or anti-nodal points (or lines).   The pattern of nodes and anti-nodes remains fixed in space and is called a stationary (or standing) wave*.   See here for animation illustrating the idea.     In an unbounded medium (consider ripples on the surface of water, a long way from the edge of the pool) stationary waves of any wavelength can be established by adjusting the frequency of the oscillators producing the waves.       Of particular interest are standing waves produced by reflection of waves at boundaries.   For example, when a string under tension is disturbed at a point, waves travel away from the point where the disturbance occurred.   The waves are then reflected at the fixed ends of the wire.    Therefore, we have two waves of the same frequency travelling in opposite sense along the wire.   However, the ends of the string are fixed in place so this means a node exists at each end of the string.       For this reason, you can only have a stationary wave on the string for certain frequencies of oscillation, corresponding to certain wavelengths which "fit into" the length of the string.   When the string is caused to oscillate at one of these special frequencies, it responds with a large amplitude oscillation and is said to beresonating (or in resonance).       Similar resonance situations can occur in other oscillating systems, for example, a mass on a spring, a pendulum, air columns in musical instruments, bridges and other structures, car engines, electric circuits... the list is virtually endless!     * A rather confusing term since a wave, by definition, moves.   However, I suppose it’s easier to say "stationary wave" than "stationary distribution of nodes and anti-nodes produced by destructive and constructive interference between similar waves moving in opposite sense through a given medium" Resonance in Air Columns Resonance in Tubes Closed at One End   Stationary (or standing) waves can occur in many different situations.   One important example, especially for musicians, is the stationary waves set up in columns of air, as in, for example trumpets, saxophones, organs etc       When you blow across the top of an open water bottle, the sound you hear (if you get it right!) is due to a stationary wave being established inside the bottle.   Turbulence around the edge of the bottle produces a more or less random set of frequencies of oscillation.   These oscillations cause sound waves to travel along the bottle.   The waves are reflected at the ends (both the closed and the open* end) of the bottle.   For certain of the frequencies of oscillations, interference between these reflected waves amplifies the sound.   When this occurs, we say the air column is resonating.   These frequencies correspond to situations in which the effective distance travelled by a disturbance is a whole number of wavelengths.     The frequencies at which resonance occurs depend on   1. the length of the air column   2. the speed of sound in the column       The first of these is easy to verify for yourself:   first  blow across the open end of an empty bottle and then do the same thing when the bottle is, say, half full.     * Waves are reflected from an open end because sound travels a little more slowly inside a tube than in free air.       Imagine a small loudspeaker placed near the end of a tube which is closed at one end.   The frequency of the sound produced by the speaker is varied, starting from a very low frequency.   At a certain frequency a much louder sound is heard: this is the first resonant frequency, the fundamental frequency, fo.   The diagram below represents the oscillations of "layers" of air in the column when it is resonating at its lowest frequency.             The length of the arrow represents (approximately) the amplitude of the oscillation at that point.   Situations of resonance in air columns are often represented as shown in the next diagram.             Diagrams like the one above, to represent resonance in air columns are inspired by looking at resonating strings.   They can be confusing because the string wave is transverse and sound is a longitudinal wave.   They should be thought of as being something like a sketch graph of amplitude of oscillation of "layers" of air, against position, measured along the axis of the tube, as shown below.             At the closed end, waves are reflected with a phase change of 180°.   There is no displacement at this point: a displacement node exists at the closed end.       At the open end, the air is free to move. Here, waves are reflected with no phase change so a displacement anti-node exists at the open end.     Therefore, if waves travel twice the length of the tube in half a time period, they will arrive back at the open end in phase and resonance will occur.   This means that the fundamental frequency of resonance occurs when     N.B. When considering stationary waves on strings we concluded that the distance between two adjacent nodes is equal to half the wavelength of the waves producing the stationary wave.   This means that the distance between a node and the adjacent anti-node must be λ/4, so we arrive at the same conclusion.       The above statement is approximate because there is some air beyond the end of the tube which should be considered as part of the air column.   The length of this "extra bit of air" (often called the end correction) depends on the diameter of the tube (we will ignore this for now).     and these two expressions give us     for the fundamental (lowest) frequency of resonance.       As with the string under tension, the air column will also resonate at higher harmonics, however, the relation between these higher frequencies of resonance is a little different.   This is because the string is fixed at both ends meaning that there must be a node at both ends.   In this tube, open at one end and closed at the other, the next frequency at when resonance occurs must still have a node at the closed end and an anti-node at the open end.   The situation for the second harmonic is therefore as shown in the next diagram             Here, the waves travel twice the length of the tube in 1½ time periods   Therefore, in this case     which gives us   Therefore, in general, for a tube closed at one end, we have     and     where, n = 0, 1, 2 etc Resonance in Air Columns Resonance in Tubes Open at Both Ends Stationary (or standing) waves occur when waves of identical frequency travel in opposite sense through the same medium. Common examples include waves on a string under tension and waves in air columns (see also resonance in closed tubes).       Consider a source of sound close to one end of a tube, of length L, as shown below.   The frequency of the sound produced by the speaker is varied, starting from a very low frequency.   At a certain frequency, the fundamental frequency or first harmonic, fo a louder sound is heard.   This loud sound is due to constructive interference between the reflected waves and the oscillations produced by the speaker.           Waves are reflected from an open end because sound travels a little more slowly inside a tube than in free air.   At an open end, waves are reflected without a change of phase.   So, if waves travel twice the length of the pipe in one time period, they will return to the source in phase and constructive interference (resonance) will occur.   This means that, for this first resonance, twice the length of the tube must be equal to one wavelength, λ.   So     So, we see that the distance between two adjacent anti-nodes is equal to half the wavelength of the waves producing the stationary wave.   This agrees with the result found when considering stationary waves on strings.   The relation between frequency, wavelength and velocity gives us     So the fundamental frequency is given by     where v is the speed of sound inside the tube.       As explained here, standing waves in tubes are often represented as shown in the next diagram.             This represents the air column resonating at its lowest frequency, fo in which we see two anti-nodes and one node.       If we continue to increase the frequency, we find other resonances corresponding to shorter wavelengths.   Remembering that there must be an anti-node at each end and a node in the middle, we can drawn diagrams to represent these higher harmonics.       Second Harmonic         As the distance between two adjacent anti-nodes is equal to half the wavelength of the waves producing the stationary wave, we now have L equals one wavelength         Third Harmonic               Therefore, in general, for resonances in tubes open at both ends we can write     and the relation between the frequencies of the harmonics is     where n = 1, 2, 3 etc   Polarization of Waves When a transverse wave travels through a medium, the plane in which the displacement of the medium occurs is called the plane of polarization of the wave.   For example, if you shake the end of a stretched spring up and down vertically, the waves you produce are described as being vertically polarized waves.   From this it should be clear that the concept of polarization only has meaning for transverse waves.       Electro-magnetic radiations, like light, radio waves etc are considered to consist of linked electric and magnetic fields at 90° to each other and at 90° to the direction of propagation of the wave, as shown below.             This leaves us with a problem when referring to the plane of polarization of electro-magnetic radiations...   By convention the plane of polarization of an E-M wave is taken to be the plane of the electric field.       Normal light sources produce a mixture of waves, with their electric fields in all planes perpendicular to the direction of propagation.   They are said to produce unpolarized (or randomly polarized) light.       The process of selecting waves having their oscillations or in the case of E-M radiations, their electric fields, in a given plane is calledpolarizing the waves.     Polarization by Reflection   When light is reflected at the surface of a piece of transparent material, it is found to be partially polarized.   A greater proportion of the reflected light has its electric field parallel to the reflecting surface than in the incident beam.       When the angle of incidence is such that the reflected light and the refracted light are at 90° to each other, the reflected light is totally polarized.   It contains only waves with their electric field parallel to the reflecting surface.   The angle of incidence which produces this situation is called the polarizing angle, ip or the Brewster angle (after the Scottish physicistDavid Brewster).             The refracted light is, of course, also partially polarized as it no longer contains a totally random “mix” of polarizations.       Snell's law of refraction states that     where n is the refractive index corresponding to the two media   It is clear from the diagram that, in this special case therefore, we have   which means that     Practical Uses of Polarized Light Stress Analysis Certain plastics rotate the plane of polarization of light passing through them.   The angle through which the plane of polarization is rotated depends on the wavelength of the light.   The angle is also found to vary when the sample is placed under stress, for example by bending it.       This photograph shows a small part of a plastic set square viewed under normal conditions.         The next photo shows the appearance of the same plastic object when placed between two polarizing filters with their planes of polarization at 90° to each other (we describe this situations as "two crossed polarizers").         Remember that two crossed polarizers would normally not let any light through but as the plastic material rotates the plane of polarization, some light is transmitted.   As the angle of rotation depends on the stress in the material and the wavelength of light, we see different colours in different places.   At places where the internal stresses are greatest, the coloured bands change more rapidly (with distance).   Engineers wishing to predict where a mechanical component might fail, when placed under stress, can make a model of the component out of, for example, Perspex and observe the stress patterns (the coloured bands are more concentrated where the stresses are greatest.   The design of the model can then be modified accordingly before the actual component is manufactured.       Measuring the concentration of solutions   Certain solutions rotate the plane of polarization of light passing through them.           The angle through which the plane of polarization is rotated depends on the concentration of the solution.       Liquid Crystal Displays   LCDs are used in a great number of devices from watches and clocks to computer screens to HDTV screens...       On the under-side of the top plate of the liquid crystal container, A, are fine lines etched parallel to the plane of polarization of the top Polaroid.   Similarly there are lines etched into face B parallel to the plane of the lower Polaroid.   The liquid crystals line up with these fine lines but also tend to line up with each other so there is a gradual change in the alignment of the crystals from A to B.   The crystals change the plane of polarization of the light so that the light can pass through the lower polarizer and be reflected by the mirror: the display appears light.   When a voltage is applied, the crystals line up along the direction of the electric field and they therefore no longer allow light to pass through: the display appears dark. I have been able to provide the necessary information needed to solve any related problems under waves....let us get it done.....Good luck to you. from MR. AKINS.