Green's Function - Kirchoff Helmholtz Integral Problem

[danong]

I'm seeking help in understanding Kirchoff-Helmholtz Integral.

Actually what i am facing the problem here is,
i don't understand certain things about Green's 2nd identity which stated that two scalar function can be interchanged,
and forming the force $$F = \phi\nabla\varphi - \varphi\nabla\phi$$,

however, i understand that $$\phi\nabla\varphi$$ represents the velocity of sound vibration across the surface to an observer point.

For say, if i take $$\phi$$ as Green's function and $$\varphi$$ as Sound potential / pressure.

So the problem comes,
how would i understand $$\varphi\nabla\phi$$? distribution of sound pressure with impulse unit at the observer point?
Then why do i need to subtract it ?
Are they equivalent?

How does reciprocal theorem applies here at the $$\varphi\nabla\phi$$?
It just seems very confusing to me,
hope someone could point out as I'm really stucked in this topic for months.

 

[jvicens]

Hello,

Thank you for reaching out for help with understanding the Kirchoff-Helmholtz Integral. I can understand that Green's 2nd identity and the concept of the force F = \phi\nabla\varphi - \varphi\nabla\phi may be confusing for you. Let me try to explain it in simpler terms.

Green's 2nd identity states that for two scalar functions, \phi and \varphi, the following equation holds true:

\int_V (\phi\nabla^2\varphi - \varphi\nabla^2\phi)\,dV = \oint_S (\phi\nabla\varphi - \varphi\nabla\phi)\cdot \hat{n}\,dS

This means that the integral of the Laplacian of \phi multiplied by \varphi and the Laplacian of \varphi multiplied by \phi over a volume V is equal to the surface integral of \phi multiplied by the gradient of \varphi and \varphi multiplied by the gradient of \phi over the surface S, where \hat{n} is the unit normal vector to the surface.

In simpler terms, this identity shows that the distribution of the two scalar functions \phi and \varphi are equivalent at a given point if their Laplacians are equal. This is useful in solving problems involving the propagation of sound waves, where \phi can represent the Green's function and \varphi can represent the sound pressure.

Now, coming to the force F = \phi\nabla\varphi - \varphi\nabla\phi, this represents the force exerted on a small volume element by the surrounding medium due to the sound waves. The first term, \phi\nabla\varphi, represents the pressure gradient force, while the second term, \varphi\nabla\phi, represents the viscous force. The pressure gradient force is the force exerted by the pressure difference across the volume element, while the viscous force is the force due to the shear stress between the medium and the volume element.

In terms of the sound waves, \phi\nabla\varphi represents the velocity of sound vibrations across the surface to an observer point. On the other hand, \varphi\nabla\phi represents the distribution of sound pressure with impulse unit at the observer point. These two terms are subtracted because the force F represents