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.

 

[eljose79]

I understand your confusion with the Kirchoff-Helmholtz Integral and Green's 2nd identity. Let me try to break it down for you and hopefully it will help you understand it better.

Firstly, Green's 2nd identity states that for two scalar functions, let's call them \phi and \varphi, the following holds true:
\int_V (\phi\nabla^2\varphi - \varphi\nabla^2\phi) dV = \oint_S (\phi\nabla\varphi - \varphi\nabla\phi)\cdot d\vec{S}
In simpler terms, the integral of the Laplacian of \phi multiplied by \varphi minus 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 minus \varphi multiplied by the gradient of \phi over the surface S enclosing the volume V.

Now, let's apply this to the Kirchoff-Helmholtz Integral. This integral is used to solve the Helmholtz equation, which describes the propagation of sound waves. In this case, \phi represents the sound potential or pressure at a point in space, and \varphi represents the Green's function, which is a mathematical tool used to solve the Helmholtz equation.

When we apply Green's 2nd identity to the Kirchoff-Helmholtz Integral, we get:
\int_V (\phi\nabla^2\varphi - \varphi\nabla^2\phi) dV = \oint_S (\phi\nabla\varphi - \varphi\nabla\phi)\cdot d\vec{S} = \oint_S F\cdot d\vec{S}
where F = \phi\nabla\varphi - \varphi\nabla\phi is known as the Kirchoff-Helmholtz force. This force represents the distribution of sound pressure at a point in space due to the sound potential at that point and its surrounding points. It takes into account both the sound potential and its gradient, which represents the velocity of sound vibrations.

Now, let's address your question about \varphi\nabla\phi. This term represents the distribution of sound pressure at a point due to the sound potential at that point and

 

[charng]

The Kirchoff-Helmholtz Integral problem is a fundamental concept in the field of acoustics and is used to solve for sound pressure and velocity at a given point in space. It is based on the principle of energy conservation and is derived from Green's second identity. This integral problem involves the use of Green's function, which is a mathematical tool used to solve differential equations.

Green's second identity states that two scalar functions can be interchanged, which means that the order of differentiation does not matter. In the context of the Kirchoff-Helmholtz Integral problem, this means that the order of differentiation of the sound potential and velocity can be interchanged without affecting the solution.

The force equation, F = \phi\nabla\varphi - \varphi\nabla\phi, is a representation of the conservation of energy in the system. The term \phi\nabla\varphi represents the velocity of sound vibration across the surface to an observer point, while \varphi\nabla\phi represents the distribution of sound pressure with impulse unit at the observer point. By subtracting these two terms, we can calculate the net force acting on the system at the observer point.

Reciprocal theorem applies to the \varphi\nabla\phi term, which states that the sound pressure at a given point is equal to the sound pressure at another point multiplied by the velocity of sound vibration between the two points. This theorem is used to simplify the Kirchoff-Helmholtz Integral problem and make it more manageable.

In summary, the Kirchoff-Helmholtz Integral problem is a complex concept that involves the use of Green's function, Green's second identity, and the principle of energy conservation. It can be confusing at first, but with proper understanding and application of these principles, it can be a powerful tool in solving for sound pressure and velocity in acoustics. I hope this explanation helps in clarifying some of your doubts.