# Coherence with an infinfinite number of waves

If I had an infinite number of sine waves with frequencies between 0 and 2, and I know what amplitude each wave has, is there a way for me to predict how they interfere?

for example if I have:

frequencies=[ 0 ......................... 2]

amplitudes=sin(frequencies*(pi/1))+sin(frequencies*(pi/2))

wave=sum(amplitudes*sin((pi*2)*frequencies))

what would the wave's phase and amplitude be at any given point?

thanks.

Provided that the amplitude function $A(\omega)$ has a known Fourier transform, you can find the sum as $A(t) = \operatorname{Re} \int\!d\omega\, A(\omega) e^{i \omega t}$ . If the Fourier transform isn't known, you can estimate it numerically.
• $\omega$ is the angular frequency $2\pi f$. I used complex numbers and Euler's formula to write $\sin(\omega t)$ in a convenient way. The rest is an integral, which is a fancy way of writing a sum over frequencies $\omega$. – lionelbrits Dec 1 '13 at 17:27
• So to sum all frequencies between $\omega = 0$ and $\omega = 4\pi$, you would break the range up into intervals $\Delta \omega = \frac{4\pi}{N}$ and let $N \to \infty$. You would write your sum as $\sum_{i=0}^{N-1} (\Delta \omega) A(\omega) \sin (\omega t)$. The calculus way of writing that is simply $\int_0^{4\pi} \! d\omega\, A(\omega) \sin(\omega t) \,$. – lionelbrits Dec 1 '13 at 17:31