Acoustics

Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

13.2. The Rayleigh integrals and Green's function

In this section, monopole and dipole boundary integrals are derived in an intuitive way based on the Huygens–Fresnel principle whereby monopole and dipole point sources are summed over surfaces. A more mathematically rigorous treatment follows in Section 13.3. We have treated all problems so far in this text as boundary value problems based on solutions to the following homogeneous Helmholtz wave equation in an arbitrary coordinate system:

(\nabla^{2} + k^{2}) \tilde{p} (r) = 0,

$(\nabla^{2} + k^{2}) \tilde{p} (r) = 0,$

(13.1)

where, for example, in rectangular coordinates (r) = (x,y,z) and the Laplace operator ∇² is given in rectangular, cylindrical, and spherical coordinates by Eqs. (2.147), (2.160), and (2.181), respectively. By homogeneous, we mean that the equation describes waves that could exist, but there are no driving forces or velocities present in the equation to create them. These come later from the boundary conditions. In general, the boundary value method involves solving the homogeneous Helmholtz wave equation in a suitable orthogonal coordinate system such that it becomes a separable equation. That is, the equation is split into a set of differential equations, each with respect to one ordinate only, as described in Sections 2.9–2.11. The solutions to those equations then contain constants, which can be determined by applying boundary conditions. By a suitable coordinate system, we mean one that first must lead to a separable wave equation (if there is more than one ordinate involved) and second fits the geometry of the problem, by which we mean that boundary conditions can be applied by setting pressure or velocity to specific values at constant ordinate values. The simplest example is the pulsating sphere that is solved by setting the particle velocity at the surface of the sphere, where the radial ordinate is equal to the sphere's radius. In the limit, when the radius approaches zero, this leads to the pressure field due to a point source as defined in Eq. (4.71). Let us now recast this equation in the following form:

\tilde{p} (r | r_{0}) = j k ρ_{0} c {\tilde{U}}_{0} g (r | r_{0}),

$\tilde{p} (r | r_{0}) = j k ρ_{0} c {\tilde{U}}_{0} g (r | r_{0}),$

(13.2)

where r and r ₀ are the positions of the observation point and source respectively in an arbitrary coordinate system. The function

g (r | r_{0})

$g (r | r_{0})$

is known as the Green's function and is defined by

g (r | r_{0}) = \frac{e^{- j k (r - r_{0})}}{4 π (r - r_{0})} .

$g (r | r_{0}) = \frac{e^{- j k (r - r_{0})}}{4 π (r - r_{0})} .$

(13.3)

For example, in rectangular coordinates, we would write

g (x, y, z | x_{0}, y_{0}, z_{0}) = \frac{e^{- j k \sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}}}}{4 π \sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}}} .

$g (x, y, z | x_{0}, y_{0}, z_{0}) = \frac{e^{- j k \sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}}}}{4 π \sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}}} .$

(13.4)

The Green's function is a useful short hand for the spatial distribution due to a point source, but it does not indicate its absolute strength. In this instance, it tells us that the sound pressure varies sinusoidally as it spreads outwards from the source, and its amplitude is inversely proportional to the distance from the source. It should be noted that there is a singularity at r = r ₀. Using the Huygens–Fresnel principle we can treat a vibrating surface as an array of point sources, or rather surface elements that in the limit shrink to points. In the case of a closed surface (i.e., that which fully encloses a volume), we need not worry about the back wave when considering the external field. The volume velocity of each surface element is given by

{\tilde{U}}_{0} (r_{0}) = {\tilde{u}}_{0} (r_{0}) δ S_{0},

${\tilde{U}}_{0} (r_{0}) = {\tilde{u}}_{0} (r_{0}) δ S_{0},$

(13.5)

where δS ₀ is the area of the surface element and

{\tilde{u}}_{0} (r_{0})

${\tilde{u}}_{0} (r_{0})$

is the velocity normal to the surface at point r ₀. The radiated field is the sum of the fields due to all the point sources so that

{\tilde{p}}_{M} (r) = j k ρ_{0} c \iint {\tilde{u}}_{0} (r_{0}) g (r | r_{0}) d S_{0},

${\tilde{p}}_{M} (r) = j k ρ_{0} c \iint {\tilde{u}}_{0} (r_{0}) g (r | r_{0}) d S_{0},$

(13.6)

which is known as the monopole Rayleigh integral. Furthermore, using the relationship

{\tilde{u}}_{0} (r_{0}) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial n_{0}} \tilde{p} (r_{0})

${\tilde{u}}_{0} (r_{0}) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial n_{0}} \tilde{p} (r_{0})$

(13.7)

leads to

{\tilde{p}}_{M} (r) = - \iint \frac{\partial}{\partial n_{0}} \tilde{p} (r_{0}) g (r | r_{0}) d S_{0} .

${\tilde{p}}_{M} (r) = - \iint \frac{\partial}{\partial n_{0}} \tilde{p} (r_{0}) g (r | r_{0}) d S_{0} .$

(13.8)

Similarly, the surface can be made up of dipole point sources, each comprising two monopole point sources of opposite polarity, separated by a distance Δz ₀ that tends to zero. Let the Green's function be defined in axisymmetric spherical–cylindrical coordinates by

g (r, θ | z_{0}) = \frac{e^{- j k r_{1}}}{4 π r_{1}},

$g (r, θ | z_{0}) = \frac{e^{- j k r_{1}}}{4 π r_{1}},$

(13.9)

where

r_{1}^{2} = r^{2} + z_{0}^{2} - 2 r z_{0} \cos θ

$r_{1}^{2} = r^{2} + z_{0}^{2} - 2 r z_{0} \cos θ$

and θ is the inclination angle of the observation point relative to the z-axis, which passes through the two monopole point sources. The dipole point source is located at a distance z ₀ from the origin and r is the distance from the observation point to the origin. The gradient of the Green's function in the z direction is then given by

\frac{\partial}{\partial z_{0}} g (r, θ | z 0) |_{z_{0} = 0} = (\frac{1}{r} + j k) \cos θ \frac{e^{- j k r}}{4 π r} .

$\frac{\partial}{\partial z_{0}} g (r, θ | z 0) |_{z_{0} = 0} = (\frac{1}{r} + j k) \cos θ \frac{e^{- j k r}}{4 π r} .$

(13.10)

From Eq. (4.114), the field due to a single dipole point source has previously been shown to be

\tilde{p} (r, θ) = j k ρ_{0} c {\tilde{U}}_{0} b (\frac{1}{r} + j k) \cos θ \frac{e^{- j k r}}{4 π r},

$\tilde{p} (r, θ) = j k ρ_{0} c {\tilde{U}}_{0} b (\frac{1}{r} + j k) \cos θ \frac{e^{- j k r}}{4 π r},$

(13.11)

which after substituting Eq. (13.10) and letting b = Δz ₀ becomes

\tilde{p} (r, θ) = j k ρ_{0} c {\tilde{U}}_{0} Δ z_{0} \frac{\partial}{\partial z_{0}} g (r, θ | z_{0}) |_{z_{0} = 0} .

$\tilde{p} (r, θ) = j k ρ_{0} c {\tilde{U}}_{0} Δ z_{0} \frac{\partial}{\partial z_{0}} g (r, θ | z_{0}) |_{z_{0} = 0} .$

(13.12)

Again, using the relationships of Eqs. (13.5) and (13.7), together with

Δ_{z_{0}} {\frac{\partial {\tilde{p}}_{0}}{\partial z_{0}} |}_{Δ z_{0} \to 0} = - {\tilde{p}}_{0},

$Δ_{z_{0}} {\frac{\partial {\tilde{p}}_{0}}{\partial z_{0}} |}_{Δ z_{0} \to 0} = - {\tilde{p}}_{0},$

(13.13)

leads to

\tilde{p} (r, θ) = {\tilde{p}}_{0} \frac{\partial}{\partial z_{0}} g {(r, θ | z_{0}) |}_{z_{0} = 0} δ S_{0},

$\tilde{p} (r, θ) = {\tilde{p}}_{0} \frac{\partial}{\partial z_{0}} g {(r, θ | z_{0}) |}_{z_{0} = 0} δ S_{0},$

(13.14)

which is then integrated over the surface to give

{\tilde{p}}_{D} (r) = \iint \tilde{p} (r_{0}) \frac{\partial}{\partial n_{0}} g (r | r_{0}) d S_{0}

${\tilde{p}}_{D} (r) = \iint \tilde{p} (r_{0}) \frac{\partial}{\partial n_{0}} g (r | r_{0}) d S_{0}$

(13.15)

in any coordinate system. This is known as the dipole Rayleigh integral. We note that the derivative of the Green's function is taken with respect to the normal n ₀ to the surface because the axis of each dipole element must be normal to the surface.

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for Acoustics

Create new playlist

Sign In

Sign Up

13.2. The Rayleigh integrals and Green's function

Table of Contents for
Acoustics