Two wave superposition with different wavelength , amplitude and frequ

[ptolomeo]

Homework Statement

Hi

Two wave superposition with different wavelength , amplitude and frequency

u₁(x,t)=A₁cos(k₁x+w₁t)
u₂(x,t)=A₂cos(k₂x+w₂t)

a)Show that an amplitude modulation is obtained

Homework Equations

No relevant equations

The Attempt at a Solution

u=u1+u2=A₁cos(k₁x+w₁t)+A₂cos(k₂x+w₂t)

and now i don't know what to do, is there any way to relate A₁ and A₂

I searched in the internet and I found Fresnell method which can solve it for two waves of the tipe:
u1=u2=A₁cos(w₁t)
u2=A₁cos(w₂t)=A₁cos(w₁t+phi)

but in my case I don't think I can do that because there are 2 variables in the cos


Its my first post so thanks
pd: sorry for my bad english

 

[vela]

Try this: Draw a right triangle with the length of one leg equal to $A_1$, the length of the other leg equal to $A_2$, and with hypotenuse equal to $\sqrt{A_1^2+A_2^2}$. Then express $A_1$ and $A_2$ in terms of $A$ and an angle $\theta$, appropriately defined. That might get it into a form you can more easily work with.

I'm just making a suggestion. I haven't actually worked the problem out, so the suggestion may not turn out to be useful.

 

[ptolomeo]

Expresing in terms of A=√A²₁+A²₂

I arrive to the expresion

u(x,t)=(A/sinθ)(tan(θ)cos(k₁x+w₁t)+cos(k₂x+w₂t)) I don't see how it can help :/ but thanks

Maybe I can prove that there's a Amplitude modulation without merging the cos

 

[ptolomeo]

I think i found a way to solve it so if i make two vectors where A1 and A2 is the module of each vector I just have to sum them and then find the angle of the resulting vector.

Where θ₁=k₁x-w₁t
θ₂=k₂x-w₂t
u₁=A₁cosθ₁
u₂=A₂cosθ₂

And alfa=(θ1-θ2)/2+θ2=θ1+θ2/2