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

 

[paranormal]

Hello,

Thank you for sharing your question. The phenomenon you are describing is known as superposition, where two or more waves combine to form a new wave. In this case, the resulting wave will have a different amplitude, wavelength, and frequency compared to the individual waves.

To determine the amplitude modulation, we can use the principle of superposition which states that the resulting wave is the sum of the individual waves. In this case, the amplitude of the resulting wave will be the sum of the amplitudes of the individual waves, A1 + A2.

As for the wavelength and frequency, we can use the fact that the wavelength is inversely proportional to the wave number (k) and the frequency is directly proportional to the angular frequency (w). So, for the resulting wave, the wavelength will be the average of the individual wavelengths, (λ1 + λ2)/2, and the frequency will also be the average of the individual frequencies, (f1 + f2)/2.

I hope this helps answer your question. If you have any further inquiries, please let me know. Good luck with your studies!