Standing vs. Travelling waves in a string fixed at both ends

[Alexander83]

Hi there,
I'm trying to understand the propagation of a pulse along a string that is fixed at both ends and am confusing myself. The demonstration I'm using is a long piece of string, tied at one end to the wall and with the other end held by my hand, putting the string under tension. Here's my logic:

1. I jerk the string with my hand, disturbing the string and causing a traveling pulse to travel down the string.

2. I then immediately bring my hand back to the equilibrium position and hold it there. Both ends of the string are now fixed at both ends with the pulse traveling down it.

3. Based on the boundary conditions, the only wave functions which should be allowed on this string are standing waves, but multiple frequencies of standing waves are permitted at once.

4. Therefore, what appears to be a traveling wave pulse is really the superposition of a number of standing waves having different frequencies, phase constants and amplitudes. Although these individual standing waves do not propagate, the net effect of the different frequencies is to give the appearance of a wave that propagates.

Is this logic correct? If not, what am I missing… because clearly it is possible to have traveling wave pulses on a string in the situation I've described.

Thanks!

Chris

 

[sophiecentaur]

You will only get a standing wave after things have settled down. Waves have to travel up and down for an large (infinite) number of times before a standing wave pattern can be established (i.e the steady state condition). Any analysis into the fundamental and harmonics is implicitly assuming that the oscillations have existed over all time. It says nothing about the details of how the energy in the initial excitation impulse will be partitioned.
Boundary conditions cannot be imposed on the initial pulse because it doesn't 'know about' the existence of any termination on the string until the very leading edge of the pulse reaches it. After the initial (partial, possibly) reflection of the pulse, it will travel to the start end. If there is no dispersion or loss, this pulse will go up and down for ever. For this situation to change into a standing wave, I think that waves of different frequencies must have different speeds.
The situation for a real plucked string is not the same as for a continually excited string because the energy in a plucked string will decay. For continuous excitation, the total energy on the string will increase until the power loss balances the input power. I have a feeling that a complete analysis of how the initial pulse changes into a set of normal modes must include some element of loss. Perhaps someone can put me right on that?

 

[Alexander83]

Hi Sophie,
Thanks for your reply - it helps… though I'm still a bit confused on a few points. Here's my thought process in more detail:

1. Thinking about things mathematically, can't we treat the ends of the string as boundary conditions which have to be nodes at anytime whether the pulse has reached the end or not?

2.Following on from that, couldn't we treat the distortion of the string into the pulse as an initial condition?

3. Solutions to the wave equation on a string fixed at both ends have allowed solutions which are standing waves.

4. Per Fourier's theorem, we should be able to represent any arbitrary disturbance on the string as the superposition of an infinite number of these standing waves, having different frequencies. Therefore the pulse can be represented by a summation over all valid standing wave frequencies.

My slight confusion is that it still seems somewhat counterintuitive that you can represent a traveling wave pulse by a summation of standing wave patterns, but I think that's due to the fact that the wave patterns have different frequencies, causing there to be a net motion of the disturbance on the string.

Otherwise, if the concept of standing waves is as much of an idealization as your post suggests (requiring waves traveling up and down an infinite # of times), I can't see how they would be useful. For instance, in acoustics, standing waves are discussed heavily because the initial disturbance of an instrument string (which is not sinusoidal) results in instrument tones that can be understood based on the varying degree to which instrumental fundamental and harmonics are present.

I do agree that decay plays an important role. My understanding is that higher frequencies generally tend to decay most rapidly leading with an evolution towards a sound closer to the fundamental in say, a plucked string (as this applet would suggest: http://www.falstad.com/loadedstring/).

Does any of the above make sense? Maybe I'm just totally confused about this, but I do appreciate you taking the time to reply.

Chris.

 

[Alexander83]

Here's a video showing how a pulse on an elastic band decays into something that closely resembles the fundamental for the string. My thinking is that the traveling pulse can be modeled as due to the effects of a huge number of harmonics, with the lowest harmonics being the last to decay. From this I would infer that numerous standing wave frequencies were initially present and only near the end do we just see essentially one wave pattern.

http://resources.schoolscience.co.uk/salford/feschools/waves/quicktime/elastic2512K_Stream.qt

 

[sophiecentaur]

This is complicated and has made me think quite a lot (that's a good thing).
The scenarios that are normally considered are
1. Response to a continuous, repeated signal.
2. Response when plucked
Both of these need losses to be taken into consideration. Energy cannot be transferred from one part of the signal spectrum to another in a linear system so, when there is frequency selection of any kind (resonances etc.) some input frequencies must be dissipated whilst some are allowed to build up in energy.
1.For a continuous signal, a standing wave will build up over a (possibly large, depending on the Q factor) number of cycles of the signal. The energy in parts of the signal spectrum which correspond to the normal modes will build up but energy at other frequencies will stay at the level of the input signal values. You will end up with an energy spectrum with peaks at the normal modes - a complicated form of standing wave.
2. When the string is plucked, you are starting with the string forming a triangle - or some other shape which consists of spatial harmonics of the length of the string. (A discrete Fourier analysis of the string shape, which is assumed to be repeated infinitely). The actual shape of the string will correspond to a combination of sinusoids which correspond to the normal modes and so the string will only vibrate at those frequencies. (In your Point 4.)

I think the standard treatment of the plucked string is not suitable for your OP. You are launching an arbitrarily shaped wave / pulse onto the string and I don't think you can assume that it equivalent to the situation with a plucked string as the energy is only traveling one way from your hand. Its spectrum is not limited to frequencies of normal modes (not as you say in your point 4.). When the pulse is reflected at the far end, without dispersion, the pulse will be merely reflected. All you can say is that there will be no displacement for all frequencies at the clamp. You will only get an identifiable standing wave if a significant proportion of the energy of the pulse is reflected at both ends -( for reverb ) and the standing wave will only be identifiable for frequencies which correspond to the normal modes. The effect of multiple reflections will cause the impedance 'seen' by the clamps to be different and the energy at resonance frequencies will not be coupled out (lost) as fast in the losses in the terminations. (I'm treating it as a basic transmission line analogy because that's more familiar to me).

Another point is that the normal modes in many musical structures do not correspond to harmonics of a fundamental. I prefer to use the term Overtone, to avoid confusion. The differences between Harmonics and Overtones help to give musical instruments their characteristic sounds. Strings are relatively well behaved in this way but, even so, the end effect at bridge and fret can be detected by a good ear.

 

[Alexander83]

Thanks again for your post, Sophie. There's some interesting information there about harmonics vs. overtones that I was not aware of and I appreciate you drawing my attention to it. I'm still not entirely convinced about the point you make where:

"You are launching an arbitrarily shaped wave / pulse onto the string and I don't think you can assume that it equivalent to the situation with a plucked string as the energy is only traveling one way from your hand. Its spectrum is not limited to frequencies of normal modes (not as you say in your point 4.). "

It was my understanding based on Fourier Analysis as applied to a string satisfying the wave equation for a string fixed at both ends that any arbitrary wave generated on a string could be envisioned as a sum over all of the normal modes of vibration of the string. I.e. once I've generated the pulse by flicking the string and thereafter return my end of the string to the "fixed end" boundary condition, then the pulse is just an arbitrary displacement of the string which will bounce back and forth indefinitely with no dispersion and dissipation present.

Have a look at page three of this document (which is very similar to a textbook I have on the subject)...

http://landau.ucdavis.edu/kiskis/phy9hc_03/fourier.pdf

... according to this document, If I know the displacement of the string and the velocity of the string segments at any particular time (call this time t = 0) associated with the pulse, I should be able to express the pulse as a Fourier series. If the pulse were to include frequency components other than the normal modes, would it not violate the assertion made in the document that I should be able to represent any distortion of the string just using a sum based off of normal modes.

Thank you again for your help. It's appreciated!

Chris.

 

[sophiecentaur]

I will attempt to get my ideas across about the difference between your idea and the simple plucked string. There is a difference between normal modes of oscillation, which are essentially spatially based and frequencies of any wave that's set up on the string by another means. The wave pulse that you set off can have any frequencies in it that you care to impose on it. (No boundary conditions have been imposed on the wave that you chose to make) No energy has reached the end of the string until you finish shaking it about. You could even choose to send off a pulse that had none of the frequencies of the natural modes of the string. I think it's clear that the basics of wave theory tell us you can't just shift a frequency. We can assume no loss, so the energy can slosh up and down without forming any standing wave at all. I think you are mis-applying the Fourier idea because you are basing a spatial Fourier analysis (=Modes) to a frequency analysis (= spectrum),
Now, when you slowly pull the taught string to one side, wait for it to settle and then let it go, the boundary conditions are there from the very start and they dictate that it will be only overtone frequencies that can exist. The 'plucked string' is a special case so the results of that analysis must not dictate what will happen in other cases.

What I wrote earlier about the impedances the the string presents to the ends will mean that the consequence of the purely progressive waves that my suggested pulse would be that the energy would leak away faster than waves set up by a 'static' set up of the string prior to a simple plucking. No sustained self-resonance occurs.
I have given this some thought and I think that reconciles the contradictions that you feel you have found. You are more than welcome to find any fatal flaw. lol

 

[Alexander83]

I will attempt to get my ideas across about the difference between your idea and the simple plucked string. There is a difference between normal modes of oscillation, which are essentially spatially based and frequencies of any wave that's set up on the string by another means. The wave pulse that you set off can have any frequencies in it that you care to impose on it. (No boundary conditions have been imposed on the wave that you chose to make) No energy has reached the end of the string until you finish shaking it about. You could even choose to send off a pulse that had none of the frequencies of the natural modes of the string. I think it's clear that the basics of wave theory tell us you can't just shift a frequency.

The above I totally agree with, save for the fact that I think the distant end of the string (that's clamped in place) still has a boundary condition present here whether the wave has "sensed" it or not.

Now, when you slowly pull the taught string to one side, wait for it to settle and then let it go, the boundary conditions are there from the very start and they dictate that it will be only overtone frequencies that can exist. The 'plucked string' is a special case so the results of that analysis must not dictate what will happen in other cases.

I think this is where I'm still having an issue. I agree that the plucked string case requires that the string be at rest and that the boundary conditions are present right from the get go. However, the calculation of Fourier coefficients takes into account the velocity of the medium, so you should be able to impose the boundary conditions once the string is already in motion (ie. if I hold my hand fixed after starting the pulse down the line). There's a particular set of Fourier coefficients (see the pdf I posted) that is tabulated based on the velocity of the string elements, so it seems that one can plausibly do the Fourier analysis based on the already moving string provided you are able to fully able to specify the displacement and velocity of the string in question.

Perhaps a better way of phrasing my questions is not whether the string pulse causes standing waves, but rather whether the motion of the string pulse can be visualized as a sum over standing wave frequencies... clearly the pulse is a traveling wave and that's the best way to represent it, but can a pulse wave be mathematically represented as the effect of a large number of resonant frequencies of the string. I.e. is it truly possible, according to Fourier's theorem to represent any wave disturbance on a string held at both ends as a sum of an infinite number of resonant modes? I think it can, but you're indicating that this is restricted to just ideal cases like the plucked string.

I suspect it's my lack of a deep understanding of the material that's preventing me from grasping your argument. I'm not trying to be difficult - I truly do appreciate the time you've taken in responding to my post. Unless anyone else here cares to weigh in on this discussion, perhaps it's best that we leave it here.

Chris.

 

[sophiecentaur]

Try to remember that Modes are not Frequencies and make sure you distinguish between the two. The discrete Fourier transform* that is performed on the spatial arrangement of the plucked string will give you the amplitudes for the modes - which will correspond to frequencies which satisfy the wave equation with particular boundary conditions. That doesn't say anything about the frequencies of waves (Continuous Fourier Transform**) that could be carried on the string when driven in the way you propose. If you want to have the Boundary Conditions thing included, I could say that the boundary conditions that you impose with your waving hand need to be considered too because the hand is there whilst the wave is being formed. You launch a progressive wavewhich occupies a finite time interval, when you move your hands. Long after the input signal is cut (start end clamped), the wave pulse encounters the boundary condition imposed by the remote end clamp. But the string, when plucked, has energy stored in it and those boundary conditions only allow the string energy to be carried by pairs of waves (from the very start of the situation), traveling in different directions - that's the only way the string can behave under that circumstance.

At times like this, I try to ask myself what is the alternative to what I am proposing and that is the string just can't be made to vibrate at any frequency other than at the frequencies corresponding to the modes. That is clearly not a valid statement so what I say must be true, I conclude (null hypothesis). (I also have a problem in coming to terms with an apparent paradox).

How do we square this with real life? I think there must be something about Amplitudes that you (and I) are ignoring. When you have the energy carried by a standing wave in a low order overtone or fundamental, you will get a visible, coherent pattern with a large amplitude and which is sustained (on a real guitar string) for a long time because the energy doesn't get dissipated. I have never seen the equivalent of that, done with an impressed burst of oscillation. What would it look like? I think you (and I) are thinking in terms of the piece of rope with the hand and the guitar string and trying to compare them. Perhaps there is an equivalent situation when you strike a Cymbal which has no identifiable fundamental but a whole host of unrelated overtones yet which will sustain for quite some time.

Have you tried playing with the longitudinal waves of a slinky? They will travel back and forth for several identifiable journeys before they decay completely. Go to a Science park and play on the torsional wave demo. You can set up standing waves and you can launch bursts of waves at arbitrary frequencies - they do propagate.

* DFT is of an assumed or actually repeated spatial or temporal function, which gives a 'spectrum' of discrete harmonics
**A signal / pulse / burst of finite length transforms to a continuum of frequency components; there are no discrete 'harmonics'

P.S. the first sentence of this post is the crucial one.