Acoustics

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2.10. Freely traveling spherical wave

Sound pressure

A solution to the spherical wave Eq. (2.25) is

\tilde{p} (r) = {\tilde{A}}_{+} \frac{e^{- j k r}}{r} + {\tilde{A}}_{-} \frac{e^{j k r}}{r},

$\tilde{p} (r) = {\tilde{A}}_{+} \frac{e^{- j k r}}{r} + {\tilde{A}}_{-} \frac{e^{j k r}}{r},$

(2.134)

where

{\tilde{A}}_{+}

${\tilde{A}}_{+}$

is the amplitude of the sound pressure in the outgoing wave at unit distance from the center of the sphere and

{\tilde{A}}_{-}

${\tilde{A}}_{-}$

is the same for the reflected wave. This equation can also be written in terms of spherical Hankel functions h ₀ ⁽¹⁾(x) and h ₀ ⁽²⁾(x)

\tilde{p} (r) = - j k ({\tilde{A}}_{+} h_{0}^{(2)} (k r) - {\tilde{A}}_{-} h_{0}^{(1)} (k r)),

$\tilde{p} (r) = - j k ({\tilde{A}}_{+} h_{0}^{(2)} (k r) - {\tilde{A}}_{-} h_{0}^{(1)} (k r)),$

(2.135)

which are also known as Hankel functions of fractional order, as defined by

h_{0}^{(1)} (x) = j_{0} (x) + j y_{0} (x),

$h_{0}^{(1)} (x) = j_{0} (x) + j y_{0} (x),$

(2.136)

h_{0}^{(2)} (x) = j_{0} (x) - j y_{0} (x),

$h_{0}^{(2)} (x) = j_{0} (x) - j y_{0} (x),$

(2.137)

j_{0} (x) = \frac{\sin x}{x},

$j_{0} (x) = \frac{\sin x}{x},$

(2.138)

y_{0} (x) = - \frac{\cos x}{x},

$y_{0} (x) = - \frac{\cos x}{x},$

(2.139)

where j ₀(x) and y ₀(x) are spherical Bessel functions of the first and second kind respectively, as plotted in Fig. 2.13. The “2” in parentheses denotes an outgoing spherical wave and the “1” denotes an incoming one. These spherical Bessel functions are related to the cylindrical Bessel functions of half-integer order

J_{\frac{1}{2}} (x)

$J_{\frac{1}{2}} (x)$

and

Y_{\frac{1}{2}} (x)

$Y_{\frac{1}{2}} (x)$

j_{0} (x) = \sqrt{\frac{π}{2 x}} J_{\frac{1}{2}} (x),

$j_{0} (x) = \sqrt{\frac{π}{2 x}} J_{\frac{1}{2}} (x),$

(2.140)

Figure 2.13 Spherical Bessel functions of the first (black curve) and second (gray curve) kind.

y_{0} (x) = \sqrt{\frac{π}{2 x}} Y_{\frac{1}{2}} (x) .

$y_{0} (x) = \sqrt{\frac{π}{2 x}} Y_{\frac{1}{2}} (x) .$

(2.141)

We can see that spherical waves differ from cylindrical ones in two respects: first, the radial wavelength remains constant as they progress, as is the case with plane waves; second, although they decay in amplitude as they spread out, they adopt a direct inverse law in the far field. The latter makes sense when we consider that the area of the wave front is proportional to the square of the radial distance r. The radiated power is the intensity multiplied by the area, where the intensity is given by Eq. (1.12). The intensity, in turn, is proportional to the square of the pressure and therefore inversely proportional to the square of the radial distance. Hence the power remains constant.

If there are no reflecting surfaces in the medium, only the first term of this equation is needed, i.e.,

\tilde{p} (r) = {\tilde{A}}_{+} \frac{e^{- j k r}}{r} .

$\tilde{p} (r) = {\tilde{A}}_{+} \frac{e^{- j k r}}{r} .$

(2.142)

Particle velocity

With the aid of Eq. (2.4b), solve for the particle velocity in the r direction:

\begin{matrix} \tilde{u} (r) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial r} \tilde{p} (r) \\ = \frac{{\tilde{A}}_{+}}{ρ_{0} c} (1 + \frac{1}{j k r}) \frac{e^{- j k r}}{r} . \end{matrix}

$\begin{matrix} \tilde{u} (r) = \frac{1}{- j k ρ_{0} c} \frac{\partial}{\partial r} \tilde{p} (r) \\ = \frac{{\tilde{A}}_{+}}{ρ_{0} c} (1 + \frac{1}{j k r}) \frac{e^{- j k r}}{r} . \end{matrix}$

(2.143)

Specific acoustic impedance

The specific acoustic impedance is found from Eq. (2.142) divided by Eq. (2.143),

Z_{s} = \frac{\tilde{p} (r)}{\tilde{u} (r)} = ρ_{0} c \frac{j k r}{1 + j k r} = \frac{ρ_{0} c k r}{\sqrt{1 + k^{2} r^{2}}} / 90 ° - \tan^{- 1} k r rayls .

$Z_{s} = \frac{\tilde{p} (r)}{\tilde{u} (r)} = ρ_{0} c \frac{j k r}{1 + j k r} = \frac{ρ_{0} c k r}{\sqrt{1 + k^{2} r^{2}}} / 90 ° - \tan^{- 1} k r rayls .$

(2.144)

Plots of the magnitude and phase angle of the impedance as a function of kr are given in Figs. 2.14 and 2.15 respectively.

For large values of kr, that is, for large distances or for high frequencies, this equation becomes, approximately,

Z_{s} \approx ρ_{0} c rayls .

$Z_{s} \approx ρ_{0} c rayls .$

(2.145)

The impedance here is nearly purely resistive and approximately equal to the characteristic impedance for a plane freely traveling wave. In other words, the specific acoustic impedance for a large distance from a spherical source in free space is nearly equal to that in a tube in which no reflections occur from the end opposite the source.

Figure 2.14 Plot of the magnitude of the specific acoustic-impedance ratio |Z _s|/(ρ ₀ c) in a spherical freely traveling wave as a function of kr, where k is the wave-number equal to ω/c or 2π/λ and r is the distance from the center of the spherical source. |Z _s| is the magnitude of ratio of pressure to particle velocity in a spherical free-traveling wave, and ρ ₀ c is the characteristic impedance of air.

Figure 2.15 Plot of the phase angle, in degrees, of the specific acoustic-impedance ratio |Z _s|/ρ ₀ c in a spherical wave as a function of kr, where k is the wave number ω/c or 2π/λ and r is the distance from the center of the spherical source.

The important steady-state relations derived in this chapter are summarized in Table 2.2.

Table 2.2

General and steady-state relations for small-signal sound propagation in gases
Name	General equation	Steady-state equation
Wave equation in p or u	$\begin{matrix} \frac{\partial^{2} ()}{\partial x^{2}} = \frac{1}{c^{2}} \frac{\partial^{2} ()}{\partial t^{2}} \\ \nabla^{2} () = \frac{1}{c^{2}} \frac{\partial^{2} ()}{\partial t^{2}} \\ \frac{\partial^{2} (p r)}{\partial r^{2}} = \frac{1}{c^{2}} \frac{\partial^{2} (p r)}{\partial t^{2}} \end{matrix}$ $\begin{matrix} \frac{\partial^{2} ()}{\partial x^{2}} = \frac{1}{c^{2}} \frac{\partial^{2} ()}{\partial t^{2}} \\ \nabla^{2} () = \frac{1}{c^{2}} \frac{\partial^{2} ()}{\partial t^{2}} \\ \frac{\partial^{2} (p r)}{\partial r^{2}} = \frac{1}{c^{2}} \frac{\partial^{2} (p r)}{\partial t^{2}} \end{matrix}$	$\begin{matrix} \frac{\partial^{2} ()}{\partial x^{2}} = - \frac{ω^{2}}{c^{2}} () \\ \nabla^{2} () = - \frac{ω^{2}}{c^{2}} () \\ \nabla^{2} (p r) = - \frac{ω^{2}}{c^{2}} (p r) \end{matrix}$ $\begin{matrix} \frac{\partial^{2} ()}{\partial x^{2}} = - \frac{ω^{2}}{c^{2}} () \\ \nabla^{2} () = - \frac{ω^{2}}{c^{2}} () \\ \nabla^{2} (p r) = - \frac{ω^{2}}{c^{2}} (p r) \end{matrix}$
Equation of motion	$\begin{matrix} \frac{\partial p}{\partial x} = - ρ_{0} \frac{\partial u}{\partial t} \\ grad p = - ρ_{0} \frac{\partial q}{\partial t} \end{matrix}$ $\begin{matrix} \frac{\partial p}{\partial x} = - ρ_{0} \frac{\partial u}{\partial t} \\ grad p = - ρ_{0} \frac{\partial q}{\partial t} \end{matrix}$	$\begin{matrix} u = \frac{- 1}{j ω ρ_{0}} \frac{\partial p}{\partial x} \\ p = - j ω ρ_{0} \int u d x \\ grad p = - j ω ρ_{0} q \end{matrix}$ $\begin{matrix} u = \frac{- 1}{j ω ρ_{0}} \frac{\partial p}{\partial x} \\ p = - j ω ρ_{0} \int u d x \\ grad p = - j ω ρ_{0} q \end{matrix}$
Displacement	ξ = ∫u dt ξ = ∫q dt	$ξ = \frac{u}{j ω}$ $ξ = \frac{u}{j ω}$ $ξ = \frac{q}{j ω}$ $ξ = \frac{q}{j ω}$
Incremental density	$\begin{matrix} ρ = \frac{ρ_{0}}{γ P_{0}} p = \frac{p}{c^{2}} \\ \frac{\partial ρ}{\partial t} = - ρ_{0} \frac{\partial u}{\partial x} \end{matrix}$ $\begin{matrix} ρ = \frac{ρ_{0}}{γ P_{0}} p = \frac{p}{c^{2}} \\ \frac{\partial ρ}{\partial t} = - ρ_{0} \frac{\partial u}{\partial x} \end{matrix}$	$\begin{matrix} ρ = \frac{ρ_{0}}{γ P_{0}} p = \frac{p}{c^{2}} \\ ρ = - \frac{ρ_{0}}{j ω} \frac{\partial u}{\partial x} \end{matrix}$ $\begin{matrix} ρ = \frac{ρ_{0}}{γ P_{0}} p = \frac{p}{c^{2}} \\ ρ = - \frac{ρ_{0}}{j ω} \frac{\partial u}{\partial x} \end{matrix}$
Incremental temperature	$Δ T = \frac{T_{0}}{P_{0}} \frac{γ - 1}{γ} p$ $Δ T = \frac{T_{0}}{P_{0}} \frac{γ - 1}{γ} p$	$Δ T = \frac{T_{0}}{P_{0}} \frac{γ - 1}{γ} p$ $Δ T = \frac{T_{0}}{P_{0}} \frac{γ - 1}{γ} p$