Max amplitude of superposition of 2 waves

  • #1
rc2008
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Homework Statement
Max amplitude of superposition of 2 waves
Relevant Equations
Find amplitude of superposition of 2 waves, 4.6sin 2pi()*5.4x and 3.2sin2pi()*10x,
My answer is simply 4.6+ 3.2 = 7.8m , correct me if I am wrong.

If it's 4.6 ##\cos(2\pi*5.4x)## and 3.2 ##\sin(2\pi*10x)##, then the max amplitude should be sqrt(4.6^2 + 3.2^2) = 5.6m, correct me if I am wrong.
 
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  • #2
Hi, and :welcome: ,

I can't even read your formulas (surely you don't mean ##(\sin 2)\ \pi \ 10x## but what you do mean is a guess.

Use some more brackets and learn to use ##\TeX##

1731756369373.png

And your two claims cannot both be correct !

rc2008 said:
If it's 4.6cos2pi()*5.4x and 3.2sin2pi()*10x, then the max amplitude should be sqrt(4.6^2 + 3.2^2) = 5.6
What mathematical rule are you using here ?

And a graph of the superposition easily proves it's wrong !

##\ ##
 
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  • #3
What do you get for the amplitude of ##\cos x+\sin x\ ## ? For ##\cos x+\sin (2x)\ ## ?

##\ ##
 
  • #4
BvU said:
What do you get for the amplitude of ##\cos x+\sin x\ ## ? For ##\cos x+\sin (2x)\ ## ?

##\ ##

For ##\cos x+\sin x\ ##, i would get 2 , simply because ##sqrt(1+1)## = 2
 
  • #5
BvU said:
What do you get for the amplitude of ##\cos x+\sin x\ ## ? For ##\cos x+\sin (2x)\ ## ?

##\ ##
For ##\cos x+\sin (2x)\ ##, I would also get 2, simply because of shape of ##\cos (x) \## will coincide after some time.

For my case of
4.6 ##\cos(2\pi*5.4x)## and 3.2 ##\sin(2\pi*10x)##
, I am not sure whether they will coincide or not because 10 is not a whole numer of multiplier of 5.4
 
  • #6
rc2008 said:
For ##\cos x+\sin x\ ## , i would get 2 , simply because sqrt(1+1) = 2
sqrt (1+1) = 2 ?
Ah, you are learning ##\TeX## ! Bravo !


rc2008 said:
For ##\cos x+\sin (2x)\ ##, I would also get 2, simply because of shape of ##\cos (x) \## will coincide after some time.

For my case of
4.6 ##\cos(2\pi*5.4x)## and 3.2 ##\sin(2\pi*10x)##
, I am not sure whether they will coincide or not because 10 is not a whole numer of multiplier of 5.4


You must have seen plots of ##\sin x## and ##\cos x##.

1731771857806.png


## \cos x+\sin (2x)\ ## looks like this:

1731772042402.png

Functions coincide some times (with y=0), peaks don't !
 
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  • #7
rc2008 said:
For ##\cos x+\sin x\ ##, i would get 2 , simply because ##sqrt(1+1)## = 2
That's not true at all.
##\sqrt{1 + 1} = \sqrt 2##.

Here's my LaTeX before it gets rendered by the browser: ##\sqrt{1 + 1} = \sqrt 2##

Notice the backslash in front of the sqrt command.
 
  • #8
rc2008 said:
My answer is simply 4.6+ 3.2 = 7.8m , correct me if I am wrong.

What condition(s) has(have) to be fulfilled for this 7.8 to be the value of ##4.6 \cos(2\pi*5.4x) + 3.2\sin(2\pi*10x)## ?
And for -7.8 ?

##\ ##
 
  • #9
[itex]4.6 \cos(2\pi*5.4x) + 3.2\sin(2\pi*10x)[/itex] is periodic, because 5.4 is a rational multiple of 10. (It would be easier to write this as [itex]4.6\cos(10.8\pi x) + 3.2\sin(20\pi x)[/itex].)

It is not in general possible to reduce sines of different frequencies to an expression of the form [itex]R\sin (\alpha x + \beta)[/itex]. It might be possible to obtain the zeros of the derivative analytically, but most likely a numerical method will be required.
 
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  • #10
Mark44 said:
That's not true at all.
##\sqrt{1 + 1} = \sqrt 2##.

Here's my LaTeX before it gets rendered by the browser: ##\sqrt{1 + 1} = \sqrt 2##

Notice the backslash in front of the sqrt command.
How do you prevent the Latex from rendering, even when using tags?
 
  • #11
WWGD said:
How do you prevent the Latex from rendering, even when using tags?
For the stuff I don't want to render, I color the first # of each group black using the BBCode color tool.
 
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FAQ: Max amplitude of superposition of 2 waves

What is the principle of superposition in wave theory?

The principle of superposition states that when two or more waves overlap in space, the resultant wave at any point is equal to the sum of the individual waves at that point. This principle is fundamental in understanding how waves interact, leading to phenomena such as constructive and destructive interference.

How do you calculate the maximum amplitude of the superposition of two waves?

The maximum amplitude of the superposition of two waves can be calculated by adding their amplitudes when they are in phase (constructive interference). If wave A has an amplitude of A1 and wave B has an amplitude of A2, the maximum amplitude (Amax) when they are perfectly in phase is given by Amax = A1 + A2.

What is constructive interference?

Constructive interference occurs when two waves meet in phase, meaning their crests and troughs align. This results in a wave with a larger amplitude than either of the original waves. The maximum amplitude in this case is the sum of the individual amplitudes of the two waves.

What is destructive interference?

Destructive interference occurs when two waves meet out of phase, meaning the crest of one wave aligns with the trough of another. This can lead to a reduction in amplitude, and in cases where the amplitudes are equal, they can completely cancel each other out, resulting in a net amplitude of zero.

Can the maximum amplitude of two waves ever exceed the sum of their individual amplitudes?

No, the maximum amplitude of the superposition of two waves cannot exceed the sum of their individual amplitudes. The maximum occurs during constructive interference when the waves are perfectly in phase, and the resultant amplitude is simply the algebraic sum of the individual amplitudes. Any other phase relationship will result in a lower maximum amplitude.

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