- #1
Trevorman
- 20
- 2
Hi, I have a Fourier problem that i do not know if it is valid to do the calculations like this.
The Fourier transform looks like this
##
\hat{v}(x,\omega) = \frac{\hat{F}(\omega)}{4(EI)^{\frac{1}{4}}i \omega^{\frac{3}{2}}(\rho A)^{\frac{3}{4}}}\left[ e^{-i\left[\omega^2 \frac{\rho A}{EI} \right]^{\frac{1}{4}}x} -ie^{-\left[\omega^2 \frac{\rho A}{EI} \right]^{\frac{1}{4}}x} \right]##
and is the Fourier transform of a displacement for a wave in a beam.
The inverse Fourier transform of this equation is the displacement and is displayed below
##v(x,t)= \sum_n\hat{v} \cdot e^{i\omega t}##
What I want to calculate is the x derivative of ##v(x,t)##. Is it valid to calculate ##\frac{\partial \hat{v}}{\partial x}## and do a inverse Fourier transform to get ##v^\prime(x,t)##
In other words, is this valid
## v^\prime(x,t) = \sum_n \hat{v}^\prime \cdot e^{i \omega t}##
The Fourier transform looks like this
##
\hat{v}(x,\omega) = \frac{\hat{F}(\omega)}{4(EI)^{\frac{1}{4}}i \omega^{\frac{3}{2}}(\rho A)^{\frac{3}{4}}}\left[ e^{-i\left[\omega^2 \frac{\rho A}{EI} \right]^{\frac{1}{4}}x} -ie^{-\left[\omega^2 \frac{\rho A}{EI} \right]^{\frac{1}{4}}x} \right]##
and is the Fourier transform of a displacement for a wave in a beam.
The inverse Fourier transform of this equation is the displacement and is displayed below
##v(x,t)= \sum_n\hat{v} \cdot e^{i\omega t}##
What I want to calculate is the x derivative of ##v(x,t)##. Is it valid to calculate ##\frac{\partial \hat{v}}{\partial x}## and do a inverse Fourier transform to get ##v^\prime(x,t)##
In other words, is this valid
## v^\prime(x,t) = \sum_n \hat{v}^\prime \cdot e^{i \omega t}##