Wave Motion Vocabulary mechanical waves pulse continuous periodic wave amplitude period wavelength period wave velocity phase transverse wave longitudinal wave intensity displacement amplitude phase velocity traveling sinusoidal wave harmonic wave wave equation one dimensional wave equation principle of superposition wave fronts plane waves law of reflection interference destructive interferecne constructive interference in phase out of phase standing wave antinodes resonant frequencies natural frequencies fundamental frequency first harmonic second harmonic refraction diffraction 15-1 Characteristics of Wave Motion Water waves and waves on a cord are two common examples of wave motion Vibrations and wave motion are intimately related subjects Waves have a source- a vibration in case of sound- not only do you have a source you also have a detector a medium in which mechanical waves travel through itself vibrates we will be focused on mechanical waves in this unit There are two general features a wave can move over large distances with a particular speed each particle of the medium in which the wave travels (the water or rope) oscillates about an equilibrium point which is SHM if the wave is sinusoidal Waves carry energy from one place to another All forms of traveling waves transport energy there is a single wave bump or pulse a single pulse can be formed on a rope by a quick up and down or pulse as you move the rope up and down you produce a wave thus the source of a traveling wave pulse is a disturbance, and cohesive forces between adjacent pieces of rope causes the pulse to travel outward waves in other media are created and propagate outward in a similar fashion A continuous or periodic wave has as its source a disturbance that is continuous and oscillating the source is vibration or oscillation If the source vibrates sinusoidally in SHM then the wave itself- if the medium is perfectly elastic- will have a sinusoidal shape both in space and in time 1 RoessBoss
the high points on the wave- the maximum height is called the amplitude- it can be the crest or the trough- but it is maximum displacement from the normal/ equilibrium the total swing from a crest to trough is twice the amplitude the distance between two successive crests is called the wavelength ( ) the wavelength is also equal to the distance between any two successive identical points on the wave the frequency is the number of crests - or complete cycles- that pass a given point per unit time the period T is the time required for one complete oscillation or one complete cycle of the wave to pass a given point along the line of travel Equation Box 15-1 Wave velocity-v- is the velocity at which wave crest (or any other part of the waveform) move The wave velocity is often referred to as the phase velocity and it must be distinguished from the velocity of a particle of the medium For example for a wave traveling along a cord the wave velocity is to the right, along the cord, whereas the velocity of the particle of the cord is up or down A wave crest travels a distance of one wavelength in one period T Thus the equation is Equation Box 15-2 15-2 Wave Types When the particles of the cord vibrate up and down in a direction transverse (perpendicular) to the motion of the wave itself- Transverse Waves longitudinal wave- in which the vibration of the particles of the medium is along the same direction as the motion of the wave With longitudinal waves you have areas of compression and expansions which propagate along the spring Compression are areas where the coils are momentarily close together Expansions (rarefactions) are regions where the coils are momentarily far apart compressions and Expansions correspond to crests and troughs of transverse waves sound wave is a longitudinal wave 2 RoessBoss
A wavelength for longitudinal is the distance between successive compressions or expansions frequency is the number of compressions that pass a given point per second The wave velocity is the velocity with which each compression appears to move; and it is equal to the product of wavelength and frequency Equation Box 15-3 a longitudinal wave can be represented graphically by plotting the density of air molecules versus position Velocity of Transverse Waves The velocity of a transverse wave depends on the properties of the medium in which it travels The velocity of a transverse wave on a stretched string or cord depends on the tension in the cord and on the mass per unit length os the cord Equation Box 15-4 The greater the mass per unit length, the more inertia the cord has and more slowly the wave would be expected to propagate This equation is only valid for small displacements This was derived from Newtonian mechanics Equation Box 15-5 Velocity of Longitudinal Waves The velocity of a longitudinal wave has a form similar to that for a transverse wave on a cord 3 RoessBoss
Equation Box 15-6 In particular for a longitudinal wave traveling down a long solid rod Equation Box 15-7 Where E is the elastic modulus of the material and is its density. For a longitudinal wave traveling in a liquid or gas Equation Box 15-8 Where B is the bulk modulus and is the density This concepts can be used with pistons The net force on the compressed region of the fluid is since the uncompressed fluid exerts a force P0A to the left of the leading edge. Equation Box 15-9 Hence the impulse given to the compressed fluid which is given the speed V - eventually as we manipulate those equations around we derive 15-8 Basically these come from previous ideas- we simply use quantities that make sense for the situation that we have Other Waves Both transverse and longitudinal waves are produced when an earthquake occurs The transverse waves that travel through the body of the earth are called S waves (S for Shear) The longitudinal waves are called P waves (P for pressure) Both Longitudinal and transverse waves can travel through a solid since the atoms or molecules can vibrate about their relatively fixed positions in any direction 4 RoessBoss
But if fluid- only longitudinal waves can propagate, because any transverse motion would experience no restoring force since a fluid is readily deformable This fact was used by geophysicists to infer that a portion of the Earth s core must be liquid; longitudinal waves are detected diametrically across the earth but no transverse waves There can also be surface waves that can pass through the body of the earth- there can be surface waves that travel along the boundary between two materials A wave on water is a surface wave that moves on the boundary between air and water The motion of each particle of water at the surface is circular or elliptical, so it is a combination of transverse and longitudinal motion Below the surface there is also transverse plus longitudinal wave motion At the bottom the motion is only longitudinal (that is why with ocean waves the crest move at higher speeds- the bottom is moving slower) With earthquakes the waves that are most responsible for the damage are surface waves 15-3 Energy Transported by Waves Waves transport energy from one place to another as waves travel through a medium, the energy is transferred as vibrational energy from particle to particle of the medium For a sinusoidal wave of frequency ƒ, the particles move in SHM as a wave passes and each particle has an energy Equation Box 15-10 Where Dm is the maximum displacement (amplitude) of its motion (transverse or longitudinal) The energy transported by a wave is proportional to the square of the amplitude and the to the square of the frequency Equation Box 15-11 intensity I- of a wave is defined as the average power transferred across unit area perpendicular to the direction of energy flow 5 RoessBoss
Equation Box 15-12 Just like before there is frictional dampening that occurs and some energy is transformed into thermal energy Thus the amplitude and the intensity for a one dimensional wave the decrease will be greater than with three dimensional waves 15-4 Mathematical Representation of a traveling wave Amplitude is the maximum displacement of the wave with waves you have a wave number k do not confuse k with the spring constant k- they are different quantities k is Equation Box 15-13 The quantity ( ) is called the phase of the wave The velocity of the wave v of the wave is often called the phase velocity, since it describes the velocity of the phase (shape) of the wave and it can be written Equation Box 15-14 we can also relate this back to SHM, though this is not needed at this time 15-6 The Principle of Superposition The principle of superposition is the actual displacement is the vector sum of the separate displacements It is valid for mechanical waves as long as the displacements are not too large and there is a linear relationship between the displacement and the restoring force of the oscillating medium if the amplitude is so large that it goes beyond the elastic region of the medium and Hooke s Law is no longer operative, the superposition principle is no longer accurate 6 RoessBoss
When the restoring force is not precisely proportional to the displacement for mechanical waves in some continuous medium, the speed of the sinusoidal waves depends on the frequency the variation of speed with frequency is called dispersion 15-7 Reflection and Transmission When a wave strikes an obstacle or comes to the end of the medium it is traveling in, at least a part of the wave is reflected For two or three dimensional wave, such as a water wave we are concerned with wave fronts, by which we mean all the points along the wave forming the wave crest A line drawn in the direction of motion, perpendicular to the wave front is called a ray wave front far from their source have lost almost all their curvature and are nearly straight, as ocean waves often are, they are then called plane waves for reflection of a two or three dimensional plan wave, the angle that the incoming or incident wave makes with the reflecting surface is equal to the angle made by the reflected wave This is the law of reflection- the angle of reflection equals the angle of incidence the angle of incidence is defined as the angle the incident ray makes with the perpendicular to the reflecting surface (or the wave front makes with the tangent to the surface the angle of reflection is the corresponding angle for the reflected wave 15-8 Interference Interference refers to what happens when waves pass through the same region of space at the same time it is an example of the superposition principle If their wave amplitudes are opposite of each other then they pass and result in destructive interference- the amplitudes decrease If their amplitudes are in sync with each other then they will have greater amplitudes after the interference and this is called constructive interference In phase- refers to when waves match either other in crest and trough relations- leads to constructive interference Out of phase- refers to when waves do not match and therefore leads to destructive interference If the two waves are completely out of phase it can lead to a full destructive interference 15-9 Standing Waves; Resonance standing wave- is one that does not appear to be traveling- like one with a rope attached to a wall It vibrates up and down and the medium stays in one place 7 RoessBoss
The points of destructive interference - where the cord remains still at all times- are called nodes points of constructive interference- where the end oscillates with maximum amplitude- antinodes the nodes and antinodes remain in fixed positions for a given frequency standing waves can occur at more than one frequency the frequency at which standing waves are produced are the natural frequency or resonant frequencies of the cord, and the different standing wave pattern are different resonant modes of vibration to determine the resonant frequencies, we first note that the wavelengths os the standing waves bear a simple relationship to the length L of the string the lowest frequency is called the fundamental frequency- it corresponds to one antinode (or loop) the other natural frequencies are called overtones When they are integral multiples of the fundamental they are also called harmonicswith the fundamental begin referred to as the first harmonic, and the next the second harmonic, etc Equation Box 15-15 The integer n labels the number of the harmonic we can relate this to frequency by Equation Box 15-16 we see that each resonant frequency is an integer multiple of the fundamental frequency 15-10 Refraction when a wave strikes a boundary some of the energy is reflected and some is transmitted or absorbed when a two or three dimensional eave traveling in one medium crosses into a medium where its velocity is different the transmitted wave may move in a different direction than the incident wave this is known as refraction you basically have a change in velocity 8 RoessBoss
You see this with the straw in the cup- it appears bent- its actually the waves bending The law of refraction is Equation Box 15-17 Earthquake waves refract within the earth as they travel though rock of different densities- velocity is different 15-11 Diffraction Waves spread as they travel and when they encounter an obstacle they bend around it somewhat and pass into a region behind it This is called diffraction the amount of diffraction depends on the wavelength of the wave and on the size of the obstacle if the wavelength is much larger than the object the waves bend around then almost as if they were not there for larger objects there is more of a shadow region behind the obstacle where we might not expect the waves to penetrate but they do at least a little bit rule of thumb- only if the wavelength is smaller than the size of the object will there be a significant shadow region it is worth noting that this rule applies to reflection from an obstacle as well A rough guide to the amount of diffraction is Equation Box 15-18 the waves can bend around obstacles and thus can carry energy to areas behind obstacles- is clearly different from energy carried by material particles 9 RoessBoss