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Borkenhagen Clemens
Borkenhagen Clemens
16:05 Oct-02-2019
Physical meaning of wave dispersion

Hello everyone,

My name is Clemens Borkenhagen and I am at the beginning of my PhD. Within this thesis I am working with ultrasonic guided waves. The field of wave propagation is quiete new for me.

At the moment I am struggling with the phenomena "dispersion". There are many disciplines, in which this word occurs (e.g. Chemistry, Optics, Phonons and Photons, etc.). This makes it quiet hard for me to understand the physical meaning of dispersion in case of ultrasonic guided waves and where it comes from.

My special questions are:
What is the physical background (reason) for wave dispersion? Is it the medium, that interacts with the propagating wave or do I have to understand it from an atomic level?

I have already started to read the textbook of Rose (Ultrasonic Guided Waves in Solid Media, 1999). I have already found other textbooks, in which the effect of dispersion is explained only in a mathematical sense (central message: dispersion occurs, because several boundary conditions have to be fulfiled). For me it is hard to imagine, how the medium and the wave interact physically, so that the speed of propagation depends on the frequency or wavelength. In my opinion, there has to be a physical reason for this dependence, which I would like to figure out. I have not found a suitable textbook or reference until yet, that explain to me the fundamental physics, that are behind of wave dispersion.

I hope you can help me with this problem. I am looking forward to hearing/reading from you soon.

Best regards and thanks

Clemens Borkenhagen

 
 Reply 
 
Wieslaw Bicz
Wieslaw Bicz
22:54 Oct-02-2019
Re: Physical meaning of wave dispersion
In Reply to Borkenhagen Clemens at 16:05 Oct-02-2019 (Opening).

In the case of sound waves the dispersion in unlimited medium existing only in the case of frequencies significantly higher than 1000000000 Hz, when the wave is interacting with the material structure. Dispersion can be very strong in materials with borders, especially if the dimensions of the object, where the wave is propagating are comparable with the wave length. The reason for it is interference of waves reflected at the boundaries with the main beam and the creation of different wave modes and delays (see: Schoch or Goos Haenchen effect) or due to the interaction with the boundary. Such waves are called "guided waves" since they cannot propagate freely, are limited (guided) by the boundaries. In this case dispersion is the effect of interaction of wave front with boundaries and interference effects. In some types of layers special types of waves can exist, that are also showing dispersion (see: Love waves).
I hope, this helps.

 
 Reply 
 
Alex Haig
Alex Haig
23:21 Oct-02-2019
Re: Physical meaning of wave dispersion
In Reply to Borkenhagen Clemens at 16:05 Oct-02-2019 (Opening).

I’ll have a go at a non-mathematical explanation. Before I start, I’d like to invite criticisms and corrections for what I write. This is how I understand it…

I’ll start very simple – sorry if this is patronising! When you observe a ripple travelling on a pond, you are seeing energy being carried across the surface. The particles of the water do not travel with the wave - they just oscillate in place and pass the disturbance (i.e. the energy) on to their neighbours. You can track a crest of the wave as it travels, which means you can see for yourself the speed of the ripple. So, firstly, sound waves carry energy and, secondly, you can observe the speed of their crests.
When we talk about the speed of the crests we generally use the term ‘phase velocity’. This is because a crest is just an identifiable phase angle in a cycle and what we’re observing is how fast that phase angle is moving (in other words, it’s the velocity that we need to track across the surface for the phase to be unchanging).

For many wave phenomena the energy being delivered is travelling at the same speed as the crests, which is intuitive to understand, but that’s not always the case. It’s actually only the case if the sound has the same phase velocity for all frequencies within its bandwidth. Unfortunately, guided waves tend to have frequency dependent phase velocities.

A quick side note… Understanding bandwidth is a crucial first step in understanding dispersion (and guided waves in general). If you haven’t learned about Fourier Series or discrete Fourier Transforms before, I recommend going over that topic first because a lot becomes clearer after that. An introductory course on digital signal processing is a good way to do that – the course does not have to have anything to do with guided waves or NDT. I recommend this one, particularly if you know a little about coding with Python already: https://www.coursera.org/learn/audio-signal-processing. As a side note, I highly recommend learning Python too! Another course I recommend is MIT’s “Physics III: Vibrations and Waves”: https://ocw.mit.edu/courses/physics/8-03sc-physics-iii-vibrations-and-waves-fall-2016/.

Back to the explanation… So, let’s say we have a wave travelling with 2 frequencies in its bandwidth (a simplistic version of what really happens where the bandwidth is a continuum of frequencies). Let’s also say they do not travel at the same speed. What occurs next is the two frequencies superpose – over time and over distance they interfere constructively and destructively. If you were to observe this wave, it’s now harder to track a crest moving along than the single frequency version, but you would be able to see the zones of high amplitude (constructive interference) separated by zones of low amplitude (destructive interference) moving along. Since the two frequencies have different phase velocities, you might also notice that the zones are not travelling at the same speed as the individual crests. This is because the energy is not being delivered at the same velocity as the phase velocity. This is known as the group velocity.

Now let’s consider something more realistic. Say you transmit a brief pulse of sound - something that looks, for example, like 8 cycles where the amplitude smoothly builds up and then down over the 8 cycles. That pulse has a bandwidth of frequencies and let’s say the phase velocity varies over that bandwidth. If you observe the wave travel over time, the faster frequencies will travel further than the slower ones. Since the bandwidth is a continuum, what you get is a smooth distortion of the pulse as it travels. The energy gets spread out. This is dispersion. The more the velocity varies in the bandwidth or the further the wave travels, the more distortion (i.e. dispersion) you get.
In non-destructive testing, we usually don’t like our signals distorting as they travel, so in guided waves (where dispersion is common) we put a lot of effort into reducing, managing or correcting for dispersion.

I hope this help!

Alex

2
 
 Reply 
 
Alex Haig
Alex Haig
23:29 Oct-02-2019
Re: Physical meaning of wave dispersion
In Reply to Alex Haig at 23:21 Oct-02-2019 .

I read the comment from Wieslaw and realised I may not have answered your real concern. I’m glad he has. Yes indeed, the frequency dependent velocity comes from the interaction of the travelling sound within the waveguide’s boundaries. If you consider the dimensions of the waveguide relative to wavelength, then it’s different for every frequency. This means not only does the phase velocity vary with frequency, but the resultant vibrational shape of the wave mode varies with frequency too!

 
 Reply 
 
Clemens Borkenhagen
Clemens Borkenhagen
11:11 Oct-04-2019
Re: Physical meaning of wave dispersion
In Reply to Alex Haig at 23:29 Oct-02-2019 .

Hello,

Thank you for your instant and extensive replies. The phenomenon "dispersion" is now a bit clearer to understand for me. However, there are some questions left.

Is there a simple way to understand or imagine the physical interaction of guided waves with the boundaries of the wave guide? What exactly happens there? Is dispersion the effect due to the interaction of the travelling wave with reflections (that leads in a next step to interference)?

Is there a way to understand dispersion considering what happens to each atom of the wave guide when a pulse of disturbance passes trough? A long time, I understand dispersion in that way, that different frequencies of an ultrasound pulse lead to a more or less stimulation of the atomic (i.e. resonance effects), that determines the speed with which energy is transferred. Is this right?

Another question is, does dispersive effects only come from the interaction of the traveling sound wave with boundaries? I have read a puplication (Wolfgang Sachse: "On the determination of phase and group velocities of dispersive waves in solids"), in which several causes of dispersions are listed:
1. Dispersion because of the presence of specimen boundaries (geometric dispersion)
2. Dispersion because of the frequency dependence of material constants, such as mass density, elastic moduli, etc. (material dispersion)
3. Dispersion because of the scattering of waves by densely distributed fine inhomogenities in a material (scattering dispersion)
4. Dispersion because of the absorption or dissipation of wave energy into heat or other forms of energy (dissipative dispersion)
5. Dispersion because of the dependence of the wave speed on the wave amplitude (nonlinear dispersion)

It seems to me, that there is no standardized interpretation of dispersion. Or there are
many effects associated with dispersion. That is the reason, why I am so confused and why i am struggling to understand the physical meaning of dispersion.

I know there are a lot of questions, but it would be grateful if you can reply or even recommend suitable literature to me.

Best regards and thanks a lot

Clemens Borkenhagen

 
 Reply 
 
John Norman
Consultant, owner of business
NTS Ultrasonics Pty Ltd, Australia, Joined Oct 2012, 114

John Norman

Consultant, owner of business
NTS Ultrasonics Pty Ltd,
Australia,
Joined Oct 2012
114
10:17 Oct-07-2019
Re: Physical meaning of wave dispersion
In Reply to Clemens Borkenhagen at 11:11 Oct-04-2019 .

Hi Clemens.

There is nothing like "jumping off the deep end". Fortunately, you are going to work with guided waves, otherwise things would get really complicated when trying to understand dispersion..

For ultrasonic guided waves, your concern is geometric dispersion. In general, dispersion occurs when the ultrasonic velocity in material changes with frequency. In a guided wave situation, the guided waves are formed from the superposition of reflected versions of the initial wave from the upper and lower surfaces. The resulting wavefront which forms the guided wave progresses at a velocity determined by the geometry, mainly the material thickness, material velocity, and incident angles. If we were in the same room, I could wave my hands and draw sketches to show you how I visualize what bis happening.

The other types of dispersion you mention fall into the realm of solid state physics, in the general area of phonon (lattice vibrations) dispersion. Many texts on solid state physics will give you mathematic formula and graphs of dispersion curves, and lots of information, and if you are like me, you will not be any the wiser. I think it comes down to inter-atomic forces, which are non-linear with distance, i.e. like non-linear springs. Because of this non-linearity, lattice vibrations (phonons or ultrasonic pulses) travel at slightly different speeds depending on frequency (and also amplitude). I think that is the idea, a real Physicist could explain this better than I can.

John Norman

1
 
 Reply 
 
Fred
Sales,
Eddyfi, France, Joined Aug 2016, 12

Fred

Sales,
Eddyfi,
France,
Joined Aug 2016
12
11:20 Oct-07-2019
Re: Physical meaning of wave dispersion
In Reply to Borkenhagen Clemens at 16:05 Oct-02-2019 (Opening).

Clemens,

The nature of the dispersion for guided waves comes from the fact that they are a superimposition of bulk wave (longitudinal and transverse) reflexions and mode converted at each surface of the guide. The result is a sum of all these contributions including amplitude and phase information. These amplitude and phase are strongly dependent on frequency and introductory wave angles of propagation inside the guide. Where you have strong superimposed results, you obtain a point on the phase velocity dispersion curve for the structure. Elsewhere there is strong cancellation. So each frequency in the bandwidth of your transducer is going to travel at a different velocity and your pulse duration is going to change

 
 Reply 
 
Clemens Borkenhagen
Clemens Borkenhagen
10:57 Oct-14-2019
Re: Physical meaning of wave dispersion
In Reply to Fred at 11:20 Oct-07-2019 .

Thanks for the detailed replies.

Based on your explanations, I try to reply some aspects in my own words:

1. The type of dispersion (material, geometric, etc.) depends on the wavelength. There is another type of dispersion, when the wavelength is comparable with atomic scales than when the wavelength is comparable with a whole wave guide (rail, pipeline, etc.).

2. In case of guided wave testing of rails, the wavelenght is in scale of my wave guide. Therefore dispersion in my case is based on interference phenomenon when incoming waves and reflected waves interact with each other.

3. The interaction of bulk waves (longitudinal and transversal waves) only occurs because the ultrasound pulse is initiated under a specific angle. If the ultrasound pulse is initiated in direction of the wave guide (=0°), the result is a 100% longitudinal wave without dispersion. And if the ultrasound pulse is initiated perpendicular to the surface (=90°), the result is a 100% transversal wave without dispersion.

If there are any mistakes, please correct me. Thanks a lot.

 
 Reply 
 
J.B.
J.B.
20:09 Oct-14-2019
Re: Physical meaning of wave dispersion
In Reply to Clemens Borkenhagen at 10:57 Oct-14-2019 .

While i was reading all the comments and advicex I just thougth. I would be completely confused.
I will not add more of that.

For me in the past - o.k. I am a little aged - Krautkamer's book "Ultrasonic Testing of Materials" was always a good source of information to understand Ultrasound. I guess you are from Germany, there is even a Version in German (https://www.springer.com/de/book/9783662109106 - which is the original, the English version is only a translation). This book is addressed to NDT inspectors, not so much for scientists, so explanations are more adequat for NDT staff.

My second idea was to draw attention to people who do simulation. Most popular code is currently CIVA - it is commercially available and well accepted, but a code addressing more the physics of waves is AFIT/EFIT once initiated by Prof. Langenberg from GHKassel. Currently I guess the brain behind this code in Rene Marklein (https://de.linkedin.com/in/ren%C3%A9-marklein-80477124), who did a lot of publications and lot of the coding work was directed by him.

Just searching if there is anything in the net for public use I came across the following GITHUB page (https://github.com/mmolero/SimNDT) called SimNDT. Author Miguel Moreno who now works for Innerspec as I saw in Researchgate (https://www.researchgate.net/profile/Miguel_Molero) where you can find more. He has also another repository at Github concerning EFIT: https://github.com/mmolero/efit2d-pyopencl, you find a paper parallel to this on Researchgate - https://www.researchgate.net/publication/280495627_GPU_computing_with_OpenCL_to_model_2D_elastic_wave_propagation_Exploring_memory_usage. And finally M.Molero did his Thesis about dispersion: "Caracterización de materiales cementicios mediante la dispersión ultrasónica" (http://digital.csic.es/handle/10261/18629 sorry - in Spanish)
about Molero see also https://sites.google.com/site/miguelmolero/Software & https://es.linkedin.com/in/moleromiguel

Could be a way for you to get deeper into the topic just with the help of accessable simulation software.

 
 Reply 
 
Vadim
R & D, Director
Wave Process Simulation System Laboratory, Russia, Joined Jun 2010, 4

Vadim

R & D, Director
Wave Process Simulation System Laboratory,
Russia,
Joined Jun 2010
4
15:50 Oct-24-2019
Re: Physical meaning of wave dispersion
In Reply to Clemens Borkenhagen at 10:57 Oct-14-2019 .

zoom image
Dear Clemens,

An interesting nonlinear dispersion occurs when a wave moves through a medium with non-analytical nonlinearities (microcracks, debondings, delaminations and microstructural material damages).
Here, with different velocities pass compression and tensile wavefronts. The speed depends on whether the defect is open or closed.
In a first approximation, we can talk about an "acoustic diode", which transmits positive and negative pulses with different speeds.

This is well illustrated by figures 6 and 7 in the article https://www.researchgate.net/publication/232708284

All these data were obtained thanks to CAE IMPULSE, which was used not only in Russia,
but also in Germany (Fraunhofer Institute for Nondestructive Testing IZFP) and Finland (VTT Technical Research Centre).
https://www.researchgate.net/publication/266617723

Best regards,
Vadim
 
 Reply 
 
Clemens Borkenhagen
Engineering, Doctoral student
University of Applied Sciences Augsburg, Germany, Joined Jul 2019, 2

Clemens Borkenhagen

Engineering, Doctoral student
University of Applied Sciences Augsburg,
Germany,
Joined Jul 2019
2
12:02 Nov-18-2019
Re: Physical meaning of wave dispersion
In Reply to Vadim at 15:50 Oct-24-2019 .

Hello everyone again,

while I am reading trough all the detailed replies again, I would be very glad, if there are some literature references due to the physical meaning of dispersion. My understanding of the dispersion is getting better and better, but there are still some gaps. It is hard for me to understand the effect of dispersion, because there are so much different cases, where dispersion occurs or come from (see my text above with the different types of dispersion). Is it right, that the type of dispersion depends on my physical application? So there is another reason for dispersion in case of guided wave than for water waves for example?

Best regards,

Clemens

 
 Reply 
 
J.B.
J.B.
17:16 Nov-18-2019
Re: Physical meaning of wave dispersion
In Reply to Clemens Borkenhagen at 12:02 Nov-18-2019 .

Maybe you will find here more:
Have you checked the following papers?

H. Lamb: "On Waves in an Elastic Plate"
Proceedings of the Royal Society of London. Series A, vol. 93, S. 114–128, 1917
https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1917.0008

D. C. Worlton: Experimental Confirmation of Lamb Waves at Megacycle Frequencies.
Journal of Applied Physics 32, 967 (1961); https://doi.org/10.1063/1.1736196

J. L. Rose, Ultrasonic waves in solid media, Cambridge University Press, 1999

Identification of Damage Using Lamb Waves: From Fundamentals to Applications
Zhongqing Su, Lin Ye
Springer Science & Business Media, 01.09.2009 - 346 pages

1
 
 Reply 
 
Wieslaw Bicz
Engineering,
PBP Optel sp. z o.o., Poland, Joined Feb 2009, 262

Wieslaw Bicz

Engineering,
PBP Optel sp. z o.o.,
Poland,
Joined Feb 2009
262
16:55 Nov-25-2019
Re: Physical meaning of wave dispersion
In Reply to Clemens Borkenhagen at 12:02 Nov-18-2019 .

You must be aware, that the so called dispersion of sound waves, that occurs in plates is fully different effect, than the dispersion, that is caused by reaction of wave with matter (as for example is the case in light waves in glass). If you send a short pulse and analyse the received signal including all paths you will not observe, that the sound speed (phase velocity) depends on frequency. This dependance can be observed only in the case, if you generate longer wave packets and observe group velocity. You will also observe a kind of filtering functions of such plate if you do it with different frequencies. I would tell, that it is a bit misleading to name this kind of occurence dispersion, as your doubts are showing. But it is a habit, that is well established.

1
 
 Reply 
 
John Norman
Consultant, owner of business
NTS Ultrasonics Pty Ltd, Australia, Joined Oct 2012, 114

John Norman

Consultant, owner of business
NTS Ultrasonics Pty Ltd,
Australia,
Joined Oct 2012
114
07:47 Nov-26-2019
Re: Physical meaning of wave dispersion
In Reply to Clemens Borkenhagen at 12:02 Nov-18-2019 .

Hi again, Clemens.
You are getting some terrific answers to your enquiry. One reference I often go to for matters related to ultrasonic theory is "Ultrasonic Measurements for Process Control" written by Lawrence C. Lynnworth (Academic Press 1989). In chapter 3 there is a good discussion of the different types of dispersion - material based and geometry based. You might find a copy of this book in a library.

In an earlier post you mentioned in your point 3 that a longitudinal wave introduced at the end of a waveguide sends a longitudinal wave down the rod. Not so straight forward. The behaviour of a travelling wave depends on the waveguide dimensions relative to the wavelength. You can still get partition of the introduced longitudinal wave into shear and longitudinal components and the subsequent interferences which lead to dispersive guided waves.

Regards
John Norman

1
 
 Reply 
 
Clemens Borkenhagen
Engineering, Doctoral student
University of Applied Sciences Augsburg, Germany, Joined Jul 2019, 2

Clemens Borkenhagen

Engineering, Doctoral student
University of Applied Sciences Augsburg,
Germany,
Joined Jul 2019
2
15:44 Dec-03-2019
Re: Physical meaning of wave dispersion
In Reply to John Norman at 07:47 Nov-26-2019 .

Thank you for your replies. I will read trough the references, that are given to me. Hopefully I will understand dispersion even better.

Best regards

Clemens

 
 Reply 
 

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