Dom_Ldn said: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?
Ray Vickson said: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 said: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?
The superposition of two cosine waves refers to the combination of two waves with the same frequency and amplitude, resulting in a new wave that is the sum of the individual waves. This is a fundamental principle in the study of wave mechanics and is commonly seen in many natural phenomena, such as the interference of light waves and the vibrations of sound waves.
The superposition of two cosine waves is calculated by adding the individual waves at each point in time. This can be done by taking the sum of the amplitudes of the two waves at each point, and then using the formula for cosine to calculate the resulting wave.
The phase difference between two cosine waves in superposition determines the resulting shape of the new wave. If the two waves have the same phase, they will add constructively and create a larger amplitude. If they have opposite phases, they will cancel each other out and create a smaller amplitude. This phase difference is also important in determining the frequency and wavelength of the resulting wave.
Yes, the superposition of two cosine waves is a useful tool for modeling many real-world phenomena. For example, it can be used to understand the behavior of electromagnetic waves in optics, the behavior of sound waves in acoustics, and the behavior of water waves in oceanography.
The superposition of two cosine waves has a wide range of applications in various fields of science and engineering. Some examples include signal processing, image processing, communication systems, and quantum mechanics. It is also a fundamental concept in the study of wave mechanics and is used to analyze and understand many natural phenomena.