Summing Waves Using Complex Notation

[cpmiller]

Homework Statement

Carry out the addition of two waves Ψ=Ψ₁+Ψ₂ where
Ψ₁ = Asin[(k+δk)x - (w+δw)t]
Ψ₂ = Acos[(k-δk)x - (w-δw)t]
by means of the complex-number representation and interpret the result.
(Hint: You may find it convenient to rewrite the sine as a cosine by introducing a phase shift in the argument.

Homework Equations

Re{e ^(iΘ) } = cos Θ = [e ^(iΘ) + e ^(-iΘ) ]/2
Im{e ^(iΘ) } = sin Θ = [e ^(iΘ) - e ^(-iΘ) ]/(2i)

The Attempt at a Solution

I rewrote Ψ₁ using a Pi/2 as a phase shift. So I got

Ψ₁ = Acos[(k+δk)x - (w+δw)t + (Pi/2)]

Then I applied the above relevant equation for cosine being the real part of the complex equation to get
Ψ₁ = A Re{e ^{i[(k+δk)x - (w+δw)t + (Pi/2)]} }
Ψ₂ = A Re{e ^{i[(k-δk)x - (w-δw)t]} }

So summing yields:

Ψ₁+Ψ₂ = A Re{e^{i[(k+δk)x - (w+δw)t + (Pi/2)]} + e^{i[(k-δk)x - (w-δw)t]} }

Pulling out the like terms of e^ikx and e^-iwt yields:

Ψ = A Re { [e ^{i[δkx - δwt + Pi/2]} + e^{-i[δkx - δwt]} * e^i(kx-wt) }

"Simplifiying" yields:

Ψ = A Re { [e^iPi/2 * e^{i[δkx - δwt]}+ e^{-i[δkx - δwt]}] * e^i(kx-wt) }

So we have

Ψ = A Re { [-e ^{i[δkx - δwt]} + e ^{-i[δkx - δwt]} ] * e ^i(kx-wt) }

Now I'm not sure what to do with the part in the bolded brackets. I could call it 2i Sin (δkx - δwt) which would be an imaginary part, so taking the real value of an imaginary part yields 0. Which would physically mean that my waves would destructively interfere, but I'm not sure that I can stick the 2i in like that. I wouldn't have a problem putting a 2 in, but I'm not sure if it's okay to multiply by i, or if I need to have the i already there.

Thanks for wading through all this algebra with me!

 

[tiny-tim]

Welcome to PF!

Hi cpmiller! Welcome to PF!

(try using the X² tag just above the Reply box )

Ψ = A Re { [-e^i[δkx - δwt] + e^-i[δkx - δwt] ] * e^i(kx-wt) }

Now I'm not sure what to do with the part in the bolded brackets. I could call it 2i Sin (δkx - δwt) which would be an imaginary part, so taking the real value of an imaginary part yields 0. …

No, Re {[iB] e^i(kx-wt)} ≠ Re {[iB]} Re {e^i(kx-wt)}

 

[cpmiller]

Hi cpmiller! Welcome to PF!

(try using the X² tag just above the Reply box )

Thanks for the response I went back and used your suggestion to try to make my original post a bit more "user friendly." Your answer helped a lot and I managed to figure the problem out this morning!

 

[tiny-tim]

Thanks for the response I went back and used your suggestion to try to make my original post a bit more "user friendly." Your answer helped a lot and I managed to figure the problem out this morning!

ooh, that's neat! :tongue2:

Please accept this present of a pi and an omega for future efforts: π ω