Expressing the addition of two sinusoidal waves this form.

In summary, to express x in the form x = Re{Aeiαeiωt}, you need to first expand the functions cos(ωt) and sin(ωt) using the equations cos x = 1/2 e^ix + 1/2 e^-ix and sin x = − i/ 2e^ix + i/2 e^−ix. Then, manipulate the resulting equations to get e^ix and e^-ix in terms of sin(ωt) and cos(ωt). Finally, apply the Re() function to the resulting equation to get the real part of the argument in the form x = Re{Aeiαeiωt}.
  • #1
Armin

Homework Statement


Express the following in the form x = Re{Aeeiωt}

(a) x= cos(ωt) + sin(wt)
(b) x= sin(ωt +π/4) + cos(ωt)
(c) x= 2cos(ωt+π/3) + (√3)sin(ωt)-cos(ωt)

Homework Equations


cos x = 1/2 e^ix + 1/2 e^-ix
sin x = − i/ 2e^ix + i/2 e^−ix

The Attempt at a Solution



To be honest, I have no clue where to start. I do not know what the Re (Short for Real) asks for. There is no imaginary number in the functions but there are imaginary numbers in the form it needs to be expressed. Any help to point me in the right direction will be appreciated.

-A
 
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  • #2
Hi Armin:

First, you need to correct some typos. You have "w" is some places where you meant "ω".

Second, you need to know that sin(ωt) also has a form involving 1/2 e^ix and 1/2 e^-ix. This should be another relevant equation under (2).

Third, you need to do a bit of manipulation to get e^ix and e^-ix as expressions in terms of the sin and cos.

Then you should be able to see the use of Re{...}. This function with a complex argument gives the real part of the argument.

Hope this helps.

Regards,
Buzz
 
  • #3
Buzz Bloom said:
Hi Armin:

First, you need to correct some typos. You have "w" is some places where you meant "ω".

Second, you need to know that sin(ωt) also has a form involving 1/2 e^ix and 1/2 e^-ix. This should be another relevant equation under (2).

Third, you need to do a bit of manipulation to get e^ix and e^-ix as expressions in terms of the sin and cos.

Then you should be able to see the use of Re{...}. This function with a complex argument gives the real part of the argument.

Hope this helps.

Regards,
Buzz

I did what you told me to and I got (1/2+i/2)e-iωt+(1/2-i/2)eiωt

and I do know that eiωt=cosωt+isinωt

But I do not know what the Re[..] does.
 
  • #4
Armin said:
I do not know what the Re[..] does.
The Re() function extracts the real part of its complex argument. Re(x+iy)=x.
 
  • #5
I would start from the other end of the problem. Expand eeiωt using cos and sin, then apply Re() to it.
 

1. What is the equation for expressing the addition of two sinusoidal waves?

The equation for expressing the addition of two sinusoidal waves is: A*sin(ωt + φ) + B*sin(ωt + φ), where A and B are the amplitudes of the two waves, ω is the angular frequency, and φ is the phase shift.

2. How do you graph the addition of two sinusoidal waves?

To graph the addition of two sinusoidal waves, plot the individual waves on the same set of axes and then add the corresponding values at each point. The resulting graph will be the sum of the two waves.

3. Can two sinusoidal waves with different frequencies be added together?

Yes, two sinusoidal waves with different frequencies can be added together. The resulting wave will have a frequency equal to the average of the two individual frequencies.

4. What happens when two sinusoidal waves with opposite phases are added together?

When two sinusoidal waves with opposite phases are added together, they cancel each other out and the resulting wave has an amplitude of zero. This is known as destructive interference.

5. How does the addition of two sinusoidal waves differ from the addition of two regular waves?

The addition of two sinusoidal waves is different from the addition of two regular waves because sinusoidal waves have a specific shape and are characterized by their amplitude, frequency, and phase, whereas regular waves can have any shape and do not have these specific characteristics.

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