Summing cosines of different amplitude

In summary, the conversation discusses the method of summing multiple cosines with different amplitude and phase shift but same angular frequency. The proposed approach is to use complex numbers and solve for the constants E and phi. However, this method does not simplify the expression and ends up with a messy arctan function. The speaker then mentions using geometry and trigonometric identities to simplify the expression instead.
Hi there

I am trying to sum many cosines of different amplitude and phase shift, but same ang. frequency (it's not a coursework question). My first thoughts are to sum them two at a time (to simplify matters?), probably using complex numbers. I tried doing it symbolically in MATLAB but it wasn't able to simplify things. Supposing the ang. frequency is 1, I know that the solution can be written:

$$\displaystyle{ C\cos (t + \delta ) + D\cos (t + \varepsilon ) = E\cos (t + \varphi)}$$

where I would have to solve for E and phi. Or equivalently:

$$\displaystyle{Ae^{i(t + \delta )} + e^{i(t + \varepsilon )} = Be^{i(t + \varphi )}}$$

where I would have to solve for B and phi.

Then I split things into two equations (one using real part/cosines, other using imag. part/sines), and eliminate B. Unfortunately this approach doesn't seem to help, as I just end up with a messy arctan of sums of sines and cosines (of different amplitudes -- i.e. back to original problem!).

Any suggestions as to a more fruitful approach?

Thank you.

Ok, nevermind, got it figured out! Just used a bit of geometry

Hi there,

It seems like you have a good understanding of the problem and have already attempted some approaches. One possible method that may help simplify things is to use the trigonometric identity:

\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)

Using this identity, you can rewrite the equation as:

C\cos(t)\cos(\delta) - C\sin(t)\sin(\delta) + D\cos(t)\cos(\varepsilon) - D\sin(t)\sin(\varepsilon) = E\cos(t)\cos(\varphi) - E\sin(t)\sin(\varphi)

Then, you can group the cosine and sine terms separately and solve for E and phi separately. This may help simplify the equations and make it easier to solve for the unknowns. Another approach could be to use the exponential form of cosine and solve for the unknowns using complex numbers. I would suggest experimenting with different approaches and seeing which one yields the simplest solution. Good luck with your problem!

1. What does "summing cosines of different amplitude" mean?

"Summing cosines of different amplitude" refers to the process of adding together multiple cosine functions with different amplitudes. This results in a new function that combines the individual cosines into a single wave-like pattern.

2. How is the amplitude of a cosine function determined?

The amplitude of a cosine function is determined by the vertical distance between the maximum and minimum values of the function. It represents the strength or magnitude of the wave.

3. Why is it important to consider different amplitudes when summing cosines?

Considering different amplitudes when summing cosines allows for a more complex and realistic representation of a wave. In real-world scenarios, waves often have varying amplitudes, and by including this in our calculations, we can better understand and model these phenomena.

4. What is the difference between summing cosines of different amplitude and summing cosines of the same amplitude?

When summing cosines of different amplitude, the resulting function will have a more complex and varied shape compared to when summing cosines of the same amplitude. This is because different amplitudes result in different wave strengths, leading to a more intricate overall pattern.

5. Can summing cosines of different amplitude be used to model real-world phenomena?

Yes, summing cosines of different amplitude is a commonly used technique in physics and engineering to model various natural and man-made phenomena. It allows for a more accurate representation of complex waves, such as sound waves, ocean waves, and light waves.

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