- #1
chiraganand
- 111
- 1
Hi, I have read about the rayleigh sommerfeld integral and its a surface integral. Why is it difficult to calculate the integral directly?
Why Is the Rayleigh Sommerfeld Integral Challenging to Compute Directly?
Click For Summary
Discussion Overview
The discussion revolves around the challenges of directly computing the Rayleigh-Sommerfeld integral, which is a surface integral used in wave theory. Participants explore the complexities involved in the computation, particularly focusing on oscillatory components and potential methods for simplification.
Discussion Character
- Technical explanation
- Debate/contested
Main Points Raised
- One participant notes that the Rayleigh-Sommerfeld integral is challenging due to its oscillating part, specifically the factor ##e^{ik|\vec{R}_{2}-\vec{r}_{1}|}##, which complicates direct computation.
- Another participant suggests that various methods, such as asymptotic expansions or expressing the integral in terms of special functions, could be explored to facilitate the calculation.
- There is mention of difficulties in obtaining an exact primitive for the integral, indicating that simplifications may be necessary.
Areas of Agreement / Disagreement
Participants express a shared understanding of the integral's complexity, but there is no consensus on a specific method for direct computation or on the extent to which simplifications can be applied.
Contextual Notes
The discussion highlights the oscillatory nature of the integral and the potential reliance on special functions, but does not resolve the specific mathematical steps or assumptions involved in the computation.
- #2
Ssnow
Science Advisor
- 573
- 182
Can you give a reference for this question ?
Ssnow
Ssnow
- #3
berkeman
Admin
- 69,479
- 24,999
@chiraganand was last at PF in Sept. 2020, but may still be receiving e-mail notifications of updates to his threads. Hopefully he will be able to respond. 
- #4
chiraganand
- 111
- 1
Oh wow I had moved on from this one, but thanks for replying! This is the the reference. so there are a lot of different ways to make it easier to calculate but my main question was how would one go about solving it without any simplifications and at which step would one get stuckSsnow said:
- #5
Ssnow
Science Advisor
- 573
- 182
Ok, now I understand the question, the principal problem is that the surface integral is not a simple surface integral but it has an oscillating part given by the oscillatory complex factor ##e^{ik|\vec{R}_{2}-\vec{r}_{1}|}##. This is the reason you can try to have an asymptotic expansion using different ways as the stationary phase expansion or others ... In addition you have the variable up to the exponent and also in the denominator- You can try to express it in terms of special functions as ##\int_{0}^{t}\frac{e^{x}}{x^{\alpha}}dx## but to have an exact primitive I don't think it is simple ...
Ssnow
Ssnow
Similar threads
- · Replies 6 ·
- · Replies 0 ·
- · Replies 5 ·
- · Replies 14 ·
- · Replies 3 ·
- · Replies 2 ·
Insights
The Art of Integration
- · Replies 10 ·
- · Replies 10 ·
- · Replies 8 ·