Finding Amplitude in Harmonic Function: Solving for Time t

[Niles]

Homework Statement

Hi all.

I have the following harmonic function:

$$V(t)=A\cos(\omega t)\exp(-Ct),$$


where C is a constant, and A is the amplitude. I need to find the time t, where the amplitude is A/2. This gives me:

$$V(t)=A\cos(\omega t)\exp(-Ct) = \frac{A}{2},$$

but how do I solve this equation?

Thanks in advance.


Niles.

 

[HallsofIvy]

Well, the obvious first step is to cancel the "A"s: $cos(\omega t)e^{-Ct}= 1/2$. Next, I think I would write the cosine in exponential form: $cos(\omega t)= (e^{it}+ e^{-it})/2$ so $cos(\omega t)e^{-Ct}= (e^{(-C+ i\omega)t}+ e^{(C-i\omega)t})2= 1/2$

 

[Niles]

Ahh, great.

If I was given a function on the form:

$$V(t)=(A\cos(\omega t)+B\sin(\omega t)\exp(-Ct),$$

then writing the sines and cosines as exponentials would be the way to go too. But am I even correct to say that the time t when the amplitude of the oscillation of V(t) is half of the original amplitude is when V(t) = A/2, where A is the amplitude?

 

[HallsofIvy]

Yes, you said "find the time find the time t, where the amplitude is A/2". If the initial amplitude is A, then half of it is A/2.

 

[TaiwanCountry]

Need numerical solve.