Waves phenomena in Physics
Waves - Introduction
Waves are everywhere! In the air, in water, even in solid objects, visible and invisible, waves carry information - sometimes useful (like speech or TV data) and sometimes purely incidental (waves in the sea, or noise in the street).
Visible waves are probably the easiest to think about, and many types of mechanical oscillation generate visible waves. Have a look at the example below: a group of people jump up and sit back down, in sequence...
...note that something (we could call it information) travels from left to right, but the people themselves only stand up and sit down - no-one swaps seats. Here the medium in which the wave propagates is 'people'...no people=no waves. The wave is mechanical - no electrical or acoustic stuff is happening, just a physical movement. In general, mechanical waves are transmitted by vibrating particles (in this case the particles are the people), but although they oscillate (stand up / sit down) the particles do not move from their normal position.
Transverse Waves
In the mexican wave example shown above, the movement of the people is vertical (up and down) whereas the wave travels horizontally (from left to right). This is a transverse wave - the vibration direction is perpendicular (at right angles) to the direction in which the wave travels or propagates. Transverse waves are very common - waves in water and on strings are two good examples..
A transverse wave can also be set up on a slinky spring - have a look at this animation:
Not all waves are transverse, and we'll come back to the slinky in order to look at an important second wave type. Before we do that, have a look at another transverse example, showing a particle model of a transverse wave. These particles could be metal molecules in a solid vibrating plate, or water molecules showing ripples traveling across a pond.
You can setup a clear transverse wave with a long piece of rope...lay it out along the ground, and flick one end rapidly up and down. This doesn't work so well if someone holds the other end and you stretch it, as a higher tension in the rope means the wave speed is too high (it moves too fast) and it's hard to see what's going on. Check out the high-speed video below, and you'll see what happens when we slow things down.
Wave Types
Waves - Introduction
Waves are everywhere! In the air, in water, even in solid objects, visible and invisible, waves carry information - sometimes useful (like speech or TV data) and sometimes purely incidental (waves in the sea, or noise in the street).
Visible waves are probably the easiest to think about, and many types of mechanical oscillation generate visible waves. Have a look at the example below: a group of people jump up and sit back down, in sequence...
...note that something (we could call it information) travels from left to right, but the people themselves only stand up and sit down - no-one swaps seats. Here the medium in which the wave propagates is 'people'...no people=no waves. The wave is mechanical - no electrical or acoustic stuff is happening, just a physical movement. In general, mechanical waves are transmitted by vibrating particles (in this case the particles are the people), but although they oscillate (stand up / sit down) the particles do not move from their normal position.
Transverse Waves
In the mexican wave example shown above, the movement of the people is vertical (up and down) whereas the wave travels horizontally (from left to right). This is a transverse wave - the vibration direction is perpendicular (at right angles) to the direction in which the wave travels or propagates. Transverse waves are very common - waves in water and on strings are two good examples..
A transverse wave can also be set up on a slinky spring - have a look at this animation:
Not all waves are transverse, and we'll come back to the slinky in order to look at an important second wave type. Before we do that, have a look at another transverse example, showing a particle model of a transverse wave. These particles could be metal molecules in a solid vibrating plate, or water molecules showing ripples traveling across a pond.
You can setup a clear transverse wave with a long piece of rope...lay it out along the ground, and flick one end rapidly up and down. This doesn't work so well if someone holds the other end and you stretch it, as a higher tension in the rope means the wave speed is too high (it moves too fast) and it's hard to see what's going on. Check out the high-speed video below, and you'll see what happens when we slow things down.
Wave Types
Longitudinal Waves
On page 1, we looked at transverse waves using a slinky animation. In contrast, some waves (such as sound) are longitudinal. In a longitudinal wave the particles vibrate parallel to the direction the wave is travelling in...not at 90 degrees as for waves on a string.
A longitudinal wave can be shown using a slinky spring; have a look at the animation below:
The information content in this wave comes down to the question 'how close together are the coils of the spring?'. At rest, the coils are a certain distance apart, and when the end of the slinky is shaken (along its length) then a compressed region travels along. The red line shows graphically 'how compressed' the slinky is - compression is on our y-axis, and distance on our x-axis.
Like waves on a string, it helps to slow things right down when looking at longitudinal waves on a slinky. Have a look at the video here:
The wave loses power as it travels from left to right, partly because the strings we suspended the slinky from absorb some of the wave energy. However, you shold still be able to see the wave reach the end of the spring, and if you look carefully you'll notice a small reflection where the wave bounces from the end and starts to travel from right to left. If you're really patient and continue to watch, you'll see that after some time things get quite complicated, and that different parts of the spring appear to be streched (science-speak: rarefied) or compressed at different points in time.
The animation below shows a particle model of a longitudinal wave - it could for instance represent a bunch of air molecules in the presence of a sound wave. The air molecules vibrate in the direction of wave travel and form a series of compressions (high pressure) and rarefactions (low pressure), where the molecules are squashed together and pulled apart respectively.
Sticking with the sound-wave example, we might be interested in 'how loud' such a wave is. Loudness has to do with ampltitude or intensity - 'how compressed / how rarefied' the air molecules are. This is often measured using the decibel scale.
Decibel Scale
The human ear in capable of hearing very quiet (low intensity) sounds and extremely loud (high intensity) sounds. The ratio of intensities between 'silence' and 'oooow, that hurts my ears' is about 1:100 million million. To make a sound 'twice as loud', you need to mulitply its intensity by about 10...so an intensity of 1,000 is twice as loud as an intensity of 100, but half as loud as an intensity of 10,000. (The units of acoustic intensity are watts per square meter or W m-2).
It makes things easier if a logarithmic scale is used, called the decibel scale. In decibel terms, a doubling in loudness corresponds to an increase in 10dB - it doesn't matter whether that increase is from 10dB to 20dB or 100dB to 110dB. How does this work? Let's see...
The logarithmic scale
In the graphs above, imagine the x-axis represents the percieved loudness of a sound, and the y-axis represents the acoustic intensity needed to create that loudness. Our '10x' rule means that as the overall level increases, we need more and more intensity to get small changes in loudness. On the left-hand graph, where intensity is plotted on a linear ( W m-2) scale, this relationship is clear. On the right-hand graph, where intensity is plotted on a logarithmic (dB) scale, the curve becomes a straight line.
To see why this is, we need to get familiar with the idea of a logarithm. Just about every piece of audio equipment (microphones, loudspeakers, sound cards, amplifiers, mixers, etc) will have specifications expressed logarithmically (i.e. in dBs).
The idea of logarithms is fairly straightforward (even though at first they can look like a mind-bender!); they are simply a way of describing numbers which vary by very large amounts, as numbers which vary by relatively small amounts. Have a look at this:
The quietest sound the average person can hear has an intensity of about 1 picowatt per square metre (1x10 -12 W m-2), and this is defined as the reference intensity level which is equivalent to 0 decibels (0dB).
The decibel scale
Intensity levels (IL) are measured in bels (B)
IL = log10 intensity of soundintensity at threshold of hearing (1pW m-2) = log10II0
1 decibel (dB) = 10 bels (B)
IL (dB) = 10log10II0
Example
A sound has intensity 1 W m-2. What is the intensity level in dB?
IL (dB) = 10log10 (1 W m-210-12 W m-2) = 120dB
Decibel Scale
The human ear in capable of hearing very quiet (low intensity) sounds and extremely loud (high intensity) sounds. The ratio of intensities between 'silence' and 'oooow, that hurts my ears' is about 1:100 million million. To make a sound 'twice as loud', you need to mulitply its intensity by about 10...so an intensity of 1,000 is twice as loud as an intensity of 100, but half as loud as an intensity of 10,000. (The units of acoustic intensity are watts per square meter or W m-2).
It makes things easier if a logarithmic scale is used, called the decibel scale. In decibel terms, a doubling in loudness corresponds to an increase in 10dB - it doesn't matter whether that increase is from 10dB to 20dB or 100dB to 110dB. How does this work? Let's see...
The logarithmic scale
In the graphs above, imagine the x-axis represents the percieved loudness of a sound, and the y-axis represents the acoustic intensity needed to create that loudness. Our '10x' rule means that as the overall level increases, we need more and more intensity to get small changes in loudness. On the left-hand graph, where intensity is plotted on a linear ( W m-2) scale, this relationship is clear. On the right-hand graph, where intensity is plotted on a logarithmic (dB) scale, the curve becomes a straight line.
To see why this is, we need to get familiar with the idea of a logarithm. Just about every piece of audio equipment (microphones, loudspeakers, sound cards, amplifiers, mixers, etc) will have specifications expressed logarithmically (i.e. in dBs).
The idea of logarithms is fairly straightforward (even though at first they can look like a mind-bender!); they are simply a way of describing numbers which vary by very large amounts, as numbers which vary by relatively small amounts. Have a look at this:
The quietest sound the average person can hear has an intensity of about 1 picowatt per square metre (1x10 -12 W m-2), and this is defined as the reference intensity level which is equivalent to 0 decibels (0dB).
The decibel scale
Intensity levels (IL) are measured in bels (B)
IL = log10 intensity of soundintensity at threshold of hearing (1pW m-2) = log10II0
1 decibel (dB) = 10 bels (B)
IL (dB) = 10log10II0
Example
A sound has intensity 1 W m-2. What is the intensity level in dB?
IL (dB) = 10log10 (1 W m-210-12 W m-2) = 120dB
Wave Types
Noise Pollution
Now we have a unit to measure with - the dB - we might want to think about what we want to measure! We've noted that sound equipment is often specified using dBs, and sometimes this equipment is responsible for creating unwanted sounds - noise - in our environment.
Environmental noise pollution doesn't just come from neighbours sound systems, car stereos, pubs and clubs etc - it can result from traffic noise, trains, planes, factories, ice-cream vans and mobile ring tones too!. People in the UK live pretty close together on average, compared to many other countries, and this means that we can easily drive each other nuts - not least by making noises that other people find objectionable.
Noise is measured using a sound level meter which gives readings in dB.
There are many regulations and laws to control the amount of noise that is acceptable, and the enforcement of these laws often comes down to Environmental Health Officers working for your local council. Ideally, noise can be 'designed-out' of a situation during the planning stage, but very often planning goes wrong and then sorting out noise pollution can get very expensive.
For example, people living next to a busy road or railway line often experience noise. If this noise is too disturbing, there are several methods to make things quieter. These include placing barriers between the house and the noise source to help block the sound, or fitting double glazing to stop the sound travelling through the windows of the house.
Noise pollution and its effect on learning in schools and colleges has become a big issue recently, and acoustics plays an increasingly important part in the design of new buildings for this purpose.
The many sources of envrionmental noise that has to be prevented from entering a concert hall.
Hearing Loss
Noise isn't just annoying - sometimes it can be dangerous. Before regulations were imposed to control noise levels in factories, people often became deaf over time due to noise exposure at work. You may know older relatives who have trouble hearing - everyone's hearing deteriorates with age, but noise exposure can make this much, much worse.
Today, workplace noise is controlled in law, and people who work in noisy environments often wear ear defenders to cut down noise levels as shown in the picture below.
So - you might think that 'noise-induced hearing loss' is a thing of the past. Unfortunately this is not true - exposure to a high level of any noise can cause temporary deafness, and repeated exposure over a period of time can cause permanent hearing loss. This includes exposure due to personal sound equipment (MP3 players over headphones etc) as well as music exposure at clubs and gigs.
Whilst many people might consider it a little eccentric to go for a big night out wearing industrial hearing defenders, small ear-plugs are widely available and are very widely used by people whose hearing is essential to them - musicians, mix engineers, broadcasters and so on. The days of the deaf sound-man should be over...you might want to think about your own noise exposure...
Wave Types Test
Question 1
What form of wave is shown below:
LongitudinalTransverseWavefrontAmplitudeQuestion 2
Which of the following is a transverse wave?
Hair WaveShock WaveSound WaveRadio WaveQuestion 3
Express the number 5 in the form Log10x
Log105Log1050Log10100000Log101000000Question 4
Log [a/b] is equal to:
log a - log blog a + log blog a / log blog a - bQuestion 5
A sound generates 0.5 W m-2. What is the intensity level in dB?
50dB108.2dB117.0dB120.0dBWave Types
Noise Pollution
Now we have a unit to measure with - the dB - we might want to think about what we want to measure! We've noted that sound equipment is often specified using dBs, and sometimes this equipment is responsible for creating unwanted sounds - noise - in our environment.
Environmental noise pollution doesn't just come from neighbours sound systems, car stereos, pubs and clubs etc - it can result from traffic noise, trains, planes, factories, ice-cream vans and mobile ring tones too!. People in the UK live pretty close together on average, compared to many other countries, and this means that we can easily drive each other nuts - not least by making noises that other people find objectionable.
Noise is measured using a sound level meter which gives readings in dB.
There are many regulations and laws to control the amount of noise that is acceptable, and the enforcement of these laws often comes down to Environmental Health Officers working for your local council. Ideally, noise can be 'designed-out' of a situation during the planning stage, but very often planning goes wrong and then sorting out noise pollution can get very expensive.
For example, people living next to a busy road or railway line often experience noise. If this noise is too disturbing, there are several methods to make things quieter. These include placing barriers between the house and the noise source to help block the sound, or fitting double glazing to stop the sound travelling through the windows of the house.
Noise pollution and its effect on learning in schools and colleges has become a big issue recently, and acoustics plays an increasingly important part in the design of new buildings for this purpose.
The many sources of envrionmental noise that has to be prevented from entering a concert hall.
Hearing Loss
Noise isn't just annoying - sometimes it can be dangerous. Before regulations were imposed to control noise levels in factories, people often became deaf over time due to noise exposure at work. You may know older relatives who have trouble hearing - everyone's hearing deteriorates with age, but noise exposure can make this much, much worse.
Today, workplace noise is controlled in law, and people who work in noisy environments often wear ear defenders to cut down noise levels as shown in the picture below.
So - you might think that 'noise-induced hearing loss' is a thing of the past. Unfortunately this is not true - exposure to a high level of any noise can cause temporary deafness, and repeated exposure over a period of time can cause permanent hearing loss. This includes exposure due to personal sound equipment (MP3 players over headphones etc) as well as music exposure at clubs and gigs.
Whilst many people might consider it a little eccentric to go for a big night out wearing industrial hearing defenders, small ear-plugs are widely available and are very widely used by people whose hearing is essential to them - musicians, mix engineers, broadcasters and so on. The days of the deaf sound-man should be over...you might want to think about your own noise exposure...
Wave Types Test
Question 1
What form of wave is shown below:
LongitudinalTransverseWavefrontAmplitudeQuestion 2
Which of the following is a transverse wave?
Hair WaveShock WaveSound WaveRadio WaveQuestion 3
Express the number 5 in the form Log10x
Log105Log1050Log10100000Log101000000Question 4
Log [a/b] is equal to:
log a - log blog a + log blog a / log blog a - bQuestion 5
A sound generates 0.5 W m-2. What is the intensity level in dB?
50dB108.2dB117.0dB120.0dB
Wave Types
Noise Pollution
Now we have a unit to measure with - the dB - we might want to think about what we want to measure! We've noted that sound equipment is often specified using dBs, and sometimes this equipment is responsible for creating unwanted sounds - noise - in our environment.
Environmental noise pollution doesn't just come from neighbours sound systems, car stereos, pubs and clubs etc - it can result from traffic noise, trains, planes, factories, ice-cream vans and mobile ring tones too!. People in the UK live pretty close together on average, compared to many other countries, and this means that we can easily drive each other nuts - not least by making noises that other people find objectionable.
Noise is measured using a sound level meter which gives readings in dB.
There are many regulations and laws to control the amount of noise that is acceptable, and the enforcement of these laws often comes down to Environmental Health Officers working for your local council. Ideally, noise can be 'designed-out' of a situation during the planning stage, but very often planning goes wrong and then sorting out noise pollution can get very expensive.
For example, people living next to a busy road or railway line often experience noise. If this noise is too disturbing, there are several methods to make things quieter. These include placing barriers between the house and the noise source to help block the sound, or fitting double glazing to stop the sound travelling through the windows of the house.
Noise pollution and its effect on learning in schools and colleges has become a big issue recently, and acoustics plays an increasingly important part in the design of new buildings for this purpose.
The many sources of envrionmental noise that has to be prevented from entering a concert hall.
Hearing Loss
Noise isn't just annoying - sometimes it can be dangerous. Before regulations were imposed to control noise levels in factories, people often became deaf over time due to noise exposure at work. You may know older relatives who have trouble hearing - everyone's hearing deteriorates with age, but noise exposure can make this much, much worse.
Today, workplace noise is controlled in law, and people who work in noisy environments often wear ear defenders to cut down noise levels as shown in the picture below.
So - you might think that 'noise-induced hearing loss' is a thing of the past. Unfortunately this is not true - exposure to a high level of any noise can cause temporary deafness, and repeated exposure over a period of time can cause permanent hearing loss. This includes exposure due to personal sound equipment (MP3 players over headphones etc) as well as music exposure at clubs and gigs.
Whilst many people might consider it a little eccentric to go for a big night out wearing industrial hearing defenders, small ear-plugs are widely available and are very widely used by people whose hearing is essential to them - musicians, mix engineers, broadcasters and so on. The days of the deaf sound-man should be over...you might want to think about your own noise exposure...
Wave Types Test
Question 1
What form of wave is shown below:
LongitudinalTransverseWavefrontAmplitudeQuestion 2
Which of the following is a transverse wave?
Hair WaveShock WaveSound WaveRadio WaveQuestion 3
Express the number 5 in the form Log10x
Log105Log1050Log10100000Log101000000Question 4
Log [a/b] is equal to:
log a - log blog a + log blog a / log blog a - bQuestion 5
A sound generates 0.5 W m-2. What is the intensity level in dB?
50dB108.2dB117.0dB120.0dBSimple Harmonic Motion
Oscillations
You may have heard the one about the opera singer breaking a wineglass with their voice...(in other versions it's a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations. Later in this section we'll show you how to break a wineglass with a loudspeaker...but first we'll try to work out some rules for things which vibrate, or oscillate. There are many forms of oscillation in the real world - oscillations determine the sound of a musical instrument, the colour of a rainbow, the ticking of a clock and even the temperature of a cup of tea. A mechanical oscillation is a repeating movement - an electrical oscillation is a repeating change in voltage and current.
Mechanical oscillations are probably easiest to think about, so we'll start there. Let's think about the movement of an object back and forth over a fixed range of positions, such as the movement of a swing in a playground, or the bouncing up and down of a weight on the end of a spring (this last one is one of those examples which teachers / scientists / engineers love, but which doesn't seem to have anything to do with the real world. As we'll see, it does!!).
The pendulum and the mass-spring system are both oscillating. How can we describe their oscillation?
Displacement
The animation shows how displacement describes the distance (and also direction...displacement can be negative as well as positive) of the object from its equilibrium position.
The displacement amplitude tells us how 'big' the oscillations are - we can use the peak value (the maximum positive displacement from the equilibrium position) or the peak-to-peak value (the distance between negative-maximum to positive-maximum).
The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.
Can you see why the first time the mass returns to the equilibrium position, we've only reached half a cycle?
Displacement on its own will sometimes do...but often it is also useful to understand the oscillation in terms of velocity or acceleration. These three things are very similar...read on.
Velocity and Acceleration
When our oscillating object reaches maximum displacement (when it is as far from equilibrium as it can get) it changes direction. This must mean that there is an instant in time when it is not moving...and so at the point of maximum displacement, its speed is zero. Then it speeds up in the opposite direction, and travels fast through the equilibrium position before starting to slow again in preparation for the next change in direction.
Velocity is just another word for speed, with the extra feature that it has direction and therefore can be negative or positive. (If you walk backwards at 4m.p.h. - difficult without falling over - then your velocity = -4m.p.h.)
So - when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity do not 'line-up' in time...one is shifted along compared to the other. This shift along in time is called a phase angle, which is an idea we'll come back to later.
At maximum displacement the object has to come to rest, and then start moving in the opposite direction. Imagine braking in a car (moving forwards) before immediately moving off fast in reverse - you'd feel rapid deceleration during braking, followed acceleration in reverse gear. Like displacement and velocity, acceleration can be negative or positive depending on direction - in this example, braking would cause negative acceleration, as would moving off in reverse. It sounds wierd, but the point in time where the acceleration would be (negative) maximum is the moment when the car stops moving - just as it changes direction.
Thinking about our animated mass-spring system, we can see this effect. Conversely as the object moves through its equilibrium position, the velocity is only changing very slowly and the acceleration actually becomes zero. This is shown below:
Note again - the period is the same, but things don't line up in time...there's some phase in there too.
All these masses and springs are OK - but you can experience this by playing on a swing. Look at these videos, and try to work out when displacement, velocity and acceleration are 1) a maximum, and 2) zero, for someone swinging...
Equilibrium and Restoring Forces
In our animations, the mass on a spring bounces up and down with a regular 'pattern' or waveform. Why does this happen?
Let's think some more about playground swings. What's the first thing we do to get the swing moving?
The swing is first pulled back, and then pushed forward. It then oscillates back and forth 'on its own' until it slowly comes to rest at its equilibrium position - the middle position between the two extremes of displacement. This is usually the place where an object will naturally rest if no external forces are applied to it.
Once the swing has been pulled away from its equilibrium position, the force of gravity will act to bring it back. This force will always act in a direction towards the equilibrium position, and is known as a restoring force. If the swing is pulled higher into the air (larger displacement amplitude) then there will be a larger restoring force acting on it, and when the pusher lets go it will travel further. This shows that the restoring force is proportional to the distance of the swing from the equilibrium position.
The animation below shows these forces in action:
So why doesn't the swing just return directly to the equilibrium position, and stay there?
The restoring forces are large enough to make the swing over-shoot and travel 'up the other side'; this is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to over-shoot equilibrium and travel 'up the other side'. The process keeps repeating and an oscillation is formed. If no other forces are applied to the system, the swing will keep swinging back and forth forever, with a constant period and constant amplitude.
Does this happen in real life? (Silly question - of course not). Then why not?
In all real-life oscillating systems, damping comes into play. Damping is all about the loss of energy, often due to resistance or friction. The air displaced by the swinger adds some resistance, and even if the swing were in a vacuum (and the swinger were, therefore, dead) the pivots at the end of the chain generate some friction with their supports and slowly, almost immeasurably, heat up. This uses energy, which has to come from somewhere...and so the amplitude of oscillation decreases.
If you have no pusher, and have to propel yourself on the swing, you have to put some effort into it to keep the swing moving. This effort is overcoming damping. If there were no damping (and also if you could switch off your heart, brain and other energy-consumptive body functions) you could swing forever without needing to eat!
Simple Harmonic Motion
Oscillations
You may have heard the one about the opera singer breaking a wineglass with their voice...(in other versions it's a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations. Later in this section we'll show you how to break a wineglass with a loudspeaker...but first we'll try to work out some rules for things which vibrate, or oscillate. There are many forms of oscillation in the real world - oscillations determine the sound of a musical instrument, the colour of a rainbow, the ticking of a clock and even the temperature of a cup of tea. A mechanical oscillation is a repeating movement - an electrical oscillation is a repeating change in voltage and current.
Mechanical oscillations are probably easiest to think about, so we'll start there. Let's think about the movement of an object back and forth over a fixed range of positions, such as the movement of a swing in a playground, or the bouncing up and down of a weight on the end of a spring (this last one is one of those examples which teachers / scientists / engineers love, but which doesn't seem to have anything to do with the real world. As we'll see, it does!!).
The pendulum and the mass-spring system are both oscillating. How can we describe their oscillation?
Displacement
The animation shows how displacement describes the distance (and also direction...displacement can be negative as well as positive) of the object from its equilibrium position.
The displacement amplitude tells us how 'big' the oscillations are - we can use the peak value (the maximum positive displacement from the equilibrium position) or the peak-to-peak value (the distance between negative-maximum to positive-maximum).
The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.
Can you see why the first time the mass returns to the equilibrium position, we've only reached half a cycle?
Displacement on its own will sometimes do...but often it is also useful to understand the oscillation in terms of velocity or acceleration. These three things are very similar...read on.
Velocity and Acceleration
When our oscillating object reaches maximum displacement (when it is as far from equilibrium as it can get) it changes direction. This must mean that there is an instant in time when it is not moving...and so at the point of maximum displacement, its speed is zero. Then it speeds up in the opposite direction, and travels fast through the equilibrium position before starting to slow again in preparation for the next change in direction.
Velocity is just another word for speed, with the extra feature that it has direction and therefore can be negative or positive. (If you walk backwards at 4m.p.h. - difficult without falling over - then your velocity = -4m.p.h.)
So - when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity do not 'line-up' in time...one is shifted along compared to the other. This shift along in time is called a phase angle, which is an idea we'll come back to later.
At maximum displacement the object has to come to rest, and then start moving in the opposite direction. Imagine braking in a car (moving forwards) before immediately moving off fast in reverse - you'd feel rapid deceleration during braking, followed acceleration in reverse gear. Like displacement and velocity, acceleration can be negative or positive depending on direction - in this example, braking would cause negative acceleration, as would moving off in reverse. It sounds wierd, but the point in time where the acceleration would be (negative) maximum is the moment when the car stops moving - just as it changes direction.
Thinking about our animated mass-spring system, we can see this effect. Conversely as the object moves through its equilibrium position, the velocity is only changing very slowly and the acceleration actually becomes zero. This is shown below:
Note again - the period is the same, but things don't line up in time...there's some phase in there too.
All these masses and springs are OK - but you can experience this by playing on a swing. Look at these videos, and try to work out when displacement, velocity and acceleration are 1) a maximum, and 2) zero, for someone swinging...
Equilibrium and Restoring Forces
In our animations, the mass on a spring bounces up and down with a regular 'pattern' or waveform. Why does this happen?
Let's think some more about playground swings. What's the first thing we do to get the swing moving?
The swing is first pulled back, and then pushed forward. It then oscillates back and forth 'on its own' until it slowly comes to rest at its equilibrium position - the middle position between the two extremes of displacement. This is usually the place where an object will naturally rest if no external forces are applied to it.
Once the swing has been pulled away from its equilibrium position, the force of gravity will act to bring it back. This force will always act in a direction towards the equilibrium position, and is known as a restoring force. If the swing is pulled higher into the air (larger displacement amplitude) then there will be a larger restoring force acting on it, and when the pusher lets go it will travel further. This shows that the restoring force is proportional to the distance of the swing from the equilibrium position.
The animation below shows these forces in action:
So why doesn't the swing just return directly to the equilibrium position, and stay there?
The restoring forces are large enough to make the swing over-shoot and travel 'up the other side'; this is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to over-shoot equilibrium and travel 'up the other side'. The process keeps repeating and an oscillation is formed. If no other forces are applied to the system, the swing will keep swinging back and forth forever, with a constant period and constant amplitude.
Does this happen in real life? (Silly question - of course not). Then why not?
In all real-life oscillating systems, damping comes into play. Damping is all about the loss of energy, often due to resistance or friction. The air displaced by the swinger adds some resistance, and even if the swing were in a vacuum (and the swinger were, therefore, dead) the pivots at the end of the chain generate some friction with their supports and slowly, almost immeasurably, heat up. This uses enery, which has to come from somewhere...and so the amplitude of oscillation decreases.
If you have no pusher, and have to propel yourself on the swing, you have to put some effort into it to keep the swing moving. This effort is overcoming damping. If there were no damping (and also if you could switch off your heart, brain and other energy-consumptive body functions) you could swing forever without needing to eat!
Simple Harmonic Motion
Oscillations
You may have heard the one about the opera singer breaking a wineglass with their voice...(in other versions it's a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations. Later in this section we'll show you how to break a wineglass with a loudspeaker...but first we'll try to work out some rules for things which vibrate, or oscillate. There are many forms of oscillation in the real world - oscillations determine the sound of a musical instrument, the colour of a rainbow, the ticking of a clock and even the temperature of a cup of tea. A mechanical oscillation is a repeating movement - an electrical oscillation is a repeating change in voltage and current.
Mechanical oscillations are probably easiest to think about, so we'll start there. Let's think about the movement of an object back and forth over a fixed range of positions, such as the movement of a swing in a playground, or the bouncing up and down of a weight on the end of a spring (this last one is one of those examples which teachers / scientists / engineers love, but which doesn't seem to have anything to do with the real world. As we'll see, it does!!).
The pendulum and the mass-spring system are both oscillating. How can we describe their oscillation?
Displacement
The animation shows how displacement describes the distance (and also direction...displacement can be negative as well as positive) of the object from its equilibrium position.
The displacement amplitude tells us how 'big' the oscillations are - we can use the peak value (the maximum positive displacement from the equilibrium position) or the peak-to-peak value (the distance between negative-maximum to positive-maximum).
The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.
Can you see why the first time the mass returns to the equilibrium position, we've only reached half a cycle?
Displacement on its own will sometimes do...but often it is also useful to understand the oscillation in terms of velocity or acceleration. These three things are very similar...read on.
Velocity and Acceleration
When our oscillating object reaches maximum displacement (when it is as far from equilibrium as it can get) it changes direction. This must mean that there is an instant in time when it is not moving...and so at the point of maximum displacement, its speed is zero. Then it speeds up in the opposite direction, and travels fast through the equilibrium position before starting to slow again in preparation for the next change in direction.
Velocity is just another word for speed, with the extra feature that it has direction and therefore can be negative or positive. (If you walk backwards at 4m.p.h. - difficult without falling over - then your velocity = -4m.p.h.)
So - when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity do not 'line-up' in time...one is shifted along compared to the other. This shift along in time is called a phase angle, which is an idea we'll come back to later.
At maximum displacement the object has to come to rest, and then start moving in the opposite direction. Imagine braking in a car (moving forwards) before immediately moving off fast in reverse - you'd feel rapid deceleration during braking, followed acceleration in reverse gear. Like displacement and velocity, acceleration can be negative or positive depending on direction - in this example, braking would cause negative acceleration, as would moving off in reverse. It sounds wierd, but the point in time where the acceleration would be (negative) maximum is the moment when the car stops moving - just as it changes direction.
Thinking about our animated mass-spring system, we can see this effect. Conversely as the object moves through its equilibrium position, the velocity is only changing very slowly and the acceleration actually becomes zero. This is shown below:
Note again - the period is the same, but things don't line up in time...there's some phase in there too.
All these masses and springs are OK - but you can experience this by playing on a swing. Look at these videos, and try to work out when displacement, velocity and acceleration are 1) a maximum, and 2) zero, for someone swinging...
Equilibrium and Restoring Forces
In our animations, the mass on a spring bounces up and down with a regular 'pattern' or waveform. Why does this happen?
Let's think some more about playground swings. What's the first thing we do to get the swing moving?
The swing is first pulled back, and then pushed forward. It then oscillates back and forth 'on its own' until it slowly comes to rest at its equilibrium position - the middle position between the two extremes of displacement. This is usually the place where an object will naturally rest if no external forces are applied to it.
Once the swing has been pulled away from its equilibrium position, the force of gravity will act to bring it back. This force will always act in a direction towards the equilibrium position, and is known as a restoring force. If the swing is pulled higher into the air (larger displacement amplitude) then there will be a larger restoring force acting on it, and when the pusher lets go it will travel further. This shows that the restoring force is proportional to the distance of the swing from the equilibrium position.
The animation below shows these forces in action:
So why doesn't the swing just return directly to the equilibrium position, and stay there?
The restoring forces are large enough to make the swing over-shoot and travel 'up the other side'; this is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to over-shoot equilibrium and travel 'up the other side'. The process keeps repeating and an oscillation is formed. If no other forces are applied to the system, the swing will keep swinging back and forth forever, with a constant period and constant amplitude.
Does this happen in real life? (Silly question - of course not). Then why not?
In all real-life oscillating systems, damping comes into play. Damping is all about the loss of energy, often due to resistance or friction. The air displaced by the swinger adds some resistance, and even if the swing were in a vacuum (and the swinger were, therefore, dead) the pivots at the end of the chain generate some friction with their supports and slowly, almost immeasurably, heat up. This uses enery, which has to come from somewhere...and so the amplitude of oscillation decreases.
If you have no pusher, and have to propel yourself on the swing, you have to put some effort into it to keep the swing moving. This effort is overcoming damping. If there were no damping (and also if you could switch off your heart, brain and other energy-consumptive body functions) you could swing forever without needing to eat!
Simple Harmonic Motion
Oscillations
You may have heard the one about the opera singer breaking a wineglass with their voice...(in other versions it's a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations. Later in this section we'll show you how to break a wineglass with a loudspeaker...but first we'll try to work out some rules for things which vibrate, or oscillate. There are many forms of oscillation in the real world - oscillations determine the sound of a musical instrument, the colour of a rainbow, the ticking of a clock and even the temperature of a cup of tea. A mechanical oscillation is a repeating movement - an electrical oscillation is a repeating change in voltage and current.
Mechanical oscillations are probably easiest to think about, so we'll start there. Let's think about the movement of an object back and forth over a fixed range of positions, such as the movement of a swing in a playground, or the bouncing up and down of a weight on the end of a spring (this last one is one of those examples which teachers / scientists / engineers love, but which doesn't seem to have anything to do with the real world. As we'll see, it does!!).
The pendulum and the mass-spring system are both oscillating. How can we describe their oscillation?
Displacement
The animation shows how displacement describes the distance (and also direction...displacement can be negative as well as positive) of the object from its equilibrium position.
The displacement amplitude tells us how 'big' the oscillations are - we can use the peak value (the maximum positive displacement from the equilibrium position) or the peak-to-peak value (the distance between negative-maximum to positive-maximum).
The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.
Can you see why the first time the mass returns to the equilibrium position, we've only reached half a cycle?
Displacement on its own will sometimes do...but often it is also useful to understand the oscillation in terms of velocity or acceleration. These three things are very similar...read on.
Velocity and Acceleration
When our oscillating object reaches maximum displacement (when it is as far from equilibrium as it can get) it changes direction. This must mean that there is an instant in time when it is not moving...and so at the point of maximum displacement, its speed is zero. Then it speeds up in the opposite direction, and travels fast through the equilibrium position before starting to slow again in preparation for the next change in direction.
Velocity is just another word for speed, with the extra feature that it has direction and therefore can be negative or positive. (If you walk backwards at 4m.p.h. - difficult without falling over - then your velocity = -4m.p.h.)
So - when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity do not 'line-up' in time...one is shifted along compared to the other. This shift along in time is called a phase angle, which is an idea we'll come back to later.
At maximum displacement the object has to come to rest, and then start moving in the opposite direction. Imagine braking in a car (moving forwards) before immediately moving off fast in reverse - you'd feel rapid deceleration during braking, followed acceleration in reverse gear. Like displacement and velocity, acceleration can be negative or positive depending on direction - in this example, braking would cause negative acceleration, as would moving off in reverse. It sounds wierd, but the point in time where the acceleration would be (negative) maximum is the moment when the car stops moving - just as it changes direction.
Thinking about our animated mass-spring system, we can see this effect. Conversely as the object moves through its equilibrium position, the velocity is only changing very slowly and the acceleration actually becomes zero. This is shown below:
Note again - the period is the same, but things don't line up in time...there's some phase in there too.
All these masses and springs are OK - but you can experience this by playing on a swing. Look at these videos, and try to work out when displacement, velocity and acceleration are 1) a maximum, and 2) zero, for someone swinging...
Equilibrium and Restoring Forces
In our animations, the mass on a spring bounces up and down with a regular 'pattern' or waveform. Why does this happen?
Let's think some more about playground swings. What's the first thing we do to get the swing moving?
The swing is first pulled back, and then pushed forward. It then oscillates back and forth 'on its own' until it slowly comes to rest at its equilibrium position - the middle position between the two extremes of displacement. This is usually the place where an object will naturally rest if no external forces are applied to it.
Once the swing has been pulled away from its equilibrium position, the force of gravity will act to bring it back. This force will always act in a direction towards the equilibrium position, and is known as a restoring force. If the swing is pulled higher into the air (larger displacement amplitude) then there will be a larger restoring force acting on it, and when the pusher lets go it will travel further. This shows that the restoring force is proportional to the distance of the swing from the equilibrium position.
The animation below shows these forces in action:
So why doesn't the swing just return directly to the equilibrium position, and stay there?
The restoring forces are large enough to make the swing over-shoot and travel 'up the other side'; this is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to over-shoot equilibrium and travel 'up the other side'. The process keeps repeating and an oscillation is formed. If no other forces are applied to the system, the swing will keep swinging back and forth forever, with a constant period and constant amplitude.
Does this happen in real life? (Silly question - of course not). Then why not?
In all real-life oscillating systems, damping comes into play. Damping is all about the loss of energy, often due to resistance or friction. The air displaced by the swinger adds some resistance, and even if the swing were in a vacuum (and the swinger were, therefore, dead) the pivots at the end of the chain generate some friction with their supports and slowly, almost immeasurably, heat up. This uses enery, which has to come from somewhere...and so the amplitude of oscillation decreases.
If you have no pusher, and have to propel yourself on the swing, you have to put some effort into it to keep the swing moving. This effort is overcoming damping. If there were no damping (and also if you could switch off your heart, brain and other energy-consumptive body functions) you could swing forever without needing to eat!
Simple Harmonic Motion
Oscillations
You may have heard the one about the opera singer breaking a wineglass with their voice...(in other versions it's a greenhouse(?!), or even the specs worn by members of the audience). Well, if the singer could pull it off, they would be exploiting oscillations. Later in this section we'll show you how to break a wineglass with a loudspeaker...but first we'll try to work out some rules for things which vibrate, or oscillate. There are many forms of oscillation in the real world - oscillations determine the sound of a musical instrument, the colour of a rainbow, the ticking of a clock and even the temperature of a cup of tea. A mechanical oscillation is a repeating movement - an electrical oscillation is a repeating change in voltage and current.
Mechanical oscillations are probably easiest to think about, so we'll start there. Let's think about the movement of an object back and forth over a fixed range of positions, such as the movement of a swing in a playground, or the bouncing up and down of a weight on the end of a spring (this last one is one of those examples which teachers / scientists / engineers love, but which doesn't seem to have anything to do with the real world. As we'll see, it does!!).
The pendulum and the mass-spring system are both oscillating. How can we describe their oscillation?
Displacement
The animation shows how displacement describes the distance (and also direction...displacement can be negative as well as positive) of the object from its equilibrium position.
The displacement amplitude tells us how 'big' the oscillations are - we can use the peak value (the maximum positive displacement from the equilibrium position) or the peak-to-peak value (the distance between negative-maximum to positive-maximum).
The time period of the oscillation is the time taken for the object to travel through one complete cycle. We can measure from the equilibrium position, as in the animation, or from any other point on the cycle, as long as we measure the time taken to return to the same point with the same direction of travel.
Can you see why the first time the mass returns to the equilibrium position, we've only reached half a cycle?
Displacement on its own will sometimes do...but often it is also useful to understand the oscillation in terms of velocity or acceleration. These three things are very similar...read on.
Velocity and Acceleration
When our oscillating object reaches maximum displacement (when it is as far from equilibrium as it can get) it changes direction. This must mean that there is an instant in time when it is not moving...and so at the point of maximum displacement, its speed is zero. Then it speeds up in the opposite direction, and travels fast through the equilibrium position before starting to slow again in preparation for the next change in direction.
Velocity is just another word for speed, with the extra feature that it has direction and therefore can be negative or positive. (If you walk backwards at 4m.p.h. - difficult without falling over - then your velocity = -4m.p.h.)
So - when velocity is maximum, displacement is zero, and when displacement is maximum, velocity is zero. The period of oscillation is the same, but the plots for displacement and velocity do not 'line-up' in time...one is shifted along compared to the other. This shift along in time is called a phase angle, which is an idea we'll come back to later.
At maximum displacement the object has to come to rest, and then start moving in the opposite direction. Imagine braking in a car (moving forwards) before immediately moving off fast in reverse - you'd feel rapid deceleration during braking, followed acceleration in reverse gear. Like displacement and velocity, acceleration can be negative or positive depending on direction - in this example, braking would cause negative acceleration, as would moving off in reverse. It sounds wierd, but the point in time where the acceleration would be (negative) maximum is the moment when the car stops moving - just as it changes direction.
Thinking about our animated mass-spring system, we can see this effect. Conversely as the object moves through its equilibrium position, the velocity is only changing very slowly and the acceleration actually becomes zero. This is shown below:
Note again - the period is the same, but things don't line up in time...there's some phase in there too.
All these masses and springs are OK - but you can experience this by playing on a swing. Look at these videos, and try to work out when displacement, velocity and acceleration are 1) a maximum, and 2) zero, for someone swinging...
Equilibrium and Restoring Forces
In our animations, the mass on a spring bounces up and down with a regular 'pattern' or waveform. Why does this happen?
Let's think some more about playground swings. What's the first thing we do to get the swing moving?
The swing is first pulled back, and then pushed forward. It then oscillates back and forth 'on its own' until it slowly comes to rest at its equilibrium position - the middle position between the two extremes of displacement. This is usually the place where an object will naturally rest if no external forces are applied to it.
Once the swing has been pulled away from its equilibrium position, the force of gravity will act to bring it back. This force will always act in a direction towards the equilibrium position, and is known as a restoring force. If the swing is pulled higher into the air (larger displacement amplitude) then there will be a larger restoring force acting on it, and when the pusher lets go it will travel further. This shows that the restoring force is proportional to the distance of the swing from the equilibrium position.
The animation below shows these forces in action:
So why doesn't the swing just return directly to the equilibrium position, and stay there?
The restoring forces are large enough to make the swing over-shoot and travel 'up the other side'; this is when a restoring force in the opposite direction takes over and pushes the swing back down. Again, the force is large enough for the swing to over-shoot equilibrium and travel 'up the other side'. The process keeps repeating and an oscillation is formed. If no other forces are applied to the system, the swing will keep swinging back and forth forever, with a constant period and constant amplitude.
Does this happen in real life? (Silly question - of course not). Then why not?
In all real-life oscillating systems, damping comes into play. Damping is all about the loss of energy, often due to resistance or friction. The air displaced by the swinger adds some resistance, and even if the swing were in a vacuum (and the swinger were, therefore, dead) the pivots at the end of the chain generate some friction with their supports and slowly, almost immeasurably, heat up. This uses enery, which has to come from somewhere...and so the amplitude of oscillation decreases.
If you have no pusher, and have to propel yourself on the swing, you have to put some effort into it to keep the swing moving. This effort is overcoming damping. If there were no damping (and also if you could switch off your heart, brain and other energy-consumptive body functions) you could swing forever without needing to eat!
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