Accuracy of the Far-Field Approximation for the Underwater Sound Radiated When Immersed Steel and Lead Piles (with Non-reflective Toes) are Driven by a Harmonic Axisymmetric Force

Abstract

Steel and lead piles immersed in seawater and their toes made non-reflective (for clarity of the results) are driven by an axisymmetric harmonic force. Axial and radial vibrations travel from head to toe. Radial vibration, which radiates sound into the surrounding medium, is computed using Membrane thin-shell vibration theory. At 1 kHz, a typical construction steel pile (radius 38 cm, wall thickness 2.5 cm) has a phase velocity of 4943 m/s and in cold seawater (sound speed 1485 m/s), the Mach wave radiates at $$a\hbox {cos}(1485 /4943) = 73^{\circ }$$ from the pile axis. According to ray theory, a Mach wave is received at any position providing a $$73^{\circ }$$ line from that position intersects the pile (at what was the emitting point). The vibration energy at this emitting point has travelled there from the pile head. For a given slant range (R) to a receiver, ray theory predicts that a Mach wave is not received below a minimum colatitude (COMIN). For a typical steel pile, it is found that as R increases beyond the pile length (L), COMIN increases rapidly from $$0^{\circ }$$ , passes through $$68^{\circ }$$ at $$10 \times L$$ , and asymptotes to $$73^{\circ }$$ as R increases further. For the same pile made of lead, the phase velocity at 1 kHz is 1332 m/s, and no Mach wave is radiated. For both steel and lead piles, radiated SPLs were calculated using both exact and far-field approximate radiation theories as functions of colatitude, at slant ranges from 10 to 1000 m. For the steel piles, the far-field approximate model (which omits the Mach wave) underestimates SPL by up to 20 dB if the receiver’s colatitude exceeds COMIN. For the lead piles however, the far-field approximate model is accurate to within 1 dB.