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

13.5. Boundary integral method case study: radially pulsating cap in a rigid sphere

In this section, we shall apply the boundary integral method to a pulsating cap in a sphere to illustrate its application to an elementary acoustical problem that has already been treated in Section 12.6 using the boundary value method. The geometry of the problem is shown in Fig. 12.16. From Eq. (13.26), we can write the pressure field as a surface integral:

\begin{matrix} \tilde{p} (r, θ) = \int_{0}^{2 π} \int_{0}^{π} g (r, θ | r_{0}, θ_{0}) |_{r_{0} = R} \frac{\partial}{\partial r_{0}} {\tilde{p} (r_{0}, θ_{0}) |}_{r_{0} = R} R^{2} \sin θ_{0} d θ_{0} d ϕ_{0} \\ - \int_{0}^{2 π} \int_{0}^{π} {\tilde{p} (r_{0}, θ_{0}) |}_{r_{0} = R} \frac{\partial}{\partial r_{0}} {g (r, θ | r_{0}, θ_{0}) |}_{r_{0} = R} R^{2} \sin θ_{0} d θ_{0} d ϕ_{0}, \end{matrix}

$\begin{matrix} \tilde{p} (r, θ) = \int_{0}^{2 π} \int_{0}^{π} g (r, θ | r_{0}, θ_{0}) |_{r_{0} = R} \frac{\partial}{\partial r_{0}} {\tilde{p} (r_{0}, θ_{0}) |}_{r_{0} = R} R^{2} \sin θ_{0} d θ_{0} d ϕ_{0} \\ - \int_{0}^{2 π} \int_{0}^{π} {\tilde{p} (r_{0}, θ_{0}) |}_{r_{0} = R} \frac{\partial}{\partial r_{0}} {g (r, θ | r_{0}, θ_{0}) |}_{r_{0} = R} R^{2} \sin θ_{0} d θ_{0} d ϕ_{0}, \end{matrix}$

(13.71)

where the Green's function in axisymmetric spherical coordinates is given from Eq. (13.62) by

{g (r, θ | r_{0}, θ_{0}) |}_{r_{0} = R} = \frac{- j k}{4 π} \sum_{n = 0}^{\infty} (2 n + 1) P_{n} (\cos θ_{0}) P_{n} (\cos θ) j_{n} (k R) h_{n}^{(2)} (k r),

${g (r, θ | r_{0}, θ_{0}) |}_{r_{0} = R} = \frac{- j k}{4 π} \sum_{n = 0}^{\infty} (2 n + 1) P_{n} (\cos θ_{0}) P_{n} (\cos θ) j_{n} (k R) h_{n}^{(2)} (k r),$

(13.72)

and its normal gradient is given by

\frac{\partial}{\partial r_{0}} {g (r, θ | r_{0}, θ_{0}) |}_{r_{0} = R} = \frac{- j k}{4 π} \sum_{n = 0}^{\infty} (2 n + 1) P_{n} (\cos θ_{0}) P_{n} (\cos θ) j_{n}^{'} (k R) h_{n}^{(2)} (k r),

$\frac{\partial}{\partial r_{0}} {g (r, θ | r_{0}, θ_{0}) |}_{r_{0} = R} = \frac{- j k}{4 π} \sum_{n = 0}^{\infty} (2 n + 1) P_{n} (\cos θ_{0}) P_{n} (\cos θ) j_{n}^{'} (k R) h_{n}^{(2)} (k r),$

(13.73)

where the derivative of the spherical Bessel function is given by Eq. (12.31). We see from Eq. (13.71) that we have a superposition of two fields. The first integral (monopole) represents the incident sound field due to the velocity source, formed by the cap. The normal pressure gradient, or velocity distribution, is obtained from the boundary conditions at the surface of the sphere:

\frac{\partial}{\partial r_{0}} \tilde{p} {(r_{0}, θ_{0}) |}_{r_{0} = R} = {\begin{matrix} - j k ρ_{0} c {\tilde{u}}_{0}, & 0 \leq θ \leq α \\ 0, & α < θ \leq π . \end{matrix}

$\frac{\partial}{\partial r_{0}} \tilde{p} {(r_{0}, θ_{0}) |}_{r_{0} = R} = {\begin{matrix} - j k ρ_{0} c {\tilde{u}}_{0}, & 0 \leq θ \leq α \\ 0, & α < θ \leq π . \end{matrix}$

(13.74)

The second integral (dipole) represents the sound field reflected by the sphere. The surface pressure distribution, which is a function of θ ₀, is not yet known and is thus represented as a Legendre series:

\tilde{p} {(r_{0}, θ_{0}) |}_{r_{0} = R} = ρ_{0} c {\tilde{u}}_{0} \sum_{m = 0}^{\infty} A_{m} P_{m} (\cos θ_{0}),

$\tilde{p} {(r_{0}, θ_{0}) |}_{r_{0} = R} = ρ_{0} c {\tilde{u}}_{0} \sum_{m = 0}^{\infty} A_{m} P_{m} (\cos θ_{0}),$

(13.75)

where the unknown coefficients A _m have to be determined. Inserting Eqs. (13.72)–(13.75) into Eq. (13.71) yields

\begin{matrix} \tilde{p} (r, θ) = - k^{2} R^{2} ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} (n + \frac{1}{2}) P_{n} (\cos θ) j_{n} (k R) h_{n}^{(2)} (k r) \int_{0}^{α} P_{n} (\cos θ_{0}) \sin θ_{0} d θ_{0} \\ + j k R^{2} ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} (n + \frac{1}{2}) P_{n} (\cos θ) j_{n}^{'} (k R) h_{n}^{(2)} (k r) \sum_{m = 0}^{\infty} A_{m} \int_{0}^{π} P_{m} (\cos θ_{0}) P_{n} (\cos θ_{0}) \sin θ_{0} d θ_{0}, \end{matrix}

$\begin{matrix} \tilde{p} (r, θ) = - k^{2} R^{2} ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} (n + \frac{1}{2}) P_{n} (\cos θ) j_{n} (k R) h_{n}^{(2)} (k r) \int_{0}^{α} P_{n} (\cos θ_{0}) \sin θ_{0} d θ_{0} \\ + j k R^{2} ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} (n + \frac{1}{2}) P_{n} (\cos θ) j_{n}^{'} (k R) h_{n}^{(2)} (k r) \sum_{m = 0}^{\infty} A_{m} \int_{0}^{π} P_{m} (\cos θ_{0}) P_{n} (\cos θ_{0}) \sin θ_{0} d θ_{0}, \end{matrix}$

(13.76)

where the integrals can be solved using the identities of Eqs. (66) and (69) from Appendix II to give

\tilde{p} (r, θ) = k R^{2} ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} P_{n} (\cos θ) h_{n}^{(2)} (k r) (j A_{n} j_{n}^{'} (k r) - k (n + \frac{1}{2}) j_{n} (k R) \sin α P_{n}^{- 1} (\cos α)) .

$\tilde{p} (r, θ) = k R^{2} ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} P_{n} (\cos θ) h_{n}^{(2)} (k r) (j A_{n} j_{n}^{'} (k r) - k (n + \frac{1}{2}) j_{n} (k R) \sin α P_{n}^{- 1} (\cos α)) .$

(13.77)

To solve for the unknown coefficients A _n, we apply the following boundary condition to the above pressure field:

\frac{\partial}{\partial r} \tilde{p} {(r, θ) |}_{r = R} = - j k ρ_{0} c \tilde{u} (R, θ) = {\begin{matrix} - j ρ_{0} c {\tilde{u}}_{0}, & 0 \leq θ \leq α \\ 0, & α < θ \leq π . \end{matrix}

$\frac{\partial}{\partial r} \tilde{p} {(r, θ) |}_{r = R} = - j k ρ_{0} c \tilde{u} (R, θ) = {\begin{matrix} - j ρ_{0} c {\tilde{u}}_{0}, & 0 \leq θ \leq α \\ 0, & α < θ \leq π . \end{matrix}$

(13.78)

The surface velocity can be represented by the following Legendre series:

\tilde{u} (R, θ) = {\tilde{u}}_{0} \sum_{n = 0}^{\infty} B_{n} P_{n} (\cos θ),

$\tilde{u} (R, θ) = {\tilde{u}}_{0} \sum_{n = 0}^{\infty} B_{n} P_{n} (\cos θ),$

(13.79)

where the coefficients B _n are found by multiplying through by the orthogonal function P _m(cos θ) and integrating over the surface as follows:

\int_{0}^{α} P_{n} (\cos θ) \sin θ d θ = \sum_{n = 0}^{\infty} B_{n} \int_{0}^{π} P_{m} (\cos θ) P_{n} (\cos θ) \sin θ d θ

$\int_{0}^{α} P_{n} (\cos θ) \sin θ d θ = \sum_{n = 0}^{\infty} B_{n} \int_{0}^{π} P_{m} (\cos θ) P_{n} (\cos θ) \sin θ d θ$

(13.80)

and applying the identities of Eqs. (66) and (69) from Appendix II to yield

B_{n} = (n + \frac{1}{2}) \sin α P_{n}^{- 1} (\cos α) .

$B_{n} = (n + \frac{1}{2}) \sin α P_{n}^{- 1} (\cos α) .$

(13.81)

The coefficients are finally solved by applying Eq. (13.78) to Eq. (13.77) and equating the coefficients of P _n(cos θ) to give

A_{n} = - (n + \frac{1}{2}) \sin α P_{n}^{- 1} (\cos α) \frac{1 + j k R^{2} j_{n} (k R) h_{n}^{' (2)} (k R)}{R^{2} j_{n}^{'} (k R) h_{n}^{' (2)} (k R)},

$A_{n} = - (n + \frac{1}{2}) \sin α P_{n}^{- 1} (\cos α) \frac{1 + j k R^{2} j_{n} (k R) h_{n}^{' (2)} (k R)}{R^{2} j_{n}^{'} (k R) h_{n}^{' (2)} (k R)},$

(13.82)

which, after inserting into Eq. (13.77), gives

\tilde{p} (r, θ) = - j k ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} (n + \frac{1}{2}) \sin α P_{n}^{- 1} (\cos α) P_{n} (\cos θ) \frac{h_{n}^{(2)} (k r)}{h_{n}^{' (2)} (k R)} .

$\tilde{p} (r, θ) = - j k ρ_{0} c {\tilde{u}}_{0} \sum_{n = 0}^{\infty} (n + \frac{1}{2}) \sin α P_{n}^{- 1} (\cos α) P_{n} (\cos θ) \frac{h_{n}^{(2)} (k r)}{h_{n}^{' (2)} (k R)} .$

(13.83)

This is exactly the same equation as would be obtained using the boundary value method described in Section 12.6. In the far field, applying the asymptotic expression for the spherical Hankel function from Eq. (12.18) gives

\tilde{p} (r, θ) = - j k ρ_{0} c S \frac{{\tilde{u}}_{0}}{4 π r} e^{- j k r} D (θ),

$\tilde{p} (r, θ) = - j k ρ_{0} c S \frac{{\tilde{u}}_{0}}{4 π r} e^{- j k r} D (θ),$

(13.84)

where the directivity function is given by

D (θ) = \frac{\sin α}{2 k^{2} R^{2} \sin^{2} (α / 2)} \sum_{n = 0}^{\infty} \frac{j^{n + 1} {(2 n + 1)}^{2} P_{n}^{- 1} (\cos α) P_{n} (\cos θ)}{n h_{n - 1}^{(2)} (k R) - (n + 1) h_{n + 1}^{(2)} (k R)},

$D (θ) = \frac{\sin α}{2 k^{2} R^{2} \sin^{2} (α / 2)} \sum_{n = 0}^{\infty} \frac{j^{n + 1} {(2 n + 1)}^{2} P_{n}^{- 1} (\cos α) P_{n} (\cos θ)}{n h_{n - 1}^{(2)} (k R) - (n + 1) h_{n + 1}^{(2)} (k R)},$

(13.85)

and S = 4πR ²sin² α/2. The radiation impedance is given by

\begin{matrix} Z_{s} = \frac{\tilde{F}}{{\tilde{U}}_{0}} = \frac{2 π R^{2}}{S {\tilde{u}}_{0}} \int_{0}^{α} \tilde{p} (r, θ) \sin θ d θ \\ = - j ρ_{0} c \frac{\sin^{2} α}{\sin^{2} (α / 2)} \sum_{n = 0}^{\infty} \frac{{(n + \frac{1}{2})}^{2} {(P_{n}^{- 1} (\cos α))}^{2} h_{n}^{(2)} (k R)}{n h_{n - 1}^{(2)} (k R) - (n + 1) h_{n + 1}^{(2)} (k R)} . \end{matrix}

$\begin{matrix} Z_{s} = \frac{\tilde{F}}{{\tilde{U}}_{0}} = \frac{2 π R^{2}}{S {\tilde{u}}_{0}} \int_{0}^{α} \tilde{p} (r, θ) \sin θ d θ \\ = - j ρ_{0} c \frac{\sin^{2} α}{\sin^{2} (α / 2)} \sum_{n = 0}^{\infty} \frac{{(n + \frac{1}{2})}^{2} {(P_{n}^{- 1} (\cos α))}^{2} h_{n}^{(2)} (k R)}{n h_{n - 1}^{(2)} (k R) - (n + 1) h_{n + 1}^{(2)} (k R)} . \end{matrix}$

(13.86)

..................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

13.5. Boundary integral method case study: radially pulsating cap in a rigid sphere

Table of Contents for
Acoustics