25-Nav-A2 Hydrodynamics of Ships (I)_ Resistance and Propulsion · December 2019
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Exams — December 2019 — 16-Nav-A2 Hydrodynamics of Ships I: Resistance and Propulsion. Three-hour, closed-book exam; a data sheet, a propeller (Wageningen B4-55) chart and a Burrill cavitation chart are supplied. Format: Questions 1–7 are compulsory (attempt all seven), then one of Questions 8 or 9. All nine are solved below for completeness. Units follow the paper (mixed SI, with the historic Imperial-unit legend that duplicates on the supplied Burrill sheet noted where relevant).
Reference texts: Larsson & Raven, Ship Resistance and Flow (SNAME) — model-scale resistance testing, Froude/Reynolds scaling and the ITTC 1978 performance-prediction method; Lewis (ed.), Principles of Naval Architecture, Vol. II — Resistance, Propulsion and Vibration (SNAME) — propeller geometry, open-water B-series design and cavitation; Carlton, Marine Propellers and Propulsion (Butterworth-Heinemann) — Wageningen B-series charts and the Burrill back-cavitation criterion.
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
(i) A single pressure point moving in a straight line at constant speed across a deep-water free surface generates a steady wave pattern — the Kelvin wave system — that remains entirely confined within a wedge of half-angle $19.47^\circ$ ($=\arcsin(1/3)$) measured from the track, regardless of the disturbance's speed. The pattern is the superposition of two distinct wave families: transverse waves, whose crests are (nearly) perpendicular to the sailing line and which travel with the disturbance, and diverging waves, whose crests are oblique to the track, radiating outward and curving ("feathering") until they meet the transverse system at cusp points that trace out the $19.47^\circ$ wedge boundary itself.
A real ship is treated (to first order) as a small number of moving pressure singularities — a bow high-pressure system, a stern high-pressure (or shoulder) system, and sometimes a stern low-pressure system — each generating its own Kelvin wave pattern; the ship's total wave pattern, and hence its wave-making resistance, is the superposition (interference) of these individual systems.
(ii) Constructive interference occurs when the transverse-wave crest generated at the bow arrives at the stern in phase with (reinforcing) the stern's own wave system, so that the two wave trains add constructively downstream and the ship radiates a larger net wave amplitude — and hence experiences a local hump in its wave-making resistance curve. Because the deep-water transverse wavelength is $\lambda_W=2\pi V^2/g$ (per the data sheet), and the effective distance between the bow and stern wave-making centres is of order the waterline length $L$, constructive interference recurs at specific speeds (Froude numbers) for which $L$ is close to an odd multiple of a half-wavelength of the transverse system, $L\approx(2m-1)\lambda_W/2$ ($m=1,2,\dots$) — these are the classic "humps" in the resistance curve. Conversely, when $L$ is close to a whole multiple of $\lambda_W$, the bow and stern systems tend to cancel (destructive interference), producing the corresponding "hollows." Naval architects deliberately choose design Froude numbers to sit near a hollow rather than a hump where practical.
