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Is there any way to estimate the $f(\theta)$ in the following equation?

$exp(\lambda_1 \times cos^2\theta) \times exp(\lambda_2 \times cos\theta) = f(\theta) \times exp( \lambda_1+\lambda_2) $

$\lambda_1 = -\frac{(\alpha\omega\xi)^2}{v^2}$

$\lambda_2 = -\frac{i\omega\xi}{v}$

where $\alpha$ is decaying rate, $\omega$ is angular frequency, $\xi$ is distance between two points and $v$ is seismic wave velocity.

it should be mentioned that $f(\theta)$ can not obtain. try $\theta=\pi/2$.

In earthquake engineering, when seismic waves propagate through soil media, the acceleration time series recorded at two different points along the propagation direction are spatially correlated. the coherency function between these two points (defined as cross spectral density function between two points dividing by square root of multiplication of auto power spectral density of each point) are estimated using exponential function ( $exp( \lambda_1+\lambda_2)$ ). suppose a case in which the angle between seismic propagation direction and the line that connects two points is $\theta$, therefore the coherency function between two points becomes : $exp(\lambda_1 \times cos^2\theta) \times exp(\lambda_2 \times cos\theta)$. the problem is that I need to integrate from this equation with respect to frequency. it is highly interesting to sperate $\theta$ from this equation to see how $\theta$ influences the final integral. it should be mentioned that the above mentioned tasks are used to compute the response (internal force) of long span bridges under spatial variation of earthquake ground motion (SVEGM). two mentioned points are bridge supports. the response of bridge under SVEGM is obtained by several researchers only for the case that the angle between seismic wave and bridge axes is zero. the calculation of response under desired direction is my target.

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  • $\begingroup$ Please provide us some context, such as where the question comes from, what you've already tried (and specifically had difficulty with), etc. Thanks. $\endgroup$ – John Omielan Feb 11 at 20:33
  • $\begingroup$ @JohnOmielan. thank you john. I have just added more explanation to my question. $\endgroup$ – Parsa Parvanehro Feb 12 at 8:34
  • $\begingroup$ Thank you for adding the explanation. What you're asking for & trying to do looks quite interesting. Unfortunately, I don't think I can provide any help to you, but I hope somebody else here can. $\endgroup$ – John Omielan Feb 12 at 20:39

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