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Derivation Of Wave Motion With Complex Calculation

mechanics harmonic oscillator

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#1 Nishan

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Posted 28 June 2019 - 10:08 AM

Can anyone help me visualize the solution of Harmonic Oscillator Equations .

Especially when we suppose the solution to be Aeat and eventually get ,  a= + - i wot

And how does it work in the complex region of the equation.

And mainly what should I do to fully understand it .



#2 exchemist

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Posted 28 June 2019 - 10:44 AM

Can anyone help me visualize the solution of Harmonic Oscillator Equations .

Especially when we suppose the solution to be Aeat and eventually get ,  a= + - i wot

And how does it work in the complex region of the equation.

And mainly what should I do to fully understand it .

I am rusty on this but I thought the formula for a classical harmonic oscillator was x(t) = Acosωt (optionally + φ, to allow for the starting phase of the motion if it does not start at maximum displacement). 

 

This is the real part of what you have written, which is in Euler's notation and is equivalent to A(cosωt +isinωt).

 

Not sure if this will help - I last did this 40 years ago. :)  



#3 Nishan

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Posted 29 June 2019 - 02:44 AM

I am rusty on this but I thought the formula for a classical harmonic oscillator was x(t) = Acosωt (optionally + φ, to allow for the starting phase of the motion if it does not start at maximum displacement). 

 

This is the real part of what you have written, which is in Euler's notation and is equivalent to A(cosωt +isinωt).

 

Not sure if this will help - I last did this 40 years ago. :)  

No I mean when deriving we suppose x = eat where a=alpha 

then we solve the differential equation of second degree to derive x(t) = Acoswt + phi

if we could help me then

and one more thing how to use notation 



#4 exchemist

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Posted 29 June 2019 - 04:10 AM

No I mean when deriving we suppose x = eat where a=alpha 

then we solve the differential equation of second degree to derive x(t) = Acoswt + phi

if we could help me then

and one more thing how to use notation 

Do you know what a complex number is? 



#5 Nishan

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Posted 29 June 2019 - 05:38 AM

Do you know what a complex number is?

Yes, but I can't understand it when it is used to solve for 2nd degree equation.
I have attached the picture of equation

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  • rps20190629_175744.jpg

Edited by Nishan, 29 June 2019 - 06:15 AM.


#6 VictorMedvil

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Posted 29 June 2019 - 12:50 PM

Alright your First degree equation where A is the Amplitude meaning like the density of the wave if it were a sound wave it would be the amount of density that the particles experience when the wave contacts the particles. X being the position of those particles back and forth.

 

download.jpg

download-1.jpg

 

The Second degree equation is the shifting of those particles over time being x(t) or position function of time.

 

Lwave-Red-2.gif


Edited by VictorMedvil, 29 June 2019 - 01:04 PM.


#7 exchemist

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Posted 29 June 2019 - 03:24 PM

Yes, but I can't understand it when it is used to solve for 2nd degree equation.
I have attached the picture of equation

What is the title of this physics textbook and who is the author? 



#8 Nishan

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Posted 30 June 2019 - 07:54 AM

What is the title of this physics textbook and who is the author? 

"A Textbook of Physics" - Pitri Bhakta Adhikari



#9 Nishan

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Posted 30 June 2019 - 08:01 AM

Alright your First degree equation where A is the Amplitude meaning like the density of the wave if it were a sound wave it would be the amount of density that the particles experience when the wave contacts the particles. X being the position of those particles back and forth.

 

download.jpg

download-1.jpg

 

The Second degree equation is the shifting of those particles over time being x(t) or position function of time.

 

Lwave-Red-2.gif

well I meant what is happening when the imaginary part of the equation of equation turns into real part of the equation in 2,3 and 4 .

And why can we add two different solution of x and combine it to make one in eq(2)



#10 VictorMedvil

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Posted 30 June 2019 - 08:10 AM

well I meant what is happening when the imaginary part of the equation of equation turns into real part of the equation in 2,3 and 4 .

And why can we add two different solution of x and combine it to make one in eq(2)

 

Well that is the elastic motion of the particles being added to acceleration differential  basically it is saying that the acceleration is being slowed by the spring that is the pressure over time then being re-accelerated by the spring that is the pressure over time. It take a understanding of what you are working on. It's a derived from Hooke's Law.

 

maxresdefault.jpg

 

nw0381-n.gif


Edited by VictorMedvil, 30 June 2019 - 08:28 AM.


#11 Nishan

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Posted 30 June 2019 - 08:26 AM

Well that is the elastic motion of the particles being added to acceleration differential  basically it is saying that the acceleration is being slowed by the spring that is the pressure over time.

 

maxresdefault.jpg

I know that . I think you are not getting my question . I want to understand math . I have posted a picture of math in book " A textbook of Physics" above .

I want to know especially 

x= A1(coswt + i sinwt ) + A2 ( coswt - i sinwt )

  = (A1 + A2 ) coswt + ( A1i - A2i ) sinwt

  = A3 coswt + A4 sinwt 

here we supposed an imaginary number to be A4

   Again 

  A3 = a sin B

  A4 = a cosB

It shows that either cos B or A4 is imaginary .... Is'nt it?

then we do x = asin B cos wt + a sin wt cos B 

when cos B = imaginary then B is imaginary so is sin B

   then finally finding solution we do 

            x = a sin ( wt + B )

     which I thought should have been imaginary soulution . but it works well .

So I think I was missing something here . I ho[e you get me now. 



#12 Nishan

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Posted 30 June 2019 - 08:29 AM

I want to understand math . I have posted a picture of math in book " A textbook of Physics" above .

I want to know especially 

x= A1(coswt + i sinwt ) + A2 ( coswt - i sinwt )

  = (A1 + A2 ) coswt + ( A1i - A2i ) sinwt

  = A3 coswt + A4 sinwt 

here we supposed an imaginary number to be A4

   Again 

  A3 = a sin B

  A4 = a cosB

It shows that either cos B or A4 is imaginary .... Is'nt it?

then we do x = asin B cos wt + a sin wt cos B 

when cos B = imaginary then B is imaginary so is sin B

   then finally finding solution we do 

            x = a sin ( wt + B )

     which I thought should have been imaginary soulution . but it works well .

So I think I was missing something here . I ho[e you get me now. 

 

 

In the picture above we add 2 solutions of x give a general solution of x how is that possible.



#13 VictorMedvil

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Posted 30 June 2019 - 08:41 AM

I want to understand math . I have posted a picture of math in book " A textbook of Physics" above .

I want to know especially 

x= A1(coswt + i sinwt ) + A2 ( coswt - i sinwt )

  = (A1 + A2 ) coswt + ( A1i - A2i ) sinwt

  = A3 coswt + A4 sinwt 

here we supposed an imaginary number to be A4

   Again 

  A3 = a sin B

  A4 = a cosB

It shows that either cos B or A4 is imaginary .... Is'nt it?

then we do x = asin B cos wt + a sin wt cos B 

when cos B = imaginary then B is imaginary so is sin B

   then finally finding solution we do 

            x = a sin ( wt + B )

     which I thought should have been imaginary soulution . but it works well .

So I think I was missing something here . I ho[e you get me now. 

 

 

In the picture above we add 2 solutions of x give a general solution of x how is that possible.

 

The two Imaginary parts of the equation cancel, thats what your missing it took me a second to notice that but A1i - A2i =  (A- A2) , i - i = 0


Edited by VictorMedvil, 30 June 2019 - 09:06 AM.


#14 Nishan

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Posted 30 June 2019 - 08:50 AM

The two Imaginary parts of the equation cancel, thats what your missing it took me a second to notice that but A1i - A2i = 0 

But isn't that :

when, A1i - A2i = 0

B = 90o

 

how could we get there for other real values of B


Edited by Nishan, 30 June 2019 - 08:51 AM.


#15 VictorMedvil

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Posted 30 June 2019 - 08:53 AM

But isn't that :

when, A1i - A2i = 0

B = 90o

 

how could we get there for other real values of B

 

 Sorry i mistyped that it isn't 0 it is A1-A2 only the I's cancel. i-i = 0 , A1 - A2  = B


Edited by VictorMedvil, 30 June 2019 - 08:59 AM.


#16 Nishan

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Posted 30 June 2019 - 09:37 AM

 Sorry i mistyped that it isn't 0 it is A1-A2 only the I's cancel. i-i = 0 , A1 - A2  = B

Can you tell me why/how?



#17 Dubbelosix

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Posted 30 June 2019 - 10:15 AM

When I find time I'll take you through a standard oscillator.





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