Waves Vocabulary mechanical wave pulse continuous periodic wave amplitude wavelength period frequency wave velocity phase transverse wave longitudinal wave intensity displacement wave number phase velocity traveling sinuoidal wave harmonic wave wave equation one dimensional wave equation principle of wave fronts plane waves law of reflection superposition angle of incidence interference destructive interference constructive interference in phase out of phase standing wave nodes antinodes natural frequencies resonant frequencies fundamental frequency overtones first harmonic second harmonic refraction diffraction 15-1 Characteristics of Wave Motion When you toss a rock into a still pond, you see circular waves form Waves can travel along a cord or a slinky Waves stem from a vibration The medium through which such mechanical waves travel itself vibrates This chapter deals with mechanical waves- like waves on water one must also consider the particles that are involved- these also oscillate about an equilibrium point Mechanical Waves 1. a wave can move over large distances with a particular speed Each particle of the medium in which the wave travels oscillates about an equilibrium point which is simple harmonic is the wave is sinusoidal Thus although a wave is not matter the wave pattern can travel in matter Waves carry energy from one place to another 1 RoessBoss physics for scientist and engineers
All forms of traveling waves transport energy A single wave bump is called a pulse a single pulse can be formed on a rope by a quick up and down motion The pulse travels along the rope the source of the wave is a disturbance and cohesive forces between adjacent pieces of rope causes the pulse to travel A continuos or periodic wave- has as its source a disturbance that is continuos and oscillating- can also be thought of as vibrations a source of a wave is a vibration vibration propagates outward- waves if the source vibrates sinusoidally then it is Simple Harmonic Motion (SHM) Wave Diagram and Vocabulary The high point is called the crest The amplitude is the maximum height of a crest or the depth of a trough relative to the normal or the mid point in the wave Wavelength- (lambda) Trough to trough or Crest to Crest Frequency ƒ (also can be v greek letter nu) the number of times the wave passes a point (counting crests/ complete cycles) Period T- the time required for one complete oscillation or one complete cycle of the wave to pass a given point along the line of travel wave velocity v- is the velocity at which wave crests (or any other part of the waveform) move 2 RoessBoss physics for scientist and engineers
Wave Velocity can also be referred to as the phase velocity- it must be distinguished from the velocity of a particle of the medium itself Example- wave along a cord, wave velocity is the right, velocity of particles is up or down A wave crests travels a distance of one wavelength, in one period. Thus the wave velocity v is equal to / T and since 1/T=ƒ Equation 15-1 (sinusoidal waves) 15-2 Wave Types Transverse and Longitudinal if a wave travels down a cord and transverses up and down then this is called a transverse wave (your typical wave) a longitudinal wave is on in which the vibration of the particles of the medium is along the same direction as the motion of the wave We use the terms compression and Expansion to label their crest and troughs A good example of longitudinal wave is sound the concepts of frequency, wavelength, period are all the same with longitudinal as they are with Sound waves need a medium to propagate and are actually variations in the density of the medium, alternating regions of expansion and compression 3 RoessBoss physics for scientist and engineers
Air there are lots of large spaces between the molecules- which makes it harder for density variations to propagate, compared with water, steel, or the thin walls of an apartment When you graph the compression and expansions of a longitudinal wave the graph looks like that of a transverse wave (sinusoidal) Velocity of Transverse Waves the velocity of a wave depends on the properties of the medium in which it travels On a cord it depends on the tension of the cord and on the mass per unit length of the cord Velocity of Longitudinal Waves For longitudinal waves it depends on the Elastic modulus of the material and its density You can derive this from the idea of impulse and momentum- which makes sense since you have to strike a drum head to make it vibrate which begins the longitudinal waves 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 can travel through solid since the atoms or molecules can vibrate about their relatively fixed positions in any direction In fluid only longitudinal waves can propagate- because any transverse motion would experience no restoring force since a fluid is readily deformed 4 RoessBoss physics for scientist and engineers
This fact was used by geophysicists to discern the center of the earth and what it was constituted of Since only longitudinal waves are detected diametrically and not transverse, then the core must be liquid Remember that when they determine the center of an earthquake they triangulate the waves and then can find the core A wave on Water is a surface wave- it moves on the boundary between two mediums- air and water The reason waves crest or topple over at the coast is due to the water dragging at the bottom- that s why you can tell where the breaks in the bottom are from looking at the places where the waves are breaking 15-3 Energy Transported by Waves Waves transport energy from one place to another as it travels through a medium it transfers energy from one vibrating molecule to another You can have a very complex formula for three dimensional waves (speakers, Earth, etc) The energy transported by a wave is proportional to the square of the amplitude, and to the square of the frequency Equation 15-2- Side note The average rate of energy transferred it the average power- just look at how the idea is connecting to previous material 5 RoessBoss physics for scientist and engineers
Equation 15-3- Not Needed Just Connection point The Intensity I of a wave is defined as the average power transferred across unit are perpendicular to the direction of the energy flow Equation 15-4- Not Needed Just Connection point If you have a wave that travels in three dimensions then you have to take in account all the area (sphere 4πr^2) The Power output is constant then the intensity decreases as the inverse square of the distance from the source If the distance doubles then the intensity is reduced to 1/4 of the earlier value The amplitude of a wave also decreases with distance The amplitude must then decrease as 1/r The situation is different for one dimensional wave The are remains constant in a one dimensional wave- amplitude remains constant, the amplitude and intensity also do not decrease Keep in mind we do know that with SHM there is a dampening seen- This is due to frictional forces- you have some energy transformed into thermal energy 6 RoessBoss physics for scientist and engineers
15-4 Mathematical Representation of a Traveling Wave Displacement- how far away from a given point (not distance) amplitude- maximum displacement Reminder Equation- Angular frequency Equation 15-5- Connection Point wave number k is not the same thing as the spring constant k phase- where a wave is in motion phase velocity- describes the phase (shape of the wave) and can be written in terms of and k (wave number Equation 15-6- Connection Point- not needed Traveling sinusoidal wave- harmonic wave 15-5 The Wave Equation Equation of motion for a wave- wave equation involves calculus due to the type of motion that is occurring in the wave 15-6 The Principle of superposition 7 RoessBoss physics for scientist and engineers
when two or more waves pass thought the same region of space at the same time it is found that for many waves the actual displacement is the vector (algebraic) sum of the separate displacements This is called the principle of superposition valid for mechanical waves as long as the displacement are not too large and there is a linear relationship between the displacement and the restoring force of the oscillating medium When the restoring force is not precisely proportional to the displacement for mechanical waves in some continuos medium, the speed of sinusoidal waves depends on the frequency The variation in frequency is called dispersion Basically- not all waves are pretty and perfect. Calculus aids alot in determining various type of information for waves that are influenced by other factors. 15-7 Reflection and Transmission When a wave strikes an obstacle or comes to the end of a medium, part of the wave is reflected We can also call this an echo- like when you yell in a canyon If the wave is traveling down a rope and the rope is fixed to the wall, then the wave is reflected perfectly- it is the inverse to what it came in as the inversion is said to be a phase change of 180 it is as if the phase shifted by 1/2 or 180 If it is not fixed to a surface, then the pulse overshoots its displacement is momentarily greater than that of the traveling pulse- there is no phase change that occurs 8 RoessBoss physics for scientist and engineers
Keep in mind that when a wave hits a surface like a wall (Newton s laws) You will have some of the energy absorbed by the wall Also part of it turns into thermal energy Wave fronts- for two or 3 dimensional waves (water) we look at wave fronts These are 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 the wave fronts far from the source will appear as if they have completely lost their curvature and are nearly straight- plane waves (ocean wave) Law of Reflection Law of Reflection-the angle of reflection equals the angle of incidence the angle the incident ray makes with the perpendicular to the reflecting surface (or the wave front makes with a tangent to the surface angle of reflection is the corresponding angle for the reflected wave 15-8 Interference 9 RoessBoss physics for scientist and engineers
interference refers to what happens when two waves pass through the same region of space at the same time it is an example of the superposition principle When the waves overlap the resultant displacement is the algebraic cum of their separate displacements (principle of superposition) the two wave amplitudes are opposite of one another as they pass by and the result is called destructive interference (out of phase) The two waves pass in phase of one another and the resultant is greater than either pulse then it is called constructive interference You can have a partial of this type of interference 15-9 Standing Waves; Resonance standing waves- there are two traveling waves that will produce a standing wave because it does not appear to be traveling- you can see the nodes very clearly You have to vibrate the cord at just the right frequency in order to obtain the standing wave nodes- there the cord remains still the entire time Antinodes- top of crest or bottom of troughs- maximum amplitude These will remain in a fixed position for a given frequency Standing waves can occur at more than one frequency the lowest frequency of vibration producing a standing wave gives rise to nodes that are the end points of the string The frequencies at which standing waves are produced are the natural frequencies or resonant frequencies 10 RoessBoss physics for scientist and engineers
standing waves represent the same phenomenon as the resonance of a vibrating spring or pendulum- the only difference is that the spring or pendulum has only one resonant frequency- cord has an infinite number of resonant frequency To determine the resonant frequencies you need to know the wavelength of the lowest frequency There is a relationship to the L which is the length of the cord Equation 15-5 The lowest frequency is called the fundamental frequency and corresponds to one antinode or loop The other natural frequencies are the overtones- they are integral multiples of the fundamental They are also called the harmonics- with the fundamental being referred to as the first harmonic, the next one has two loops and is called the second harmonic or the first overtone, etc Wavelength Equation 15-6 Frequency 11 RoessBoss physics for scientist and engineers
Equation 15-7 15-10 Refraction When any wave strikes a boundary, some of the energy is reflected and some is transmitted or absorbed When one crosses a boundary into a medium where its velocity is different, the wave may move in a different direction than the incident wave- refraction you have seen this with the bending straw Equation Diagram of Refraction angle of refraction angle of incidence We can think of this as a proportion between the two mediums Equation 15-8 12 RoessBoss physics for scientist and engineers
15-11- Diffraction When waves encounter an obstacle they bend around the objectdiffraction the amount of diffraction depends on the wavelength of the eave and on the size of the obstacle Equation 15-9 only if the wavelength is smaller than the size of the object will there be a significant shadow region 13 RoessBoss physics for scientist and engineers