Acoustics

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Part XXXVII: Radiation theorems, radiation in rectangular-spherical coordinates, mutual impedance

13.13. The Bouwkamp impedance theorem [36]

To find the radiation impedance of a piston in an infinite baffle, we used an expression for the near-field pressure and integrated the pressure over the surface of the piston to find the total force. However, the far-field pressure is generally given by a much simpler expression. According to Bouwkamp's impedance theorem, if the acoustic medium is loss free, we can obtain the radiation resistance of a piston, or combination of pistons, of any shape, vibrating in an infinite baffle or free space (assuming antisymmetry) as follows: We integrate the square of the far-field pressure over a hemispherical surface, while letting the radius tend to infinity, and then divide the result by the specific impedance of free space to obtain the total radiated power. We then obtain the acoustic radiation resistance by dividing the power by the square of the volume velocity of the piston(s). Furthermore, we can obtain the radiation reactance by integrating the square of the far-field pressure over certain complex values of the spherical inclination angle θ. In general, the far-field pressure is given in spherical coordinates (r, θ, ϕ) by

\tilde{p} (r, θ, ϕ) = \frac{j k ρ_{0} c {\tilde{U}}_{0}}{2 π r} D (θ, ϕ),

$\tilde{p} (r, θ, ϕ) = \frac{j k ρ_{0} c {\tilde{U}}_{0}}{2 π r} D (θ, ϕ),$

(13.269)

although, in the case of an axisymmetric source such as the circular piston, there is no ϕ dependency. The total radiated power W is given by

W = {| \frac{{\tilde{U}}_{0}}{\sqrt{2}} |}^{2} R_{A R} = \frac{1}{ρ_{0} c} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} {| \frac{\tilde{p} (r, θ, φ)}{\sqrt{2}} |}^{2} {r^{2} \sin θ d θ d φ |}_{r \to \infty},

$W = {| \frac{{\tilde{U}}_{0}}{\sqrt{2}} |}^{2} R_{A R} = \frac{1}{ρ_{0} c} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} {| \frac{\tilde{p} (r, θ, φ)}{\sqrt{2}} |}^{2} {r^{2} \sin θ d θ d φ |}_{r \to \infty},$

(13.270)

where R _AR is the acoustic radiation resistance of the source and

{\tilde{U}}_{0}

${\tilde{U}}_{0}$

is its total volume velocity. Hence the specific radiation resistance is given by

R_{s} = S R_{A R} = \frac{k^{2} ρ_{0} c S}{4 π^{2}} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} {| D (θ, ϕ) |}^{2} \sin θ d θ d ϕ .

$R_{s} = S R_{A R} = \frac{k^{2} ρ_{0} c S}{4 π^{2}} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} {| D (θ, ϕ) |}^{2} \sin θ d θ d ϕ .$

(13.271)

Also, the specific radiation reactance is given by

X_{s} = S X_{A R} = - j \frac{k^{2} ρ_{0} c S}{4 π^{2}} \int_{0}^{2 π} \int_{\frac{π}{2} + j 0}^{\frac{π}{2} + j \infty} {| D (θ, ϕ) |}^{2} \sin θ d θ d ϕ .

$X_{s} = S X_{A R} = - j \frac{k^{2} ρ_{0} c S}{4 π^{2}} \int_{0}^{2 π} \int_{\frac{π}{2} + j 0}^{\frac{π}{2} + j \infty} {| D (θ, ϕ) |}^{2} \sin θ d θ d ϕ .$

(13.272)

It is fairly straightforward to verify this result by inserting the directivity function of Eq. (13.102) together with k _w = k sin θ into Eqs. (13.271) and (13.272). In this way, the expressions for the radiation impedance given by Eqs. (13.118) and (13.117) can be duplicated, bearing in mind that sin(π/2 + j∞) = cos j∞ = cosh ∞ = ∞. Of course, this theorem is not limited to radiators with uniform surface velocity. Bouwkamp's expression [36] includes the square of average surface velocity divided by the square of the velocity at some reference point, although we have omitted this here. We will use this theorem to derive an expression for the radiation impedance of a rectangular piston in an infinite baffle.

We can extend Bouwkamp's impedance theorem to a pressure (resilient) source, where the far-field pressure is given by

\tilde{p} (r, θ, ϕ) = \frac{j k S {\tilde{p}}_{0}}{4 π r} D (θ, ϕ)

$\tilde{p} (r, θ, ϕ) = \frac{j k S {\tilde{p}}_{0}}{4 π r} D (θ, ϕ)$

so that the total radiated power is given by

W = {| \frac{{\tilde{p}}_{0}}{2 \sqrt{2}} |}^{2} G_{A R} = \frac{1}{ρ_{0} c} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} {| \frac{\tilde{p} (r, θ, φ)}{\sqrt{2}} |}^{2} {r^{2} \sin θ d θ d φ |}_{r \to \infty},

$W = {| \frac{{\tilde{p}}_{0}}{2 \sqrt{2}} |}^{2} G_{A R} = \frac{1}{ρ_{0} c} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} {| \frac{\tilde{p} (r, θ, φ)}{\sqrt{2}} |}^{2} {r^{2} \sin θ d θ d φ |}_{r \to \infty},$

where G _AR is the acoustic radiation conductance of the source and

{\tilde{p}}_{0} / 2

${\tilde{p}}_{0} / 2$

is the driving pressure on one side only of the baffle or plane. Hence the specific radiation conductance is given by

G_{S} = \frac{G_{A R}}{S} = \frac{k^{2} S}{4 π^{2} ρ_{0} c} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} {| D (θ, ϕ) |}^{2} \sin θ d θ d φ

$G_{S} = \frac{G_{A R}}{S} = \frac{k^{2} S}{4 π^{2} ρ_{0} c} \int_{0}^{2 π} \int_{0}^{\frac{π}{2}} {| D (θ, ϕ) |}^{2} \sin θ d θ d φ$

and the specific radiation susceptance is given by

B_{S} = \frac{B_{A R}}{S} = j \frac{k^{2} S}{4 π^{2} ρ_{0} c} \int_{0}^{2 π} \int_{\frac{π}{2} + j 0}^{\frac{π}{2} + j \infty} {| D (θ, ϕ) |}^{2} \sin θ d θ d φ .

$B_{S} = \frac{B_{A R}}{S} = j \frac{k^{2} S}{4 π^{2} ρ_{0} c} \int_{0}^{2 π} \int_{\frac{π}{2} + j 0}^{\frac{π}{2} + j \infty} {| D (θ, ϕ) |}^{2} \sin θ d θ d φ .$

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Part XXXVII: Radiation theorems, radiation in rectangular-spherical coordinates, mutual impedance

13.13. The Bouwkamp impedance theorem [36]

Table of Contents for
Acoustics