25-Nav-A2 Hydrodynamics of Ships (I)_ Resistance and Propulsion · Undated paper

Question 4 of 10: Transverse Wave Interference and the Corresponding Froude Number

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

National Exams — May 2019 — 16-Nav-A2 Hydrodynamics of Ships I: Resistance and Propulsion. Three-hour, closed-book exam; a data sheet, a Wageningen B4-55 propeller chart and a Burrill cavitation chart are supplied. Format: Questions 1–8 are compulsory (attempt all eight), then one of Questions 9 or 10. All ten are solved below for completeness.

Reference texts: Larsson & Raven, Ship Resistance and Flow (SNAME) — model-scale resistance testing, Froude/Reynolds scaling, boundary-layer estimates and the ITTC 1978 performance-prediction method; Lewis (ed.), Principles of Naval Architecture, Vol. II — Resistance, Propulsion and Vibration (SNAME) — propeller geometry, wave-pattern interference, open-water B-series design and wake-induced blade loading; Carlton, Marine Propellers and Propulsion (Butterworth-Heinemann) — Wageningen B-series charts, the Burrill back-cavitation criterion and open-water model testing.

Question 4: Transverse Wave Interference and the Corresponding Froude Number (10 marks)

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.

Given. A slender body of length $L$ moving at constant speed $V$, generating a Kelvin wave system; transverse wavelength condition $L_{WT}=\dfrac{2L}{3}$.

Find. The transverse/divergent wave pattern and interference condition at the bow and stern, and the Froude number corresponding to $L_{WT}=2L/3$.

Approach. The data sheet gives the deep-water transverse wavelength as a function of speed, $L_{WT}=2\pi V^2/g$; set this equal to the given fraction of hull length and solve for $V$, then non-dimensionalize by $\sqrt{gL}$ to get $F_n$. Separately, reason about the phase relationship between the bow and stern transverse-wave systems to classify the interference.

Set the wavelength condition. $$L_{WT}=\frac{2\pi V^2}{g}=\frac{2L}{3}\ \Rightarrow\ V^2=\frac{gL}{3\pi}.$$
Non-dimensionalize to get the Froude number. $$F_n=\frac{V}{\sqrt{gL}}\ \Rightarrow\ F_n^2=\frac{V^2}{gL}=\frac{1}{3\pi}\ \Rightarrow\ F_n=\frac{1}{\sqrt{3\pi}}=\boxed{0.326}.$$
Classify the interference. The ratio of hull length to transverse wavelength is $$\frac{L}{L_{WT}}=\frac{L}{2L/3}=\frac{3}{2}=1.5,$$ a half-integer number of wavelengths between the bow and stern transverse-wave-crest systems. A half-integer ratio puts the stern system's first crest in the trough of the bow system's train at the stern — destructive (cancelling) interference between the two transverse-wave trains, producing a local hollow (favorable dip) in the wave-making-resistance curve at $F_n\approx0.326$, rather than the reinforcing hump produced at integer ratios.

Figure 1 — Bow and stern transverse-wave-crest systems (blue) with the diverging wave wedges (orange) at the classic ~19.5° half-angle. With $L=(3/2)L_{WT}$, the stern system's leading crest (green) falls in the trough of the bow system's train, giving destructive interference at $F_n=0.326$.

Quantity	Result
Length-to-wavelength ratio $L/L_{WT}$	1.5 (half-integer ⇒ destructive)
Corresponding Froude number	$F_n=1/\sqrt{3\pi}\approx0.326$

Topic. Kelvin wave pattern and transverse-wave interference (humps and hollows).

Key relations. $L_{WT}=2\pi V^2/g$; $F_n=V/\sqrt{gL}$.

Why this works. The bow and stern each generate essentially the same Kelvin wave system; their transverse components superpose along the hull's track. Whether the two systems reinforce (integer $L/L_{WT}$, resistance hump) or cancel (half-integer $L/L_{WT}$, resistance hollow) depends purely on how many wavelengths fit between them, which is exactly what the Froude number controls through $L_{WT}\propto V^2$.

Common pitfall. Confusing the transverse-wave half-angle (which stays fixed at $\arcsin(1/3)\approx19.47°$ for the diverging wave wedge in deep water, independent of speed) with the interference condition, which is speed (Froude-number) dependent.

Source. Lewis (ed.), PNA Vol. II, Ch. 1 (wave-making resistance, humps and hollows); Larsson & Raven, Ship Resistance and Flow, Ch. 2.

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