How Waves Overlap, and Why Common Sense Works! Principle of Superposition for Linear Equations
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- เผยแพร่เมื่อ 3 ต.ค. 2024
- Why do wave amplitudes add when they overlap? More specifically, why does it make sense to add the displacement of each wave at every point in space and time in order to find the resultant wave? We know that waves of the same type will interfere with each other in this way.
In this video, we will look at the Principle of Superposition. It explains why when two waves overlap, we can simply add their displacements at each point to find the resultant wave. This is what we would find reasonable due to our common sense. But common sense isn't always correct - so why is it accurate here?
To understand this, we need to realize that the wave equation (the classical governing equation for describing all sorts of waves) is linear in wave displacement. In other words, the displacement of a wave (u) only appears as a single factor of u everywhere in the equation. No other powers of u, no functions of u such as ln(u). This linearity ensures that if we take any two known solutions of the wave equation, then adding them together produces yet another solution to the wave equation - in this case the resultant wave. However, if the wave equation was nonlinear (i.e. had powers of u other than 1), then this would not be the case. The sum of two existing solutions would NOT be a solution in itself.
Luckily, the universe seems to behave linearly very often, and any linear system can use the principle of superposition to find solutions that are formed by summing other existing solutions. As a result, wave interference becomes an easy-to-understand topic, but this has a much deeper reason than common sense!
The Principle of Superposition actually has a lot more mathematical detail and rigor behind it. Linear systems that follow the Principle of Superposition can be defined in terms of two properties: the linear functions must be additive and homogeneous. In other words, the function of two variables added together must be equal to the sum of the functions of the two individual variables (additivity), and a constant multiplied by the function of a variable must be equal to the function of the constant multiplied by the variable.
For more information on the Superposition Principle, as well as an idea of other systems where this applies, check out the following link: en.wikipedia.o...
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Common sense is not so common
- someone said
Parth I am very glad I subscribed 👍. And Yes I Have a question now,
curious about why don't the U-wave displacements just precisely "cancel" ? Guess its the same question of when noise cancellation headphones cancel out sound by rotating wave 180 Degrees the sound must become heat by conservation of energy (COE) .. but why not for photons at QM level ?
Are we missing something in terms of alignment ?
@@marccox8977 Using common terms, we call the high points crests, and the low points troughs. When a crest and trough overlap, they cancel. When a crest or trough overlaps with another crest or trough, the result is a larger crest or trough.
I’m a sound design guy, adding waves, using FFTs for synthesis and modulating frequency is what I understand.
Yes I Have a question now I'd like you to consider pls. Basically I'm
curious about why don't the U-wave displacements just precisely "cancel" ?
Guess it's the same question of when noise cancellation headphones cancel out sound by rotating wave 180 Degrees the sound must become heat by conservation of energy (COE) ..
? For Parth but why not for photons/quanta at QM level ?
Or, Are we missing something in terms of alignment ?
Is adding waves common sense? It surprised me the number of medical professionals and signal processing specialists who hadn't realized that a photoplethysmogram (PPG) signal (used in pulse oximeters, the thing they clip on your finger in hospitals) with its characteristic double hump was mostly just a sine wave at the current heart rate and its harmonics at progessively lower amplitudes added togther. Or that it's basically a filtered ECG/EKG signal, very nearly related by a double integral. Many didn't believe it after I told them until they did the math. Your ciculatory system acts quite a bit like a haemodynamic equivalent of a classic RLC filter.
I didn't know the thing they clip on your fingers is called a PPG either..it measures your heart rate and blood pressure right? What do you mean by the harmonics of your heart rate? Like additional subfrequencies of the main frequency of your heart rate?
Wait, but isn't the filtering happening at the reading by the PPG?
I was just going to say the exact same thing!
Honest.
@@danteregianifreitas6461 While it is true that the devices do quite a bit of post-procesing for heart rate and oximetry, the PPG signal itself is just a measure of the intensity of light on a photodiode. The source of the light is one or more LEDs (different wavelengths, only one on at a time) and it is either passed through a body part or reflected depending on the device design. The signal varies with localized pressure as more red blood cells get crammed into the area while your heart is actively discharging. It's related to blood volume and vascular cross-sectional area as well as charge in electric and hydraulic systems. The difference between the raw EKG and the PPG waveforms is mostly due to arterial hardness acting as a capacitor, vasodilation as a resistance and the inertia of the roughly 5kg of blood in your body adds an inductive term. There's also a phase delay due to the speed of sound in blood. Your bloodpressure is the voltage in this analogy. If you aren't familiar with the general relations, wikipedia has a decent article on the correspondence between electric, hydraulic, mechanical and thermal dynamics (including a handy table of all the equations and their correspondences) if you search for "hydraulic analogy".
@@protocol6 oh, I see what you mean now. You're saying the vascular system is analogous to an electric circuit and the heart would be a voltage source, right? And what is desired is the fundamental frequency of the heartbeat. So there would be a filter from the vascular system's passband and from the sensor's passband.
Thank you so much for that, finally understood why my book says triumph of linearity after writing the third solution to 2nd order linear homogenous differential equation by adding the previous two solutions.
Next video on convolution, why convolution is used almost everywhere wherever waves are involved.
I would look it the other way around. Why does waves appear everywhere there is convolution?
And to answer that, it's that exponential functions (and, therefore, waves) are the fundamental solutions to linear time-invariant ODE's, which can be modeled by the convolution of the input with the impulse response.
Because convolution and impulse like delta Dirac, gives you a wave for things like reverb in audio, you record clap or something short transienty and it gets the acoustics of the room.
Honestly thank you so much for the explanation but my dumb brain didnt understand even 1% of this explanation sorry for the disappointment 😅😅
It would be very cool to see the difference between half spin and integer spin particles as it pertains to superposition.
I think some of the confusion is that there is a velocity wave, not just a displacement wave. The difference between the case where the L and R waves exactly cancel and that of just a stationary string is that the former has a velocity wave that stores the energy and yields the L and R displacement waves as time evolves.
Yes, the energy oscillates between kinetic and potential.
Best explanation I have heard 🎉 thank you 🙏
PARTH CAN WE GET A VIDEO ON QUANTUM FIELD THEORY!!!!
Really nice explanation TY & the colours are nice
Plzz make a video on yang mills mass gap problem
Thanks a lot for your video. Happy Thanksgiving!
Hi Parth,
Many thx for these videos.
Could you make a video on why wave functions must be complex.
So far, bo body has got it right.
Many thx
This video is good.
May be I haven't studied enough but everything made sense untill 2:20
Thank you so much
Thanks for your explanation..sir please tell me the unit of j in quantum mechanics and how is it calculated from equation of continuity???
Edit: I watched the previous vid and now the c^2 term makes sense. Thanks!!
Original comment: I gotta go watch the previous vid: I have no idea where the c^2 term comes from... I assume it has to do with the speed of propagation? Or the energy of the wave? But how would c^2 have a bearing on a wave in a string or other medium which isn't light in a vacuum?
Sir!, Then what's the difference between interference and superposition?
Two waves of same amplitude traveling in opposite directions superpose. If the amplitude of the resultant wave is zero where does all the energy go? Please explain.
@Parth G Yes I Have a question now I'd like you to consider in a video pls. Basically I'm
curious about why don't the U-wave displacements just precisely "cancel" ?
Guess it's the same question of when noise cancellation headphones cancel out sound by rotating wave 180 Degrees the sound must become heat by conservation of energy (COE) .. but why not for photons/quanta at QM level ?
-Or- Are we missing something in terms of alignment ?
-Or- is the energy involved in the cancellation so slow particle formation isn't possible ? If so, What does
that imply about "virtual" / "vacuum" particles - is the vacuum actually a lot quieter than some propose ?
The energy of a wave is proportional to the square of u. In other words, if u doubles in constructive superposition, the energy quadruples, and if u cancels out in destructive superposition, the energy disappears. When we consider the wave as a whole, the constructive and destructive interferences would balance each other out and the energy would double. However, this seems to be a stochastic phenomenon and seems insufficient as a basis for the conservation of energy. I would like to know the mathematical principle that guarantees the correct doubling of energy no matter how the waves are superposed.
The energy is proportional to the amplitude rather than u. The energy is carried by the full wavelength of the wave. While the displacement of individual segments is affected by the superposition, the amplitudes of the waves involved is not. Ultimately, conservation of energy is implied by one of the symmetries of our universe (as all conservation laws are). In particular, conservation of energy is implied by time symmetry. If you're interested in knowing more, look up Noether's theorem.
hi, how do u explain destructive interference wrt conservation of energy? where that energy will go?
Hi Parth :) I write short stories -mostly as a hobby - and I had an idea earlier that I wondered if it'd be possible to run by you, please? Kinda relates to quantum physics and I don't really know who to ask to see if it's absolutely the most outlandish thing ever or has just enough of an element of (truth? fact? likelihood? I don't know haha) something to be able to give it a go.
No worries if it's not your type of thing :)
I love you. I didn't really understand wave equations last semester and this has helped a lot. Although at 3:20, shouldn't the 2nd partial derivative with respect to X have -k^2 in front of it not -w^2?
Yeah i don't think the function he showed is a solution
what does G stand for in PARTH G
I have a question that originates from around 2:24. Why are the terms in parentheses not (kx+wt) and (kx-wt), where 'w' represents omega, here? The way it is currently written seems to make the x-variable the one that causes the "motion". I appreciate any help in this matter. Thank you.
I'm not certain, but I think it may just be a matter of convention. We typically work with complex exponentials because they're mathematically convenient. At the end of the calculations, if we want a physical result, we take the real part. This is the same in either case because cos(-x)=cos(x). It also just feels more natural to have them symmetric in time and antisymmetric in space since they flow the same way through time and in opposite directions through space.
Amazing
What are the frame conditions for allowed non linear solutions?
Hello! Greetings from Mexico, would you consider making a video about how you make your videos? It would be lovely, I would like to spread science for Spanish-speakers and I'm finding my way haha.
(I commented the same in another of your videos but I swear I'm not going to spam lol)
Why we are always taking in opposite direction not in same direction
We must bear in mind that in many circumstances using the linear wave equation is an APPROXIMATION to the physics that only works if the amplitude of the wave is very small. I feel that physics education could include more content on nonlinear systems.
or better yet, how to get to the linearized form by taking the limit of a parameter, from an intractable nonlinear problem
@@geoffrygifari3377 in my undergraduate class in fluid mechanics (which few physics students learn about) we did exactly that, starting with the equations for fluid motion, and using perturbation theory to derive the acoustic wave equation.
@@DavidMFChapman yeah i guess physics students are directed to learn those which are most crucial to contemporary physics research. in depth fluid/continuum mechanics and thermo are usually for engineers.... there's just too much physics out there
@@geoffrygifari3377 there is, yet I feel much could be learned if potential theorists cut their teeth on more tangible phenomena.
While you are correct about the former point, dealing with non-linear systems in a systematic way is mathematically extremely demanding. It would overtax all but the theoretically most gifted students in most universities. I once decided to get a book about perturbation theory for classical mechanics from the library. I took it back after skimming through the first chapter. Reading it was like wading through molasses. Every single step seemed to take an eternity to understand and it was obvious that there were no simple techniques in there that one could learn just for the fun of it.
Had fun
Can you recommend a book that deeply explains the modern physics mathematics ?
What do you mean by that? What mathematics do you need to understand modern physics? That depends. How deep do you want to go? Classical mechanics and non-relativistic quantum mechanics can be understood with linear algebra and multivariate calculus (including algebra of complex numbers and complex calculus). For general relativity you will have to learn tensor algebra and a modicum of differential geometry (but not the level of differential geometry that mathematicians are concerned with). If you want to learn quantum field theory I would suggest you take a few lessons on lie algebras and non-commutative algebra in addition to developing very solid skills with complex integrals. If you want to be at the bleeding edge of any of these fields, then you better read deep into the fields of differential geometry and algebraic geometry because all of physics seems to pivot around them in one way or another. This is not a one book exercise. It's dozens, if not hundreds of books that you will have to "read into" in the university library for partial answers and insights. That's OK, to go from beginner's classical mechanics to bleeding edge relativity or QFT will take you five to eight years full time, anyway, assuming that you are gifted in math and theoretical physics. If you are not, like me, then you will never get there. In that case you can still do very useful experimental physics with little more than high school algebra and calculus.
This is what is occurring with superposition. It is actually wave collisions under controlled situations.
@
Respected parth,🌟 will everyday objects such as a ball When passed through a double slit experiment in space where there is no light and air (NO INTERACTION) so no measurement...will ball produce an inference pattern on the screen?
Thank you sir 🌟
Yes, even macroscopic objects will behave quantum mechanically, but you won't be able to measure it with a simple "screen" technique. There are better techniques for that, though, and experimental physicists have been pushing the mass limit for which this can still be verified quite a bit.
Easy answer asin(bx+c)+asin(bx+c)=2asin(bx+c)
How do wave functions interact? Are they similar to the classical waves?
The mathematics is identical.
The important thing to realize is that in linear wave equations (whether classical or in quantum mechanics) partial solutions do _not_ interact. They go "through each other" without modifying each other. Linear superposition is therefor the absence of interaction. Once we add nonlinearity (even small amounts of it), the mathematical treatment of these equations becomes very demanding, but interesting things like shocks and solitons appear, which we do see in nature (in acoustics and gravity (water surface) waves, for instance).
What do you think of Veritassium’s latest video about power in an electrical circuit being due to the field and not moving electrons (current)? Can you debunk him?
It's done in the comments on that video.
There is nothing to debunk. It's roughly third semester physics knowledge that energy transport is in the field and not in the electrons in conductors. Having said that, the video was a total bust and it seems that he made a version 2.0 where he tries to explain it better.
❤️❤️
The demonstration is circular.
🙏💐💐💐
Ooooo waves
Superposition => Conservation of energy
That's not true. One can have non-linear energy conserving systems. They just don't have a nice solution theory. Almost all of them happen to be chaotic. See non-integrable Hamiltonian system.
o
Parth..... love the explanation. the problem is that none of partial differential equations are valid at real rates of power. Which is what you pointed out about Ohms law......I have the solution so call me
?
I have the real solution you should 📞 me instead
Wow
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