The Fundamental Principle of Standing Waves: Is it all about phase?

[Studiot]

So what you are saying is that a lower energy will, say, only produce the first harmonic but a higher energy will produce, say, the first second and third harmonic? If we then were to combine these, would it be like the superposition of waves?

Basically.
If you start with only enough energy to generate the fundamental then that is all you get.
Using more energy allows some first, second, third etc to be generated. Some systems only allow odd harmonics or even harmonics.

This idea is fundamental. It comes out in many ways so for instance, when you study the quantum theory the lowest energy configuration is occupied first by a system and then the next and so on.

With a propagating wave, would this simply be the number of waves per unit length?

A propagating wave with harmonics is still apropagating wave, it just has a different wave shape.

Herein lies a wonderful tieup between maths and physics. The maths says that the basic wave is sinusoidal. However Fourier theory also says that almost any repetitive waveshape can be represented or built up from a fundamental (of the same frequency) plus suitable % of harmonics.

If we look at the soft boundary, is this example like the one from before - where we move a string around in air (no rigid boundary). There is therefore no phase change in this reflection BUT a standing wave is still produced?

Don't forget that the animation is of an ideal string. An ideal string cannot generate standing waves if it has a free end.

Try to get hold of the concepts first, but keep going.

 

[jsmith613]

Basically.
Don't forget that the animation is of an ideal string. An ideal string cannot generate standing waves if it has a free end.

In post #24 SC says "Remember, at a 'loose' end, there is an antinode so the phase of the reflection will be unchanged - a maximum is formed."
Based on what you just said "An ideal string cannot generate standing waves if it has a free end." would the motion of the string be totally dependant on the force applied by the thing moving it (say the hand). If the amplitude is 0.5m as provided by the hand, it will always be 0.5m.
If we presume a hard-boundary, then the maximum amplitude could be 1 m? right?

I shall actually try the experiment and see if it is true but I wanted to check the theory before the experiment

thanks

 

[jsmith613]

I did the experiment and this is what I think i observed:
I took the string and waved it about round and round. There was no apparent standing wave. The maximum amplitude was dependant totally on how I move the string.
There was a node where I move the string and another about a wavelength away (2 nodes) and from what I could see, two anti-nodes BUT I may be wrong!

I then tried producing a standing wave: one node end at my hand, the other supported by an object - did not work - shame!

 

[sophiecentaur]

On a practical note: I have to point out that harmonics are exact multiples of a fundamental frequency. In resonant systems - standing waves and others - the higher order frequencies at which resonance occurs can, in no way, be relied upon to be exact multiples of the fundamental. They should be referred to as overtones.
The frequency difference between a harmonic and an overtone may not be much but it is enough to give a musical instrument its characteristic sound. Also, if you ever need to buy a quartz crystal for use in an oscillator, you will find that some crystals are specified to be operated at overtone frequencies and that the overtone frequency will usually not be harmonically related to the crystal fundamental. The 'node' pattern will be as you'd expect to find it but, because of the effects at the ends, the exact positions may well be shifted.

 

[jsmith613]

On a practical note: I have to point out that harmonics are exact multiples of a fundamental frequency. In resonant systems - standing waves and others - the higher order frequencies at which resonance occurs can, in no way, be relied upon to be exact multiples of the fundamental. They should be referred to as overtones.
The frequency difference between a harmonic and an overtone may not be much but it is enough to give a musical instrument its characteristic sound. Also, if you ever need to buy a quartz crystal for use in an oscillator, you will find that some crystals are specified to be operated at overtone frequencies and that the overtone frequency will usually not be harmonically related to the crystal fundamental. The 'node' pattern will be as you'd expect to find it but, because of the effects at the ends, the exact positions may well be shifted.

does this imply that every instrument has a slightly different difference between the the harmonic and the overtone - this also, presumably, depends on the amount of energy put into the system?

 

[Bob S]

Thanks Studiot for the url to the Kettering demo of traveling wave refections form hard and soft boundaries.

Here is a discussion and illustration on harmonics and nodes on guitar strings:

http://en.wikipedia.org/wiki/Guitar_harmonics

Bob S

 

[sophiecentaur]

does this imply that every instrument has a slightly different difference between the the harmonic and the overtone - this also, presumably, depends on the amount of energy put into the system?

As long as you are operating in the linear region, the amplitude should make no difference. But the impedance of the driving source (and the loading of the detector) could affect things subtly.

 

[jsmith613]

As long as you are operating in the linear region, the amplitude should make no difference. But the impedance of the driving source (and the loading of the detector) could affect things subtly.

According to the dictionary impedance,
"the impedance of a load is the resistance of the load and is its reactance". WHat has this to do with harmonics and overtones

 

[sophiecentaur]

According to the dictionary impedance,
"the impedance of a load is the resistance of the load and is its reactance". WHat has this to do with harmonics and overtones

Fair question----
Putting power into a system and taking it out (measurement) involves connecting to that system. Any practical source of power (amplifier) has an impedance which is not 'ideal' (i.e. not zero or infinite). This constitutes a 'load' on the resonator and can affect the phase and amplitude of the reflected waves. This is equivalent to altering the characteristics of the resonator and can change the resonant frequency. An example of this could be bending the note on a mouth organ reed by sucking hard.

 

[Studiot]

In the discussion I noted that you will not get standing waves in a string with a totally free end.

Under certain conditions it is possible to get the string end to exhibit chaotic behaviour, I think the poplar term is 'the strange attractor' for this.

Since we are talking about mechanical oscillations we need the mechanical definition of impedance.

This is the ratio of the force needed to move a body to the velocity achieved by said movement.

 

[jsmith613]

Thanks so much guys for all your help! I really appreciate it!

 

[jsmith613]

I know I seemed to close the topic but I have a really quick last question on this point:

If I superimpose to longitudinal waves on each-other (drawn as wave-fronts) do i simply get an double amplitude if they are in phase and have the same frequency and amplitude as each other.

Here is what I mean:

Wave 1: ¦¦¦¦¦¦¦¦¦¦
Wave 2: ¦¦¦¦¦¦¦¦¦¦
(ignore the bits in-between each wave front ¦ - the middle bit)

Superimposed wave: ¦ ¦ ¦ ¦ ¦ ¦

is that correct

 

[Studiot]

There is always some quantity that you can plot as an X - Y graph for a longitutinal wave.

In the case of sound waves, which are longitudinal waves, that quantity is pressure.
A plot of pressure along any straight line taken as the x-axis will show the typical sinusoidal variation.

Longitudinal waves can participate in constructive and destructive interference and standing waves, just as transverse ones. You add the quantity (eg pressure) varying as above, rather than particle displacement amplitude todescribe the whole wave.

 

[jsmith613]

There is always some quantity that you can plot as an X - Y graph for a longitutinal wave.

In the case of sound waves, which are longitudinal waves, that quantity is pressure.
A plot of pressure along any straight line taken as the x-axis will show the typical sinusoidal variation.

Longitudinal waves can participate in constructive and destructive interference and standing waves, just as transverse ones. You add the quantity (eg pressure) varying as above, rather than particle displacement amplitude todescribe the whole wave.

Why do I use pressure? not speed or displacement?

 

[sophiecentaur]

Whatever direction the displacement takes place in, you can still get a resultant when two waves coincide. I think your diagrams may be confusing wave fronts with displacement.
You can use pressure (which corresponds to displacement) or speed (average speed, if it's a gas). The pressure max will be at a velocity min, of course. You can still do the sums but it's normal to consider displacement as it's zero at a wall.

 

[jsmith613]

Whatever direction the displacement takes place in, you can still get a resultant when two waves coincide. I think your diagrams may be confusing wave fronts with displacement.
You can use pressure (which corresponds to displacement) or speed (average speed, if it's a gas). The pressure max will be at a velocity min, of course. You can still do the sums but it's normal to consider displacement as it's zero at a wall.

Wavefronts are point on a wave that join up all adjacent points.
Displacement is the maximum distance traveled from resting point.
On my diagram, the point between two lines represents the distance between two new waves - ah I see why I got confused!

How then would I draw a displacement graph for a longitudinal wave or does this not exist?

 

[Studiot]

Why do I use pressure? not speed or displacement?

You must have done the kinetic theory by now.

Sound is a pressure wave.
Pressure is the average result of the movement of a large number of individual particles, not in any way aligned like in a transverse wave.
So, on average, the pressure ( ie particle concentration) at any location varies periodically. Individual particles may be moving in widely different patterns and speeds. Tracking the actions of any single particle will not lead to a wave.

 

[jsmith613]

I have uploaded a question here about waves and superposition http://www.mediafire.com/file/bg6raawvbv8mbts/A%20loudspeaker%20is%20connected%20to%20a%20signal%20generator1.doc

here are my answers. Could you please check them:

"Questions"

a) Explain why the minima never have a zero value. [2]
b) As the microphone is moved towards the metal plate, the amplitudes at the minima gradually decrease. Suggest why this happens. [2]

"My answers"
a) the reflected wave will have less energy than the wave produced by the speaker. This means the amplitudes of the both waves cannot destroy one another (interfere totally deconstructivl) as the two amplitudes are not of equal magnitude

b) The sound wave produced by the sound emitted from the microphone will have lost energy as it propagates towards the microphone - hence the amplitudes of the waves will be smaller.
The path difference decreases as the microphone moves closer to the metal plate, so the energy level of both waves will be more similar, as will the amplitudes, so the deconstructive interference becomes more obvious.

How do you rate these answers? are they even correct?

 

[Studiot]

How do you rate these answers? are they even correct?

You mean apart from the obvious howler in (b) ?

sound emitted from the microphone

(a) The pressure variation in the sound is about an average value. Obviously the pressure can never actually be negative . Can it be zero?

(b) You got the general idea. Obviously the power in the reflected wave can never be greater than the incident. The power emitted by the loudspeaker is spread out through a larger volume as it travels away. This may not be an inverse square law since it depends upon the directionality of the speaker. So yes close to the speaker there is a much bigger disparity between the power of the direct and the reflected waves than close to the reflector.

 

[jsmith613]

sound emitted from the microphone

Ye kinda big mistake!

quickly thinking back to this
Wave 1: ¦¦¦¦¦¦¦¦¦¦
Wave 2: ¦¦¦¦¦¦¦¦¦¦
(ignore the bits in-between each wave front ¦ - the middle bit)

Superimposed wave: ¦ ¦ ¦ ¦ ¦ ¦

Would this diagram be true if I plotted against time on the x-axis.
The space between each wavefront is the wavelength. It should take about double as long to complete a wavelength if we interfere constructively so the time taken along the x-axis between each wavefront should also be double?

 

[Studiot]

This issue is often rushed over in textbooks.

For a traveling wave you can put either time or distance on the x-axis and draw a graph of some suitable quantity such as displacement or pressure against this.

So your graph describes how the wave varies in time at a fixed position in space

or

how the wave varies in space at a fixed instant in time.

Both graphs actually look pretty similar and are usually represented by a sinusoidal curve for sound mathematical reasons.

When you allow two waves to combine to form a standing wave a graph of your 'varying' property against time shows a straight line parallel to the x axis. The property does not vary with time or the wave is time independant.

A graph plotted against position shows the characteristic node/antinode sequence.

Understanding this goes back to your question about phase.

If we define the phase-difference between two spatial points as

'the time difference when our varying quantity reaches its maximum'

you can see this is zero for a standing wave.
All points in the wave reach their maximum simultaneously, at least between any two nodes.

Most textbooks omit to emphasise that the illustrations they draw for traveling waves plot time on the x axis, but when they illustrate standing waves they are plotting distance. They often place the two graphs in juxtaposition, when they are not really directly comparable.

 

[jsmith613]

If i were to address the issue using words I might then be able to imagine the graph.

What is going on in any constructive interferece is as follows: the amplitude of the two propagating waves combine to form maxima and minima. These new amplitudes, assuming the waves have the same amplitude will be double. This means that if we assume time along the x axis, it should take double as long to complete a stage in the wave than it would do with our propagating wave.
This means that if it took 0.1 second to complete each of our individual propagating waves, the length of time between each wave in our standing wave would be 0.2.

Obviousley there is not really a concept of time with a standing wave as the standing wave remains stationary so if we assume time as the distance between the beginning of one wave and the beginning of the next on both graphs, our standing wave would have a 'distance' of double the propgating wave

How's that as an explanation?

 

[jsmith613]

On a slight tangent:

http://s359.photobucket.com/albums/oo40/jsmith613/?action=view&current=Long.png

I am writing my notes for physics and came across this diagram telling me what the amplitude for a longitudinal wave is. From the equillibrium position to apparently the next point along (see where blue arrow points to and ends). Surley the displacement is the distance between an area of compression and an area of rarefaction divide by 2

OR does amplitude vary continuosley on a longitdinal wave. Is it actually the maximum displacement of each point on a wave from the equillibrium (in the diagram this is the distance between each line on the wave and the nearest equillibrium point)

 

[Studiot]

How's that as an explanation?

Puzzling.

This means that if we assume time along the x axis, it should take double as long to complete a stage in the wave than it would do with our propagating wave.
This means that if it took 0.1 second to complete each of our individual propagating waves, the length of time between each wave in our standing wave would be 0.2.

A standing wave has no period - it is time independent.

It does however have a wavelength, equal to twice the distance between adjacent nodes.

What is going on in any constructive interferece is as follows: the amplitude of the two propagating waves combine to form maxima and minima. These new amplitudes, assuming the waves have the same amplitude will be double.

Only if the two waves are in phase and going in the same direction.

came across this diagram

Can you name any real world waves that actually look like this?
What happens in the white space above and below your diagram?

 

[jsmith613]

Puzzling.

Can you name any real world waves that actually look like this?
What happens in the white space above and below your diagram?

this diagram was in my physics textbook but poorley explained. I don't know exactly what it is on about but know that I ought to know it - the book is officialy endorsed by edexcel

I would presume the what space in between the diagram was between the waves like in a slinkey

ignore the spaces above and below!

 

[sophiecentaur]

On a slight tangent:

http://s359.photobucket.com/albums/oo40/jsmith613/?action=view&current=Long.png
Surley the displacement is the distance between an area of compression and an area of rarefaction divide by 2

This is wrong. The distance between compression regions and rarefaction regions is a quarter wavelength, whatever the amplitude (maximum displacement) of a wave.
In a 'real' compression wave, like sound, the actual displacement is a small fraction of a wavelength. A set of equally spaced vertical lines could be used to show the rest positions of small regions of air (/medium) and the displacement from those lines would be hardly visible. As you can plot a variable on any axis you like, you can just as easily show the displacement using an axis at right angles to the direction of propagation - and use whatever scale factor (gain) you want to show the way the displacement varies. So a longitudinal wave can just as easily be represented by a 'wiggly' line. Or it can be shown as varying shades of grey / colour, the darkest representing high pressure and the lightest, low pressure. I think there is a lot of needless confusion about this but XY graphs are usually chosen to represent most relationships between variables because they are easy to plot onto and to read off data.

 

[Studiot]

waves like in a slinkey

Good comment.

How about the rest?

 

[jsmith613]

Good comment.

How about the rest?

you mean the other types of waves?

 

[jsmith613]

This is wrong. The distance between compression regions and rarefaction regions is a quarter wavelength, whatever the amplitude (maximum displacement) of a wave.

if the wavelength of a wave is between two compression / rarefaction regions then logically it is half not a quarter

compression rarefaction compression
(0)---------- (50) --------------(100)

on the bottom (0), (50), (100) i show the percentage of a wave that has passed at each point.

SC your argument is that

compression rarefaction compression rarefaction compression
(0) -------- (25)------- (50) ------- (75)---------- (100)
is that we pass through two complete compression regions (and begin our third) to complete a single wave?

What?? surely this is wrong?

IGNORE the --- this just allows me to place the correct bracket under the correct term

 

[Studiot]

Originally Posted by Studiot
Good comment.

How about the rest?

you mean the other types of waves?

I mean my other comments in post#59

 

[jsmith613]

I mean my other comments in post#59

How's that as an explanation?
Puzzling.

How do you mean, puzzling?
Is it wrong or just gramatticaly confusing?

A standing wave has no period - it is time independent.

It does however have a wavelength, equal to twice the distance between adjacent nodes.

when we say wavelength I presume we are accounting for amplitude (y axis) as opposed to what ever appears on the x axis, because we are discounting time!

Only if the two waves are in phase and going in the same direction.

So deconstructive inteference would occur even when waves are 1 degree out of phase.

Can you name any real world waves that actually look like this?
What happens in the white space above and below your diagram?

I believe i tackled this - the real world wave is like a slinky

the white space above and below the diagram should be ignored it has no relevance. We are only interested in the white space in between each wave front

 

[sophiecentaur]

if the wavelength of a wave is between two compression / rarefaction regions then logically it is half not a quarter

compression rarefaction compression
(0)---------- (50) --------------(100)

on the bottom (0), (50), (100) i show the percentage of a wave that has passed at each point.

SC your argument is that

compression rarefaction compression rarefaction compression
(0) -------- (25)------- (50) ------- (75)---------- (100)
is that we pass through two complete compression regions (and begin our third) to complete a single wave?

What?? surely this is wrong?

IGNORE the --- this just allows me to place the correct bracket under the correct term

Yes -totally correct. I didn't re-read what I'd written. I was, I guess, thinking in terms of potential energy rather than the sign of the pressure difference.
My comment that the wavelength is not related to the amplitude of the compression / displacement / velocity still holds, though. It's only because of the attempt to depict both displacement and displacement on the same axis that any confusion can arise. Who (in any other circumstances) would think of trying to draw a graph on just one dimension when two dimensions are available to display the information? And, if people can't use graphs to aid understanding then they have no chance of getting this process clear in their heads, I fear.

And your comment about defining wavelength by "discounting time". If a train is going past you, you would not say that its speed is altering the distance between the fronts of successive coaches, would you? (Ignore special relativity). You take a snapshot and measure it with a ruler. WaveLENGTH will have the dimension of length, surely, so why bring in the concept of time? The two variables are 'separable', as they say. This just adds confusion to people who are already confused.

There is little point, either, in complaining about naff diagrams in textbooks. You can find one of those in almost every chapter of most schoolbooks! They were sketched by an original author and then messed about with by a graphic designer and then copied into future textbooks without the information being processed by the brains of future authors.

 

[sophiecentaur]

"So deconstructive inteference would occur even when waves are 1 degree out of phase.
"
The relative phases of the two waves is different over all distances so, for "one degree out of phase" in one place will mean that you have zero phase difference somewhere else. The resulting interference pattern will just be shifted a bit.

 

[sophiecentaur]

A =A₀ cos(wt-kx)
I wrote this earlier in the thread. It describes a simple harmonic wave perfectly and it beats me why there needs to be any extra arm-waving discussion of the message of that simple equation. 'A' can be any of the variables in a wave and it can be 'in any direction'. If anyone needs answers, then just feed the numbers in and you will get them. Add two (or several) of these together and you will get a value for total A at anywhere in space or time. Modify A, a bit, to take into account spreading or attenuation of waves and you will also get an accurate answer for the resultant.
Just give the Maths a chance.

 

[Studiot]

The relative phases of the two waves is different over all distances so, for "one degree out of phase" in one place will mean that you have zero phase difference somewhere else. The resulting interference pattern will just be shifted a bit.

Would you like to rethink this?

For simplicity any two waves will add to provide a compound wave.

Special things happen if

1) The waves are of the same frequency or one frequency is a whole number multiple of the other.

2) The waves are going in the same direction. This is how you can double the amplitude by adding waves. You cannot create a standing wave this way. This is also how to obtain constructive /destructive interference.

3) The waves are going in opposite directions. This is how to create a standing wave.

All this has already been discussed in previous posts.