going to either zero or infinity. Using this simple model, a receiver located within the deep-sound channel would not be expected to receive noise arriving along paths near the horizontal since range-independent ray tracing would show that such ray paths do not exist. The beam pattern of the noise from monopole sources would therefore have a maximum at an oblique angle above and below the horizontal (typically ±10° to ±15° off the horizontal axis in temperate, deep-sea regions), with a minimum at 0 = 90°. This simple conceptual picture is not always valid in realistic (range-dependent) environments. Specifically, more sophisticated models would consider the effects of range-dependent refraction, bottom reflection and multipath arrivals. This aspect will be discussed further under the topic of the noise notch (Section 7.5).
For high-frequency noise sources (>1,000Hz), many investigators (e.g. Anderson, 1958; Becken, 1961; Von Winkle, 1963) have suggested a function of the form I(0) = I0 cosm 0, where I0 is the intensity radiated by a small area of the sea surface in the downward direction (0 = 0°) and where m is an integer. Values of m = 1,2 or 3 have been obtained, depending upon conditions and methods of measurement. Most measurements center roughly about m = 2, a value consistent with the hypothesis of a dipole source formed by the actual source and its image in the sea surface.
For low-frequency noise sources (<500 Hz), the agreement between available observations (see Figure 6.7) and the findings of this simple model (Figure 7.1) further suggests that a distribution of monopoles [m = 0; thus, I(0) = I0] adequately models distant shipping noise.
Pertinent examples from the ANDES noise model (Renner, 1995b) will illustrate how the horizontal and vertical noise directionalities are computed. The fundamental output from ANDES is the directional noise intensity per unit solid angle [Ns(0,0)]. The horizontal noise directionality [N(0)] is calculated from [Ns(0,0)] as:
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