Superposition of two cosine waves

[Dom_Ldn]

Homework Statement

Superposition of two cosine waves with different periods and different amplitudes.

Homework Equations

This is basically:
acos(y*t) + bcos(x*t)

The Attempt at a Solution

I looked at different trig functions but it seems it is not a standard solution. I've found solutions for different amplitudes (but the same periods) but am unable to find one for different amplitudes and periods.

Can anyone help?

 

[jedishrfu]

There is some discussion here where Fourier series methods are used:

https://curricula2.mit.edu/pivot/book/ph1605.html?acode=0x0200

 

[Ray Vickson]

Homework Statement

Superposition of two cosine waves with different periods and different amplitudes.

Homework Equations

This is basically:
acos(y*t) + bcos(x*t)

The Attempt at a Solution

I looked at different trig functions but it seems it is not a standard solution. I've found solutions for different amplitudes (but the same periods) but am unable to find one for different amplitudes and periods.

Can anyone help?

What are you trying to do? If you know the values of a, b, x, y you can plot a graph of y over a range of t values. If you are trying to find the maximum and minimum values of y, you have a Calculus problem whose solution would (generally) require numerical solution methods---there would no universal "formula" you could use the find the desired values. If you are trying to determine whether y(t) is periodic--and to find the period if it is---there would be still other calculations you would need to make.

So, what you should do depends on what you are attempting to achieve.

 

[Dom_Ldn]

What are you trying to do? If you know the values of a, b, x, y you can plot a graph of y over a range of t values. If you are trying to find the maximum and minimum values of y, you have a Calculus problem whose solution would (generally) require numerical solution methods---there would no universal "formula" you could use the find the desired values. If you are trying to determine whether y(t) is periodic--and to find the period if it is---there would be still other calculations you would need to make.

So, what you should do depends on what you are attempting to achieve.

I was trying to achieve a universal formula for a bichromatic wave surface elevation consisting of two waves with different amplitudes. There is a universal formula for a bichromatic wave surface elevation with the same amplitudes:

eta = H/2cos(om1*t) + H/2cos(om2*t) = H * cos((om1-om2)/2 * t) * cos((om1+om2)/2 * t)

From your reply, I am assuming that something similar doesn't exist for a bichromatic wave with two different amplitudes?

 

[Ray Vickson]

I was trying to achieve a universal formula for a bichromatic wave surface elevation consisting of two waves with different amplitudes. There is a universal formula for a bichromatic wave surface elevation with the same amplitudes:

eta = H/2cos(om1*t) + H/2cos(om2*t) = H * cos((om1-om2)/2 * t) * cos((om1+om2)/2 * t)

From your reply, I am assuming that something similar doesn't exist for a bichromatic wave with two different amplitudes?

I don't think you can read that into anything I said in my reply; I was just asking you to clarify what you wanted. However, I think there may not be any simple solution to the more general problem, but I am not sure. So, yes, indeed, I suspect there is not something similar in the general case.