How to i solve the amplitude 3sin(2x)+4sin(x)

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SUMMARY

The discussion focuses on solving the amplitude of the function 3sin(2x) + 4sin(x). The maximum amplitude achieved when these two sine waves constructively interfere is approximately 6.10, as confirmed through both analytical methods and graphical representation using ROOT software. The derivative of the function, y' = 6cos(2x) + 4cos(x), is utilized to find critical points, leading to the determination of maximum values. The conversation highlights the importance of understanding the interaction between different sine functions in amplitude calculations.

PREREQUISITES
  • Understanding of trigonometric functions, specifically sine and cosine.
  • Knowledge of calculus, particularly derivatives and critical points.
  • Familiarity with amplitude concepts in wave functions.
  • Experience with graphing software, such as ROOT or Excel.
NEXT STEPS
  • Learn how to derive and analyze trigonometric functions using calculus.
  • Explore the concept of constructive and destructive interference in wave functions.
  • Investigate the use of ROOT software for graphing complex functions.
  • Study the properties of sine and cosine functions, including their maximum and minimum values.
USEFUL FOR

Mathematicians, physics students, and anyone interested in wave mechanics or signal processing will benefit from this discussion, particularly those looking to deepen their understanding of amplitude calculations in trigonometric functions.

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how to i solve the amplitude 3sin(2x)+4sin(x)
 
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The amplitude of one component is 3; the amplitude of the other is 4. When two sine waves are added together, regardless of their frequencies or phases, they will periodically be in phase and add together constructively. They will add to an amplitude of 7 at those points.

- Warren
 
thx , is amplitude find through max/min pts?
 
Far be it from me to disagree with Chroot, but I don't see how sin(x) and sin(2x) will "constructively interfere" to give an amplitude of 3+ 4. 4sin(x)= 4 only for x= pi/2+ 2npi while 3sin(2x)= 3 only for 2x= pi/2+ 2mpi or x= pi/4+ mpi. That means we would have to have (m-2n)pi= pi/4 for some integers m and n.

The way I would do this problem is to find the maximum value of y= 3sin(2x)+ 4sin(x) as expscv suggested:
If y= 3sin(2x)+ 4sin(x) then y'= 6cos(2x)+ 4cos(x). cos(2x)= 2cos2(x)- 1 so this is y'= 12 cos2(x)+ 4cos(x)- 6= 0.

Solving that quadratic equation for cos(x), then converting that to sin(x) and sin(2x) (sin(x)= sqrt(1- cos2(x)) of course, and sin(2x)= 2sin(x)cos(x)), I get that 3sin(2x)+ 4sin(x) has a maximum value of approximately 6.10, the same thing I get by graphing y= 3sin(2x)+ 4sin(x).
 
Whoops. Integer multiple frequencies. :sheepish grin:

- Warren
 
thx both in reply, espically HallsofIvy ,
 
I got too curious. :smile:

The red and blue are the two components. The black one the sum.
 

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cool wat software is that?
 
Nice looking functions. You can do that with Excel as well, although it would be a bit less convenient.
 
  • #10
It's called ROOT. You can download it from here. It is mainly used for high energy physics.
 

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