Point-Source Speakers

Consider a spherical secondary source (``point source'') located at ${\ensuremath \underline{x}}={\ensuremath \underline{0}}$ as shown in Fig.15. Then the pressure amplitude along the listening line $(x,0,z_l)^T$ for all $x$ is given by

$$p_l(x) = p_l(0) \frac{z_l}{\sqrt{z_l^2 + x^2}} \isdefs \frac{p_l(0)}{\sqrt{1 + \left(\frac{x}{z_l}\right)^2}} \isdefs \pi_l(\theta) = \frac{p_l(0)}{\sqrt{1 + \tan^2(\theta)}} \protect$$ (9)

where $p_l(0)$ denotes the pressure amplitude observed from the point source at ${\ensuremath \underline{x}}=(0,0,z_l)^T$ , and $\theta \isdeftext \tan^{-1}(x/z_l)$ denotes the angle of the line from the source center to the line-array point at $x$ , i.e., ${\ensuremath \underline{x}}=(x,0,z_l)^T$ . This polar-pattern slice is plotted in Fig.16 for the triangular array+listener geometry such as described in §4.10 on page , with $p_l(0)\isdeftext 1$ . The top plot shows $p_l(x)$ over a 12 m width centered about a 6 m wide array, and the bottom plot shows $\pi_l(\theta)$ for $\theta\in[-\pi/2,\pi/2]$ , thereby covering the entire axis containing array.

Figure 16: Point-source polar-pattern slice for a listening-line $z_l=\lambda$ away from the source. Top: Gain $p_l$ versus position $x$ . Bottom: Gain $\pi _l$ versus angle $\theta \isdeftext \tan^{-1}(x/z_l)$ .

In addition to the gain variation $p_l(x)$ in Eq.(9), we also have phase effects that we don't have when keeping radius $r$ constant and only varying $\theta$ . The envelope $p_l(x)$ in Fig.16 is due only to spherical spreading loss according to $1/r$ . The difference in propagation distance between $p_l(0)$ and $p_l(x)$ is

$$r_d(x) \isdef r_l(x)-r_l(0) = r_l(x)-z_l = \sqrt{z_l^2+x^2}-z_l = z_l\left[\sqrt{1+\tan^2(\theta)}-1\right]. \protect$$ (10)

The resulting frequency-response re$\left\{p_l(x)e^{jkr_d(x)}\right\}$ is plotted in Fig.17 for 1 kHz (wavelength about a foot (1.126 ft)) and $p_l(0)=1$ , where $k=\omega/c=2\pi\cdot 1000\,/\,343$ is the wavenumber (spatial radian frequency), which has made its appearance for the first time in these formulas.³⁴

Figure 17: Real part of point-source frequency-response for a listening-line one wavelength away ( $z_l=\lambda$ ). Top: Gain $p_l$ versus position $x$ . Bottom: Gain $\pi _l$ versus angle $\theta \isdeftext \tan^{-1}(x/z_l)$ .

For small $\vert\theta\vert$ , we can use the approximation

$$r_d(x) \approx \frac{1}{2}\,z_l\,\tan^2(\theta) \eqsp \frac{x^2}{2z_l}.$$