hi guys , I need help in understanding the the definition of orthogonality ,

can I say that if I have two waveforms not in the same frequency , is it true to say that they are orthogonal to each other ? or maybe there's another condition.

 

The geometric interpretation of orthogonal means perpendicular, or at right angles. There are several mathematical tests for orthogonality, but the easiest is to take the dot product of two vectors and if it is zero then the vectors are orthogonal. Wait a minute you say what do vectors have to do with waves? Glad you asked that, but that is left as an exercise for the reader.

 

The geometric interpretation of orthogonal means perpendicular, or at right angles. There are several mathematical tests for orthogonality, but the easiest is to take the dot product of two vectors and if it is zero then the vectors are orthogonal. Wait a minute you say what do vectors have to do with waves? Glad you asked that, but that is left as an exercise for the reader.

thanks for your answer !

i'm not sure that I know what is dot product...
dot product is the multiplication of two vectors at a specific time period ? or am I wrong ?

 

The geometric interpretation of orthogonal means perpendicular, or at right angles. There are several mathematical tests for orthogonality, but the easiest is to take the dot product of two vectors and if it is zero then the vectors are orthogonal. Wait a minute you say what do vectors have to do with waves? Glad you asked that, but that is left as an exercise for the reader.

The OP asked for "not in the same frequency". Does the above applies to it? That would mean an occurrence every so many (miliseconds or whatever), right?

 

Orthogonal has a wider meaning that at right angles, although it include that.

Papabravo's maths is pretty good, but having mentioned the dot product of vectors, he didn't elaborate because you need to know quite a bit of maths to fully understand the statement.

Mathematically 'vectors' includes considerably more than just lines with arrowheads on them.
It includes the definite integral and any two functions, f(x) and g(x), are orthogonal if the definite integral of their product is zero over a particular range.

So if f(t) = Asin(wt+p) and g(t)=Bsin(vt+q) they are othogonal if their product over a certain range t=r to t=s is zero

Where p and q are phase angles, t is time, w and v iare the angular frequencies and A and B are amplitude constants.


\(\int\limits_r^s {f(t)g(t)dt = } \int\limits_r^s {A\sin (wt + p)B\sin (vt + q)dt = } 0\)

Examples of orthogonal functions are

Fourier polynomials (series)
Chebychev polynomials
Legendre polynomials

 

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

In simple terms the dot product of two vectors, representing waves, is a number equal to the product of the magnitudes times the cosine of the angle between them. If the dot product is zero and the magnitudes are non-zero, then the only thing left is that the cosine of the angle between them must be zero. Thus they are orthogonal. As has been pointed out the subject of orthogonality is much richer than the OP imagined and I was hoping that he would discover some of it on his own.

 

The OP did ask about waves of different frequencies.

 

Thanks guys !
That discussion helped me a lot !

 

If you want more detail post a new question in the maths section.

You cannot use rotating vector arms to deal with different frequencies.

 

If the waves are not coupled by some mechanism, say Maxwell's Equations, how can they remain orthogonal if they are independent of each other. I'm looking for an example here of waves that have different frequencies and yet are orthogonal.

 

If the waves are not coupled by some mechanism, say Maxwell's Equations, how can they remain orthogonal if they are independent of each other. I'm looking for an example here of waves that have different frequencies and yet are orthogonal.

An example will be great in oreder to understand better.

Thanks Papabravo !

 

Hello yuvi1,
When most of the people here read “Orthogonality of waves definition”, we tend to think of the orientation of the electric and the magnetic fields and the direction of travel of a radio wave.
But in your written post you ask about the orthogonality between two separate waves. This is a much different question. It would help us if we knew your age, your country and some kind of idea of your knowledge background.

The orthogonality of a radio wave is the relation between the electric and magnetic fields and the direction of travel.....they are all perpendicular to each other.

ie:

See how the red and blue waves travel sideways to the direction of travel? That’s orthogonality of a radio wave.

But the concept of orthogonality in electronics is much deeper than that. In our daily lives, most of our interactions are linear. For instance, when you push a book across the desktop, the book moves in the direction that you pushed it. The book moves in the direction of force. This makes sense too. This is how most of the world works right? This is a direct action and a direct response. It’s all inline....it’s linear action. All the force and all the movement is on the one straight line.

There is another interaction that happens in nature that is not linear, it is orthogonal action. This is a sideways force or action. Now I do not mean a linear system being hit sideways, ie....a car being hit in the side by another car. I mean that the direction of force and action turn sideways.

That means that when you push on the book, instead of the book moving straight ahead......it will move to the right hand direction. The harder you push straight on the book....the more the book moves to the right. It is very mysterious and does not make sense when you see it or experience it. To see this force first hand....play with a gyroscope. If you want to START to understand this force.....read and study gyroscopes.

But back to this sideways force for a minute. When you pushed on the book......it did not turn left.....it turned right. It only went one way. Half did not go left and half went right.....the whole force and the whole book went right. It’s a one direction force. Now pull the book towards you.....as you pull......the book will move to the left hand side. Keep pulling on the book until it is centered back in front of you. Now push on book to the right...as you do, the book will come towards you. Strange isn’t it. Now push the book to the left......and the book will move back up away from you. This is orthogonal action. This is how electrical and magnetic force is related. This is the way electricity and radio waves work. This is also the way the molecules in your body work. And in a fundamental way....this is how the universe works.

Knowing how this force works and being able to anticipate it is good enough for the study of electronics. If you want to know the process and mechanism that produces this force, you will need to study spin and physics. Even though it is a simple process.........it is a complicated concept. Electrical spin is different from mechanical spin.

I hope this helps and doesn't confuse you any more than you are. Good luck with your studies.

 

When most of the people here read “Orthogonality of waves definition”, we tend to think of the orientation of the electric and the magnetic fields and the direction of travel of a radio wave.
But in your written post you ask about the orthogonality between two separate waves.

And not of the same frequency, keep in mind. And nothing else.

 

That's a nifty animation, BR-549.

I do suggest, however you consider rewording your discussion on pushing books to include the point of application of said push.

 

studiot, what does "point of application" mean?

 

Just that what happens depends upon where you push it.
It will definitely rotate if your push is applied towards the edges, but it will not rotate if your push is applied at the centre.

 

studiot.....the book is not spinning.......please go to physics forum.

 

You just need to be clearer in what you are describing.