Complex exponential description of SHM

[savvvvvvvvvvy]

Hey,

I'm currently reading a textbook which is attempting to derive the equation for a standing wave from first principles. I understand most steps with the exception of one.

It derives a sinusoidal function {x = A \sin \omega t} from a second order ODE, but then immediately interchanges this to {exp{\pm i \omega t}}. Presumably the \pm indicates that there are two solutions, both of which may take the {cos \theta \pm i sin \theta} form from Euler's formula.

My question is how is this format compatible with a single sinusoidal function {x = A \sin \omega t}. No matter how I manipulate it, I always end up with the sum of a sine and cosine, and the sine generally comes out to be imaginary. I imagine there's some simple feature that I'm overlooking, but can't really see it at the moment. Any help would be greatly appreciated!

 

[tiny-tim]

welcome to pf!

hey savvvvvvvvvvy! welllllllllcome to pf!

(use #s )

… how is this format compatible with a single sinusoidal function ${x = A \sin \omega t}$. No matter how I manipulate it, I always end up with the sum of a sine and cosine, and the sine generally comes out to be imaginary.

difficult to tell without seeing the book,

but i'll guess that the coefficients of e^iωt are allowed to be imaginary (or complex)

 

[AlephZero]

Remember that the general equatinn for SHM is $x = A \sin \omega t + B \cos \omega t$. There must be some reason why your book choose to set $B = 0$.

If you have an expression like $x = C_1e^{i\omega t} + C_2 e^{-i\omega t}$, the constants $C_1$ and $C_2$ are complex numbers. To be precise, it should be written $x = \Re(C_1e^{i\omega t} + C_2 e^{-i\omega t})$ where $\Re$ means "the real part of".

To make the complex number form the same as the sine function, $-i e^{i\omega t} + ie^{-i\omega t} = 2 \sin \omega t$.