L20 Waves

追加: 2026-05-28

[00:00:01]hello everyone and welcome to lecture 21 of 124103 biophysical principles in this lecture we will learn about waves at the end of the lecture you will be able to name and differentiate the two main types of waves and give examples of each determine the speed of a wave identify and describe constructive and destructive interference and describe this diffraction so what is a wave a wave is a traveling disturbance it propagates from one point to the next and while it propagates it transfers energy without transferring matter think about a stadium wave or a mexican wave made by people at the football stadium the mexican wave starts from one point and it travels around the stadium without the people having to move away from their seats in in the last lecture you learned about periodic motion where you looked at an object that is oscillating about an equilibrium position and as it oscillates it traces out the sinusoidal graph the amplitude of the graph is the maximum displacement of the object from the equilibrium position while the period is the time it takes to complete one cycle of the oscillation so note that in periodic motion and the sinusoidal graph that you learned last time the x axis is given in time measured in seconds now when a particle in a medium oscillates there is some disturbance in the medium and if this disturbance propagates then we have a wave just like the stadium or mexican wave that we saw in the previous slide the wave will also have this sinusoidal behavior but because the disturbance propagates from one point to the next notice that the x-axis is no longer given in time but rather it's given in distance measured by the s-i unit meter now let's look at the different points in the wave in detail the wave will have a crest and a trough and these are the points of maximum displacement from the equilibrium of the object or particle that is oscillating and so the wave will also have an amplitude and this amplitude is equal to the maximum displacement that the object oscillating has it's measured from the equilibrium position to the crest or from the equilibrium position to the trough of the wave now the distance between two crests or two troughs or any successive points along the wave is called the wavelength the wavelength is represented by the greek symbol lambda which is read as lambda and wavelength is measured by the s-i units meters now the speed at which the disturbance propagates is given by this equation v is equal to f frequency multiplied by lambda the wavelength so here v represents the speed speed of the wave which means the speed for which the disturbance propagates f is the frequency and this is the frequency of oscillation of the particles being disturbed in the medium and lambda is the wavelength which we learned earlier as the distance between two crests or the distance between two troughs or the distance between two successive points along the wave this equation v is equal to f lambda actually came from the basic definition of speed now let's look at how speed equals distance over time becomes v is equal to f times lambda you've learned that speed is defined as distance divided by time it takes to cover that distance for the speed of the wave the distance is given as the wavelength and the time is the period of oscillation of the oscillating particle then recall that last time you learned that period is related to frequency by the inverse so frequency recall that frequency is inverse of the period and therefore speed the speed of the wave is equal to the distance which is lambda divided by time time is the period but periods is the inverse of frequency and therefore speed is equal to the wavelength lambda times the frequency note how when you are multiplying two variables the order of them doesn't matter so v is equal to lambda times f is equal it's the same as v equals f times lambda this means that in one period the wave would have traveled a distance of one wavelength meaning if you have this oscillating particle by the time the particle has oscillated one cycle the disturbance would have propagated a distance of one wavelength now let's look at an example relating speed of the wave to its wavelength and frequency i'll give you a few minutes to try and solve this example and then we will discuss the solution later you can pause the video and return when you're done so let's write down what's given in the problem we are given the frequency is equal to 200 kilohertz so that will be equal to 200 000 hertz and hertz is also the same as second to the minus one recall that the prefix kilo is equal to 10 to the three or that's equal to a thousand and so 200 kilohertz is equal to 200 000 hertz we are also given the speed of the wave and that is equal to one point three nine times ten to the three meters per second the first question asks us to find the wavelength of the wheels echolocation wave so we know that the speed of the wave is equal to the frequency multiplied by the wavelength of the wave so we rearrange this equation to isolate the wavelength lambda we can divide both sides by the frequency and so we have wavelength is equal to the speed of the wave divided by the frequency so we have wavelength is equal to 1.39 times 10 to the 3 meters per second divided by 200 000 second to the minus 1. so the unit seconds will cancel out and we have the wavelength equal to 6.95 times 10 to the minus 3 meters now a school of fish is 10 meters in front of the whale how long after the whale emits a wave is is its reflection detected so you have the whale and you have the school of fish the distance is 10 meters and we want to find the time so the whale will emit a wave a sound wave the wave will bounce off of the school of fish and its reflection will go back to the whale and it's this reflection that the whale will detect and used to determine how far the school of fish is so from the basic definition of speed which is equal to distance over time we can rearrange this equation to isolate time so time will be equal to distance divided by speed so that will be time equals 10 meters the speed of the wave 1.39 times 10 to the 3 meters per second that will give us a time of 0.007 seconds but note that this time 0.00 seconds is the time it takes for the wave to go from the whale to the school of fish if we want to find the echo look the time for the echolocation wave we will need to consider the time it takes for the wave to bounce off the fish and go back to the whale so we'll assume that the time of bouncing is negligible and so you have just this time it takes to go to the fish and go back to the whale and so we double this time 0.007 and therefore the time it takes for the reflection to be detected is equal to twice the time it takes to go to the school of fish and so that will be 0.014 seconds okay so we started with an oscillating particle in a medium and that introduces some disturbance and a wave is the propagation of this disturbance now depending on the direction of oscillation relative to the direction of propagation we can classify the wave as either a transverse wave or a longitudinal wave for a transverse wave the particles oscillate perpendicular to the direction of propagation whereas for a longitudinal wave the particles oscillate parallel to the direction of propagation now to illustrate what this means this is an example of a transverse wave it can be created for example by moving a string up and down so you have this oscillating particle and notice that as the disturbance propagates horizontally to the right the particles that make up the medium oscillate vertically up and down without horizontal motion if you focus your attention on one red particle notice that it's just going up and down in its place while the wave is propagating to the right here's a demonstration of a transverse wave using a slinky [Music] notice how [Music] is perpendicular to the direction of the vibration of the slinky on the other hand this is an example of a longitudinal wave notice that the disturbance propagates horizontally to the right and the particles making up the medium that's being disturbed oscillate also horizontally left to right without any vertical motion again the particles of the wave do not move with the wave but rather they stay in place and just oscillate or vibrate about their equilibrium position as the wave or the disturbance propagates now notice how there are regions of compression and rarefaction in the medium if we take a two-dimensional slice across the horizontal direction the two-dimensional slice will be sinusoidal the region of compression corresponds to a trough sorry the region of compression corresponds to a crest while the region of rare faction or expansion corresponds to a trough again even for longitudinal waves the distance between two successive crests or two successive troughs give you the wavelength here is a demonstration of a longitudinal wave using a slinky [Music] okay in this demonstration we're going to look at longitudinal waves most of the diagrams of waves you've seen so far have probably been transverse this is the classic sine wave that gets drawn on the chalkboard up and down and back and that's a useful diagram but it doesn't illustrate very well how sound it propagates through the air sound propagates through the air as what's called a longitudinal wave so sound is not going up and down sound is propagating through the air through air molecules that you can imagine all the molecules in the air here running into each other so those molecules get compressed and then refactored and then compressed and then refracted the molecules themselves do not travel through the air they stay where they are they just bounce off the thing next to them and that one bounces off another one and bounces off another one what we have here to demonstrate this is essentially a slinky suspended in a cage and if you can imagine each little wire of the slinky representing an air molecule what i'm going to do is start vibrating one end of the slinky and you can see that the one wire bounces off another that bounces off another and ultimately propagates that energy down the length of the spring this is what's called a longitudinal wave and as this is a much closer example of how sound propagates through the air keep in mind though that sound propagates equally in all directions so sound is not a running column going through the air this is happening in a 360 degree sphere but if you can keep in mind that this is how the sound actually propagates through the air it will help you as you end up staring at all of these diagrams of sine waves that are inevitable even through the material that we will show you keep in mind that sound is not going up and down it's actually traveling longitudinally like this and that'll help you keep straight the physics involved in sound and wave propagation in the air now we will learn what happens when a wave encounters another wave and talk about interference and what happens when a wave encounters an obstacle like a barrier or an opening by talking about diffraction consider two pulses that are traveling in opposite directions this animation here as the pulses overlap notice how their amplitudes add up a crests reinforces a crest and a trough reinforces a trough and therefore when two crests meet they amplify each other whereas when a crest meets a trough they tend to cancel each other out resulting to a smaller amplitude well if the crest and the trough if the two pulses have same amplitude then the resulting amplitude when the crest and the trough meet will be equal to zero so this adding up of amplitudes when two pulses overlap is called the principle of superposition the superposition or overlapping of waves in space is called interference when the waves overlap so that their crests coincide with each other such as this shown here in the upper diagram notice how the crests of the two waves coincide with each other we say that the waves are in phase and they will interfere constructively resulting to amplification of the amplitude but when the crest of one wave coincides with the trough of the other wave we say that the waves are 180 degrees out of face and here waves interfere destructively resulting to the cancellation of the amplitude watch this demonstration of constructive and destructive interference in this video we will demonstrate constructive and destructive interference of water waves using a pasco ripple tank the ripple tank setup is shown here on the left is a motor that oscillates with different frequencies and amplitudes above a thin layer of water is a light source and below is a mirror that reflects the ripples onto a screen this video was recorded using a high speed camera so that one second in real time is represented by 15 seconds in this video here plane waves are incident upon two openings when a plane wave passes between two openings the result is like creating two new waves these waves can add together to form new waves which is called superposition when the two waves are out of phase with one another they add together to create no wave this is called destructive interference these red lines show where destructive interference is happening when the two waves are in phase they add together to form an even larger wave whose amplitude is the sum of the amplitudes of the two waves this is called constructive interference these green lines show where constructive interference is happening you can think about how this experiment might prove that light and sound are waves we can let's look at the waves individually so the bright lines here these are crests and the regions between the crests are the troughs notice how when the crest of one wave coincides with the crest of another wave we can see here constructive interference the adding up of the amplitudes of these two waves resulting to a higher wave now when the crests of one wave meets with the trough of the other wave so this darker region here notice how there is destructive interference at this point and the resulting water wave flattens out due to the cancellation of the amplitudes of the waves now when a wave meets its reflection it could form what we call a standing wave this here illustrates what a standing wave is so you have one wave moving to the right it gets reflected by this barrier and this reflected wave interferes with the incoming wave and the blue wave at the bottom is the resulting standing wave it is called a standing wave because it appears as if the disturbance is not propagating and rather it appears like the disturbance is standing still and just oscillating up and down when a standing wave is formed you will notice points that appear to remain still at all times with zero amplitude so these points here they appear to be standing still with zero amplitudes okay and these points are called nodes these are points of destructive interference you will also notice points that appear to oscillate with maximum amplitude these points are called antinodes and these are points of constructive interference watch this demonstration of standing waves standing waves are a result of reflection and interference as i send a single pulse down to this fixed end you can see that it reflects upside down on itself if i send a rhythmic pulse it reflects on itself this is the fundamental mode sending the pulse at twice the frequency results in the second harmonic if i triple the frequency i get the third harmonic waves can also interfere in time for forming what is called a beat a beat is formed from two waves with the same amplitude propagating in the same direction with the same wave speed but with slightly different frequencies notice how this blue and pink waves appear to be out of sync it is like these two drums that are playing out of sync initially they start out playing out of sync but after some time they play in sync before playing out of sync again if we look at these drums as waves represented by these two sinusoids the blue and the red sinusoids the waves are slightly out of sync because they have different frequencies but there is a point when they are completely out of sync such that the crest of one wave coincides with the trough of the other wave and so their amplitudes add up to zero and then at one point they become perfectly in sync such as at this point where their crests coincide with each other and at that point their amplitudes add up to maximum and between these two points we have this partial addition of amplitudes and so the resulting amplitude will range from 0 to the maximum amplitude if we zoomed out this picture we will look some we will we will see something like this we can trace out the edges of the superimposed amplitudes and this envelope wave forms a new wave called the beat we can measure a period for the new wave by taking the distance between two successive troughs or two successive crests or any two successive points along the wave and this period is what we call the beat period recall that frequency is the inverse of the period so period is related to the frequency the frequency of this beat wave is called the beat frequency and the beat frequency is the number of amplitude maxima per second along this beat wave the beat frequency can be calculated as the difference between the frequencies of the two waves involved in for forming the beat wave notice how you have these two bars in this equation these bars mean absolute value an absolute value means that your answer for the beat frequency should always be positive so it doesn't matter whether which frequency you assign as f1 or f2 if you get a negative frequency beat frequency simply discard the negative sign and give a positive number for the beat frequency here is how a beat sounds like here we have two waves with slightly different frequencies one with 440 hertz and another one with 442 hertz this is how a 440 hertz sounds like and this is how a 442 hertz sound sound like if we add these two together so that they interfere or superimpose this is how the addition of 440 plus 442 hertz sound like [Music] now what happens when a wave encounters an obstacle such as shown in this figure here now what happens is so if a wave encounters an obstacle now what happens when a wave encounters an obstacle such as shown in this figure the waves bend around the obstacle the bending of the waves is called diffraction if the obstacle is a barrier the waves leave behind a shadow region as they bend around the barrier what about if there are two barriers placed side by side so that there is an opening between the two barriers what will happen to the wave as it encounters this opening watch this video and see you may have noticed this before someone is talking in a room over here and someone else is listening in a room over here you'll notice that voices sound kind of muscled through the wall but if we open the window my voice is clear as day so how does sound exit one window make a 180 degree bend and come in through this window that is the fascinating world of diffraction [Music] what happens when sound hits a window or more generally what happens when a wave hits some sort of opening let's imagine a simple wave approaching a small gap in a wall just ahead that's one fashionable wave because there's a gap some of the wave can get through but what does it look like on the other side does the wave continue in a straight line well since there's nothing containing the wave on the other side of the gap the wave expands in all directions the wave looks as if it's radiating out from a single point this is an example of huygens principle that is when a wave is disturbed each point where there is a disturbance becomes the source of a spherical wave a disturbance is simply whenever something's in the way so when there's a small gap that gap becomes a source for a spherical wave for a large gap every point along the gap becomes a source for a spherical wave in either case this is known as diffraction a wave does crazy things because it ran into something you can even try this at home just get a baking sheet and fill it with some water as such take a piece of cardboard fold it in half and tape the ends together then with two little walls we can just stick them in our that of water here creating a little gap in between the two and if you create a wave on one side of it no matter what you do uh on one side of the cardboard obstruction you can see that you get little circles coming out of the other side you'll notice also there's a part of the wave that moves along your obstruction jumping back to the window there is a part of the wave that moves along the wall so sound can easily go out one window creep along the wall and climb right back in the other one okay so that ends our lecture for waves in the next lecture we will focus more on sound waves and the maths needed to analyze sound waves