Kelvin–Havelock–Peters approximations to a classical generic wave integral

Applied Mathematical Modelling 77 (2020) 950–962

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Kelvin–Havelock–Peters approximations to a classical generic wave integral Hui Liang a, Huiyu Wu b, Jiayi He b, Francis Noblesse b,∗ a b

Technology Centre for Offshore and Marine, Singapore (TCOMS), 12 Prince George’s Park, 118411, Singapore State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China

a r t i c l e

i n f o

Article history: Received 19 February 2019 Revised 9 July 2019 Accepted 12 August 2019 Available online 20 August 2019 Keywords: Integrals with rapidly oscillatory integrands Asymptotic approximations KHP approximation Ship waves Capillary-gravity waves

a b s t r a c t A recently-proposed analytical approximation to farfield ship waves – called Kelvin– Havelock–Peters (KHP) approximation because it combines three classical asymptotic approximations given by Kelvin, Havelock and Peters – is further considered. Specifically, a more accurate KHP approximation is given for a classical generic wave integral, and is applied to steady ship waves and transient capillary-gravity waves. Both the KHP approximation given previously and the refined KHP approximation obtained here only involve elementary functions and are then fully analytical, whereas the Chester, Friedman, Ursell (CFU) asymptotic approximation involves the Airy function and its derivative. Moreover, the KHP approximations – like Kelvin’s classical approximation but unlike the CFU approximation – provide an explicit decomposition of the two systems of waves (called transverse and divergent waves in the special case of ship waves) that exist inside the cusps of the wave pattern. This formal decomposition, and the simplicity of the KHP approximations, are two notable useful features that render the fully analytical KHP approximations well suited, and arguably preferable to the CFU approximation, for practical applications. © 2019 Elsevier Inc. All rights reserved.

1. Introduction The waves created by a ship that travels at a constant speed (U, 0, 0) in calm water are commonly expressed as a linear superposition of elementary plane progressive waves. In particular, the nondimensional elevation g E/U 2 , where g denotes the acceleration of gravity, of the free surface at a point

(X, Y, 0 ) g/U 2 = h (− cos γ , sin γ , 0 ) located aft of the ship can be expressed as [1–3]

gE 1 =  π U2



q∞ −q∞

A(q ) exp[ i h ψ (q, γ )] dq,

(1a)

where 1  q∞ , γ is measured from the negative x−axis, and the phase function ψ (q, γ ) is defined as

ψ (q, γ ) = ∗



1 + q2 (cos γ − q sin γ ).

Corresponding author. E-mail addresses: [email protected] (H. Liang), [email protected] (F. Noblesse).

https://doi.org/10.1016/j.apm.2019.08.007 0307-904X/© 2019 Elsevier Inc. All rights reserved.

(1b)

H. Liang, H. Wu and J. He et al. / Applied Mathematical Modelling 77 (2020) 950–962

951

Moreover, the amplitude function A(q) in (1a) is given by a distribution of elementary plane waves over the ship hull surface [4,5]. The integral representation (1a) of ship waves is ill suited for numerical evaluation far behind the ship, where one has 1  h and the trigonometric function oscillates rapidly. However, classical asymptotic approximations to the wave integral (1a) have been given by Kelvin [6] in 1891, Havelock [7] in 1908, Peters [8] in 1949 and Ursell [14] in 1960. Kelvin’s approximation is based on the fact that the dominant contribution to the wave integral (1a) for 1  h stems from points where the phase function ψ (q, γ ) is stationary, i.e. from real roots of the equation ψ (q, γ ) ≡ ∂ ψ (q, γ )/∂ q = 0. If tan2 γ < 1/8, i.e. if −γ0 < γ < γ0 with

γ0 ≡ arcsin(1/3 ) ≈ 19◦ 28 ,

(2)

the stationary-phase relation ψ  = 0 has two real roots

q− =

1−



1 − 8 tan2 γ 4 tan γ

and q+ =

1+



1 − 8 tan2 γ . 4 tan γ

(3a)

These two real roots are associated with two systems of waves, commonly called transverse and divergent waves. If γ = ± γ0 , one has

√ q− = q+ = ± q0 with q0 ≡ 1/ 2 ,

(3b)

i.e. the two roots q− and q+ coalesce and become a stationary point of multiplicity 2 that is related to the cusps γ = ± γ0 of the Kelvin wake. Finally, if γ0 < |γ | , the two roots q− and q+ become the complex conjugate pair

q− =

1− i



8 tan2 γ − 1 4 tan γ

and

q+ =

1+ i



8 tan2 γ − 1 . 4 tan γ

(3c)

These roots correspond to complex saddle points. Kelvin’s stationary-phase approximation [6] to the wave integral (1a) is valid for −γ0 < γ < γ0 , i.e. strictly inside the cusps of the Kelvin wake, and Peters’ steepest-descent approximation [8] is valid strictly outside the cusps, i.e. for γ0 < |γ | . Havelock’s approximation [7] is only valid at the cusp lines γ = ±γ0 , where Kelvin’s and Peters’ approximations are singular. Kelvin’s and Havelock’s stationary-phase approximations show that the transverse and divergent waves, associated with the real roots q− and q+ as was already noted, that exist inside the cusp lines γ = ±γ0 decay like h−1/2 , while the waves at the cusps decay like h−1/3 , as h → ∞. Peters’ asymptotic approximation shows that ship waves decay exponentially outside the cusps, except in the vicinity of the cusps where they decay like h−1/3 , as h → ∞. Large ship waves can then exist outside the cusp lines γ = ±γ0 . Indeed, the highest waves created by a ship can be found outside the Kelvin wake at low Froude numbers [9]. Peters’ analysis [8] also provides higher-order terms in Kelvin’s and Havelock’s farfield asymptotic approximations. An asymptotic approximation that is valid both inside and outside the cusps, as well as at the cusps, of the Kelvin ship wake has been given by Chester, Friedman and Ursell (CFU) [10] in 1957 and Ursell [14] in 1960, and revisited by Borovikov [11] in 1994. The global asymptotic approximation given by Chester et al. [10] involves the Airy function and its derivative, whereas Kelvin’s, Havelock’s and Peters’ approximations only involve elementary functions and are then fully analytical. Moreover, the CFU approximation requires selection of a physically-realistic cubic root of a complex number [12], an issue that is considered in [11]. The CFU approximation is then significantly more complicated than Kelvin’s, Havelock’s and Peters’ complementary approximations. Another notable drawback of the CFU approximation is that this approximation — unlike Kelvin’s classical approximation — does not provide a formal decomposition of ship waves into transverse and divergent waves inside the cusps of the Kelvin wake. Kelvin’s formal decomposition, which explicitly defines the phases and the amplitudes of the transverse and divergent waves and thereby provides physical insight that is essential to fully understand farfield ship waves [9,13] – as well as the simplicity of Kelvin’s, Havelock’s and Peters’ analytical approximations – arguably render these approximations preferable to the CFU approximation for practical applications, as is attested by the fact that the CFU approximation has not been applied as widely as Kelvin’s, Havelock’s and Peters’ fully analytical, simpler and more practical approximations. Indeed, although the CFU global asymptotic approximation is a perfectly satisfactory ‘mathematical approximation’ – which essentially amounts to expressing the wave integral (1a) in terms of the Airy function (another, closely related, integral) – the CFU approximation arguably is not an entirely satisfactory ‘hydrodynamical approximation’ because it does not provide a formal decomposition of ship waves into transverse and divergent waves, as was already noted. The asymptotic approximations to farfield ship waves given by Kelvin [6], Havelock [7], Peters [8] and Chester et al. [10] are considered, applied or extended in a large literature, including [11–25]. In particular, Kelvin’s approximation has been used extensively to study ship waves in deep water or in water of uniform finite depth, e.g. [1,2,13,19,20,26–34]. A simple analytical approximation that combines the basic analytical approximations given by Kelvin, Havelock and Peters has recently been given by Wu et al. [3]. This approximation, called Kelvin–Havelock–Peters (KHP) approximation in [3] and identified as KHP1 hereafter, is nearly identical to Kelvin’s and Peters’ approximations inside and outside the cusp lines γ = ±γ0 , but agrees with Havelock’s approximation at the cusps (where the Kelvin and Peters approximations are

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singular as was already noted). The KHP1 approximation only involves elementary functions, like Kelvin’s, Havelock’s and Peters’ approximations, and indeed is as simple as these three basic analytical approximations. Thus, KHP1 is well suited to study ship waves inside as well as outside Kelvin’s cusp lines, as is illustrated in a recent numerical investigation of the influence of the Froude number and, for a submerged body, of the submergence depth on ship wave patterns [9]. The KHP1 approximation is more realistic than Kelvin’s and Peters’ approximations, but is less accurate than the CFU approximation, in the vicinity of the cusps of the Kelvin wake [3]. A modification of the KHP1 approximation, called KHP2 approximation hereafter, that is appreciably more accurate than the KHP1 approximation is given in the present study. The KHP2 approximation only involves elementary functions, i.e. is fully analytical, like the KHP1 approximation and Kelvin’s, Havelock’s and Peters’ approximations, and indeed is not appreciably more complicated than these approximations. The analysis given in Wu et al. [3] only considers ship waves, for which the phase function ψ in (1a) is given by (1b), whereas the generic wave integral

I (h, γ ) =



∞ −∞

A(q, γ ) exp[ i h ψ (q, γ ) ] dq

(4)

is considered hereafter. The amplitude and phase functions A(q, γ ) and ψ (q, γ ) in (4) are arbitrary complex or real analytic functions of a real variable q, respectively, and 1  h is a large real number. As in the case (1b) related to ship waves, the stationary-phase relation ψ  (q, γ ) ≡ ∂ ψ (q, γ )/∂ q = 0 is assumed to have two distinct real roots q− < q0 < q+ for γ < γ 0 or for γ0 < γ . Although the case γ < γ 0 is considered hereafter for definiteness, the analysis is also valid if the two real roots q− and q+ of the stationary-phase relation ψ  = 0 occur for γ0 < γ . Moreover, the roots q− and q+ are assumed to coalesce for γ = γ0 , where one has

q− = q0 = q+ for

γ = γ0 and ψ0 ≡ ∂ 2 ψ (q0 , γ0 )/∂ q2 = 0,

(5a)

and to become a complex conjugate pair for γ0 < γ . The condition (5a) implies that, for γ < γ0 , one has

σ ≡ sgn(ψ0 ) ≡ sgn(ψ+ ) = −sgn(ψ− ) where ψ± ≡ ∂ 2 ψ (q± , γ )/∂ q2 and ψ0 ≡ ∂ 3 ψ (q0 , γ0 )/∂ q3 .

(5b)

For γ0 < γ , the complex roots q− and q+ and the corresponding values ψ− ≡ ψ (q− , γ ) and ψ+ ≡ ψ (q+ , γ ) of the phase function ψ (q, γ ) are real in the limit γ = γ0 . One then has (ψ− ) = 0 and (ψ+ ) = 0 for γ = γ0 , and the functions (ψ− ) and (ψ+ ) have opposite signs for γ0 < γ . The generic wave integral (4) occurs in the previously cited studies related to water waves in deep and shallow water, and for many other classes of waves, including interfacial waves, waves related to ship motions in regular ambient waves, capillary-gravity waves due to an impulsive disturbance, and transient flexural-gravity waves [35–44]. Indeed, the wave integral (4) is a classical generic integral that is important in engineering and physics, notably in fluid mechanics, electromagnetism and diffraction theory, and asymptotic approximations to the integral (4) are considered in a broad literature [45–47]. 2. Classical approximations The three complementary basic asymptotic approximations to the generic wave integral (4) given by Kelvin [6], Peters [8] and Havelock [7] are now summarized. 2.1. Kelvin’s approximation Kelvin’s classical stationary-phase approximation

√

I = I + O (h K

−1

),

with

K

I =

2 π A−



h|ψ− |



exp i h ψ− − σ

iπ 4

 +

√ 2 π A+



h|ψ+ |

 exp i h ψ+ + σ



iπ , 4

(6)

to the wave integral (4) holds if the stationary-phase relation ψ  = 0 has two distinct real roots q− and q+ with q− < q0 < q+ . In (6), one has A ± ≡ A(q ± , γ ), ψ± ≡ ψ (q± , γ ) , and ψ± and σ are given by (5b). 2.2. Peters’ approximation Peters’ asymptotic approximation

I = I P + O(h−1 ) ,

with

IP =

√ 2 π A±



h|ψ± |



exp i h ψ± + i

π 4

−

α±  2

,

(7a)

to the wave integral (4) holds if the roots q− and q+ are a complex conjugate pair. Moreover, one has

q± = q− if sgn[ (ψ− )] > 0

or

q± = q+ if sgn[ (ψ+ )] > 0 ,

(7b)

 2 2  and A± ≡ A(q± , γ ) ,ψ ±

≡ψ

(q ± , γ ) and ψ± ≡ ∂ ψ (q± , γ )/∂ q . Finally, α ± in (7a) is the argument of ψ± that is determined by the identity ψ± = ψ± e i α± .

H. Liang, H. Wu and J. He et al. / Applied Mathematical Modelling 77 (2020) 950–962



953



W

H H W Fig. 1. Errors  = GW app − Gnum associated with the Havelock approximations I1 and I2 to the wave component G (h, γ ) in the Green function for steady ship waves along the cusp γ = γ0 ≈ 19◦ 28 of the Kelvin wake for distances 10 ≤ h ≤ 30 from a submerged point source.

2.3. Havelock’s approximation The Kelvin and Peters approximations (6) and (7) to the wave integral (4) are singular at a point γ = γ0 where the conditions (5a) are verified, i.e. where the stationary-phase relation ψ  = 0 has a double root and one has ψ  (q0 , γ0 ) = 0 and ψ  (q0 , γ0 ) = 0. In this case, the Havelock approximation

I=

I1H

+ O (h

−2/3

),

I1H

with

(1/3 )

=

√ 3

1/3





6



A0 exp( i h ψ0 ) ,

h ψ0

(8a)

holds. In (8a), A0 ≡ A(q0 , γ 0 ), ψ 0 ≡ ψ (q0 , γ 0 ), ψ0 ≡ ∂ 3 ψ (q0 , γ0 )/∂ q3 , and ( · ) is the Gamma function [48]. Classical methods [45–47] show that a more accurate (second-order) Havelock approximation can readily be obtained via Taylor series expansions of the amplitude and phase functions A(q, γ ) and ψ (q, γ ) about the stationary point q0. Specifically, the second-order Havelock approximation can be expressed as

I=

I2H

+ O (h

−4/3

),

I2H

with

(1/3 )

=

√ 3

1 / 3



( 1 + i H0 )



6



h ψ0

A0 exp( i h ψ0 ) .

(8b)

in agreement with [25]. Here, H0 ≡ H(q0 , γ0 ) is defined as

H0 ≡

  1 / 3 ψ  (2/3 ) A0 6 − 0  , (1/3 ) A0 6 ψ0 h ψ0

(8c)

where A0 ≡ ∂A(q0 , γ0 )/∂ q and ψ0 ≡ ∂ 4 ψ (q0 , γ0 )/∂ q4 . The refined Havelock asymptotic approximation I2H can be significantly more accurate than the classical Havelock approximation I1H , as is illustrated in Fig. 1 for the example (considered further on in Section 5) of the wave component



W

GW (h, γ ) in the Green function for steady ship waves. Specifically, Fig. 1 depicts the errors  = GW app − Gnum where the subscripts “num” and “app” identify results obtained numerically or determined via Havelock’s classical approximation I1H or the refined approximation I2H . 2.4. The CFU approximation at a cusp and Havelock’s approximation Inside, at, or outside the cusps, the Chester, Friedman, Ursell (CFU) uniform asymptotic approximation to the generic wave integral (4) given in [10,11,45–47] can be expressed as:



I = I U + O(h−4/3 ) ,

where

a0 =

λ=

I U = π e i h ( ψ+ + ψ− )/2 a0

with

2/3

− ( 3 h | ψ+ − ψ− |/4 ) 2/3

[ 3 h ( ψ± )/2 ]



γ ≤ γ0 ,

A e

±

+ A− e −i ( π /4 − α±

 /2 )

(9a)

(9b)

γ0 ≤ γ ,

⎧   +  | + A− 2 /| ψ  | ( 3 |ψ − ψ | /4 )1/6 ⎪ A 2 / | ψ + − ⎪ + − ⎨ 2 A0 (2 /| ψ0 | )1/3 ⎪ ⎪ ⎩ + + i ( π /4 − α /2 )



Ai(λ ) Ai (λ ) + i b 0 2/3 , h1/3 h

γ < γ0 , γ = γ0 , 1/6

2 /| ψ± |[ 3 (ψ± ) /2 ]

γ0 < γ ,

(9c)

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H. Liang, H. Wu and J. He et al. / Applied Mathematical Modelling 77 (2020) 950–962

⎧    −1/6 ⎪ σ A− 2 /| ψ− | − A+ 2 /| ψ+ | ( 3 |ψ+ − ψ− | /4 ) γ < γ0 , ⎪ ⎨   b0 = 2 A0 ψ  | 541/5 ψ  |−5/3 − σ A (2 /| ψ  | )2/3 γ = γ0 , 0 0 0 0 ⎪ ⎪  ⎩ −1/6 + + i ( π / 4 − α± / 2 ) − −i ( π /4 − α± /2 )  i A e −A e 2 /| ψ± |[ 3 (ψ± ) /2 ] γ0 < γ ,

(9d)

and (7b) holds for γ0 < γ . Expressions equivalent to (9) have been applied in [3,14,42]. The asymptotic approximations (11b) and (12b) to the Airy functions Ai and Ai show that the CFU approximation (9) is asymptotically equivalent to Kelvin’s approximation (6) for γ < γ 0 or Peters’ approximation (7) for γ0 < γ . At a cusp γ = γ0 , (9a) becomes



I =π U





a0 Ai(0 ) i b0 Ai (0 ) + exp( i h ψ0 ) , h1/3 h2/3

(10a)

where the values Ai(0) and Ai (0) of the Airy functions Ai(x) and its derivative Ai (x) at the origin x = 0 are given by Abramowitz and Stegun [48]

Ai(0 ) =

3−2/3 31/3 (1/3 ) = √ (2/3 ) 2π 3



and Ai (0 ) =

−3−1/3 . (1/3 )

(10b)

Expressions (10b) for Ai(0) and (9c) for a0 show that the term that involves the Airy function Ai(0) in (10a) is given by

I1U

=

(1/3 ) √ 3

1/3





6



h ψ0

A0 exp(ih ψ0 ) .

This expression for the component I1U that involves the Airy function Ai(0) in the CFU approximation (10a) at a cusp is identical to the Havelock approximation (8a). Expressions (10b) for Ai (0) and (9d) for b0 can similarly be used to determine the term I2U that involves the derivative Ai (0) of the Airy function in (10a), and one finds I2U/I1U = i H0 where H0 is given by (8c). This result shows that the component I2U that involves the derivative Ai (0) of the Airy function in the CFU approximation (10a) at a cusp is identical to the second-order term in the refined Havelock approximation (8b). 2.5. Errors The basic Havelock approximation (8a) and the refined Havelock approximation (8b) involve O(h−2/3 ) or O(h−4/3 ) errors, respectively, as h → ∞. The Kelvin approximation (6) and the Peters approximation (7) involve O(1/h) errors. The O(1/h) errors associated with the Kelvin and Peters approximations are smaller or larger than the O(h−2/3 ) or O(h−4/3 ) errors associated with the basic or refined Havelock approximations (8a) or (8b). 3. The Airy function and its derivative As was already noted, the CFU approximation involves the Airy function Ai and its derivative Ai . Approximations Aia and Aia are used in the KHP approximations given in [3] and here. Specifically, the Airy function Ai is approximated in [3] as 

⎧  2  ⎪ if x ≥ 0 , ⎨ 1 − exp(−x2 ) Ai+ (x ) + exp(−x2 ) exp − 3 x3/2 Ai0 Aia (x ) =  ⎪ ⎩1 − exp(−x2 )Ai− (x ) + exp(−x2 )√2 sin 2 |x|3/2 + π Ai0 if x ≤ 0 , 3

(11a)

4

where Ai0 ≡ Ai(0 ) , and Ai+ (x ) and Ai− (x ) denote the asymptotic approximations [48]

Ai+ (x ) =



2 x−1/4 √ exp − x3/2 3 2 π



and Ai− (x ) =

2 |x|−1/4 π sin |x|3/2 + √ 3 4 π

(11b)

to the Airy function Ai(x) in the limits x → ∞ or x → −∞. Similarly, the derivative of the Airy function is approximated in [3] as

 ⎧  ⎨ 1 − exp(−x2 ) Ai+ (x ) + exp(−x2 ) exp − 2 x3/2 Ai0 if x ≥ 0 ,  Aia (x ) =  32 √  ⎩ 1 − exp(−x2 ) Ai (x ) + exp(−x2 ) 2 cos |x|3/2 + π Ai if x ≤ 0 , − 0 3

(12a)

4

where Ai0 ≡ Ai (0 ) , and Ai+ (x ) and Ai− (x ) denote the asymptotic approximations [48] 

Ai+ (x ) =



2 −x1/4 √ exp − x3/2 3 2 π





and Ai− (x ) =



−|x|1/4 2 3/2 π cos |x| + √ 3 4 π



(12b)

H. Liang, H. Wu and J. He et al. / Applied Mathematical Modelling 77 (2020) 950–962

955

Fig. 2. Airy function Ai (x ) and related approximations (11) and (13) given in Wu et al. [3] and here for −6 ≤ x ≤ 6.

to Ai (x) in the limits x → ∞ or x → −∞. The approximation (11a) given in [3] is modified as

⎧  4  ⎪ ⎨ 1 − exp(−x2 ) Ai+ (x ) + exp(−x2 ) exp − x3/2 Ai0 if x ≥ 0 , 3 Aia (x ) =  √   ⎪ ⎩ 1 − exp(−x2 ) Ai− (x ) + exp(−x2 ) 2 sin 2 |x|3/2 + π Ai0 if x ≤ 0 . 3

(13)

4

Fig. 2 shows that the modified approximation (13), used hereafter, is more accurate than the approximation (11a) used in [3]. 4. The KHP1 and KHP2 approximations 4.1. The KHP1 approximation For the generic wave integral (4) considered here, the KHP approximation KHP1 given in [3] becomes

  √   √ 2π iπ 2 π KH + iπ − I ≈ I1KHP = √ KH A exp i h ψ − σ +  A exp i h ψ + σ if √ − + − + 4 4 h h   √ 2π iπ ± I ≈ I1KHP = √ HP A exp i h ψ + if ± ± 4 h

γ ≤ γ0 ,

(14a)

γ0 ≤ γ ,

(14b)

KH where (7b) holds for γ0 < γ . The functions KH − and + related to the combined Kelvin and Havelock approximation in (14a), and the functions HP related to the combined Havelock and Peters approximation in (14b), are defined as ±

   4/3 − μ4 / 3 KH |ψ± | + e−μ CH h1/6 /|ψ0 |1/3 , ± ≡ 1−e

(14c)

   4/3 −ν 4 / 3 HP ψ± + (1 − i ) e−ν CH h1/6 /|ψ0 |1/3 , ± ≡ 1−e

(14d)

√ where CH = (1/3 )/( 2 π 61/6 ) ≈ 0.792831679, and μ and ν are defined as

μ=

3 h | ψ+ − 4

ψ− | and ν =

3 h | ( ψ± ) | . 2

(15)

The KHP1 approximation (14) is obtained in [3] by using the approximations (11a) and (12a) for the Airy functions Ai and Ai in the CFU approximation (9a). Near a cusp, the approximations

 

1 + A 2

2

+A |ψ  | +



−

2

|ψ− |

 3 4

|ψ+ − ψ− | 

 1  + + i ( π / 4 − α± / 2 ) A e + A− e−i ( π /4 − α± /2 ) 2 are used in [3].

1/6

2

≈

3

|ψ± | 2

A+ 2 1/3

|ψ0 |1/3 ( ψ± )

≈

1/6

A− 2 1/3

|ψ0 |1/3 ≈

and

A± 2 1/3

|ψ0 |1/3

(16a)

(16b)

956

H. Liang, H. Wu and J. He et al. / Applied Mathematical Modelling 77 (2020) 950–962

4.2. The KHP2 approximation In the alternative KHP approximation KHP2 considered here, the approximations (16) are modified as

 

1 + A 2



2

+ A− |ψ  |

2

|ψ  |

+

−

 3 4

|ψ+ − ψ− |

1/6



 1  + + i ( π / 4 − α± / 2 ) A e + A− e−i ( π /4 − α± /2 ) 2

2

≈

A0 2 1/3

|ψ0 |1/3

3

( ψ± ) |ψ± | 2

and

1/6

≈

(17a) A0 2 1/3

|ψ0 |1/3

.

(17b)

Use of the approximations (17), instead of the approximations (16), in the KHP2 approximation yields numerical predictions that are nearly identical for ship waves (considered in Section 5) but are appreciably more accurate for transient capillary-gravity waves (considered in Section 6), as is shown in Appendix A. Thus, the approximations (17), which are somewhat simpler than the approximations (16), are used hereafter. The alternative KHP approximation KHP2 is then given by



I ≈ I2KHP = F−KH exp i h ψ− − i σ



I ≈ I2KHP = F±HP exp i h ψ± + i

π 4



+ F+KH exp i h ψ+ + i σ

π 4

π 4

if

γ ≤ γ0 ,

(18a)

if

γ0 ≤ γ ,

(18b)

where σ ≡ sgn(ψ0 ) and (7b) holds for γ0 < γ . The functions F−KH and F+KH related to the combined Kelvin and Havelock approximation in (18a), and the functions F±HP related to the combined Havelock and Peters approximation in (18b), are defined as

√



F±KH = 1 − e−μ

4/3

4/3  2 π A± e−μ CH∗ A0 +



1/3 [ 1 ± σ H0 ] , h| ψ± | h ψ0

(18c)

√



F±HP = 1 − e−ν

4/3

4/3  2 π A± e−ν CH∗ A0 +



1/3 ( 1 − i )[ e− 2 ν /3 + i H0 ] , 

h ψ± h ψ0

(18d)

where CH∗ ≡ (1/3 )/61/6 ≈ 1.987334053,H0 is defined by (8c), and μ and ν are given by (15). 5. Illustrative application to steady ship waves Illustrative applications to steady ship waves, previously considered in [3] for the purpose of comparing the KHP1 approximation obtained in [3] to the CFU approximation, are considered again in order to compare the KHP2 approximation obtained here to the KHP1 and CFU approximations. The waves created by a point source and an elementary ship model are successively considered. The Green function, i.e. the flow potential due to a point source, associated with steady flow around a ship that travels at a constant speed in calm water [4,5] can be formally expressed as the sum of a non-oscillatory local-flow component and a wave component GW , which can be expressed as [49]

 GW = −

∞ −∞

2 A(q ) e−(1+ q ) (z+δ ) exp[ ih ψ (q, γ )] dq ,



(19)



where the phase function ψ is defined by (1b), and h ≡ x2 + y2 = X 2 + Y 2 g/U 2 denotes the horizontal distance from the point source located at ( 0 , 0 , −δ ). The submergence depth δ ≡  g/U2 is chosen as δ = 0.01. The points of stationary phase associated with the phase function ψ , considered in the introduction, are given by (3). The top half of Fig. 3 depicts the variation of the wave potential GW ( x, y, z = 0 ), determined via accurate numerical evaluation of the integral (19) or via the corresponding asymptotic CFU approximation or the alternative approximations KHP1 and KHP2, along two transverse cuts 0 ≤ y ≤ 6 and 0 ≤ y ≤ 30 in the free surface z = 0 at distances x = −10 and x = −50 W

W W given by the CFU, KHP1 behind the point source. The related errors  = GW app − Gnum , where Gapp denotes the values of G or KHP2 approximations and GW num is determined via accurate numerical evaluation, are depicted in the bottom half of Fig. 3. The top half of Fig. 3 shows that the asymptotic approximations CFU, KHP1 and KHP2 are realistic. However, the approximations CFU and KHP2 are noticeably closer to the numerical predictions than the KHP1 approximation near the cusp of the Kelvin wake, marked as a vertical line in Fig. 3. The bottom half of this figure shows that the errors  associated with the KHP1 approximation indeed are significantly larger than the errors associated with the CFU or KHP2 approximations. In particular, the errors  associated with the KHP2 approximation are significantly smaller than the errors  associated with the KHP1 approximation outside the cusp of the Kelvin wake. Moreover, Fig. 3 shows that the KHP2 errors decrease as

H. Liang, H. Wu and J. He et al. / Applied Mathematical Modelling 77 (2020) 950–962

957

Fig. 3. The top half depicts accurate numerical evaluations, the CFU approximation, and the KHP approximations KHP1 and KHP2 to the wave component 2 GW (x, y, z) defined by (19) for x ≡ Xg/U 2 = −10 (left) or x = −50 (right), 0 ≤ y ≡ Yg/U2 ≤ 6 (left)

or 0 ≤ y ≤ 30 (right), and z ≡ Zg/U = 0. The vertical lines at W

y ≈ 3.5 or y ≈ 17.7 correspond to the cusp of the Kelvin wake. The errors  = GW app − Gnum associated with the CFU, KHP1 and KHP2 approximations are depicted in the bottom half.

the distance −x behind the point source increases, whereas the largest KHP1 are not smaller for x = −50 than for x = −10. Lastly, Fig. 3 shows that the CFU approximation is more accurate than the KHP approximations, as is expected and is noted in [3]. The farfield waves created by an elementary model of a ship, of length L, that advances at a constant speed U in calm water are now considered. This elementary ship model [3] consists of a point source near the ship bow ( 0.5, 0, 0 ) L and a point sink near the ship stern (−0.5, 0, 0 ) L . Specifically, the point source and point sink are located at X/L = ( ±0.45, 0, −δ ) where the nondimensional submergence depth δ of the source-sink pair is chosen as δ = 0.02 (about half the draft/length ratio of a typical displacement vessel). The nondimensional elevation g E/U 2 of the free surface behind this point source-sink ship model is defined by (1) where the amplitude function A(q) is given by

A (q ) = − 2 i







2 2 1 + q2 e− ( 1+q ) δ /F sin( 0.45 1 + q2 /F 2 ) with F ≡ U/ gL .

(20)

Fig. 4 depicts colored contour plots of the wave pattern created by the point source-sink ship model within the rectangular region −120 ≤ Xg/U 2 ≤ −100,30 ≤ Yg/U2 ≤ 45 for three Froude numbers. At the Froude numbers F = 1 and F = 0.5, the wave patterns obtained by means of accurate numerical evaluation or determined via the CFU, KHP1 or KHP2 asymptotic approximations cannot be distinguished. At the lower Froude number F = 0.3, the three wave patterns determined numerically or via the CFU approximation or the KHP2 approximation likewise cannot be distinguished. However, the wave pattern determined via the KHP1 approximation is slightly different in the vicinity of the Kelvin cusp line. 6. Capillary-gravity waves due to an impulsive disturbance The function F(δ , h, τ ) defined by the integral

F ≡ e and

i π /4



∞

0

ψ (q, a ) =

e−q δ ω (q )





2π qh



exp [ i τ ψ (q ) ]dq ,

q + σ 2 q3 − q a

with

where

a = h/τ ,

ω (q ) =



q + σ 2 q3

(21a) (21b)

is now considered. This integral, given in [42], is related to the ‘time-domain’ Green function for the transient capillarygravity water waves due to an impulsive disturbance δ (t − t0 ) at time t = t0 considered in [41,42], that is associated with unsteady motions of a floating body of length L [50–52]. In (21a), h and δ denote (nondimensional) horizontal and vertical distances between the flow-field point and the mirror image of the source point about the undisturbed free surface, τ

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Fig. 4. Colored contour plots of the free-surface elevation g E/U 2 related to the waves created by an elementary ship model within the rectangular region −120 ≤ Yg/U 2 ≤ −100, 30 ≤ Yg/U 2 ≤ 45. The wave patterns obtained from accurate numerical evaluation (top row), the CFU approximation (second row), and the approximations KHP1 (third row) and KHP2 (bottom row) are depicted for the three Froude numbers F = 0.3 (left column), 0.5 (center) and 1.0 (right column).



is the time after the impulsive disturbance created at t = t0 , and the parameter σ is defined as σ ≡ T /(ρ gL2 ) where T ≈ 0.073 N/m is the air-water interface tension. The stationary-phase relation ψ  ≡ ∂ ψ (q, a )/∂ q = 0 has two real roots q− and q+ that can be approximated [41] as



q− =



1 5 σ 1+ 16 a2 4a2



2



4a2 27 σ q+ = 1+ 16 a2 9σ 2

+O

2

 σ 4 

+O

,

(22a)

 σ 4 

(22b)

a2

a2

√ for a  σ . The wavenumbers q− and q+ correspond to gravity-dominant and capillarity-dominant waves. For a = a0 ≈ √ 1.086 σ , the roots q− and q+ coalesce, i.e. one has q± = q0 where

q0 =

 √

2/ 3 − 1



σ.

(23a)

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Fig. 5. The upper half depicts accurate numerical evaluations, the CFU approximation, and the approximations KHP1 and KHP2 to the free-surface term F (δ = 0.01, h, τ ) defined by (21a) as a function of 16 ≤ τ ≤ 19 for h = 1 (left) or as a function of 0.5 ≤ h ≤ 0.65 for τ = 10 (right side). The vertical lines at τ ≈ 17.6 (left) and at h ≈ 0.57 (right) correspond to the cusp. The errors  = | Fapp − Fnum | associated with the CFU, KHP1 and KHP2 approximations are depicted in the bottom half.

At the two coalesced stationary points q0 , the second derivative ψ0 ≡ ∂ 2 ψ (q0 , a0 )/∂ q2 of the phase function ψ (q, a) is nil, and the third and fourth derivatives are given by

ψ0 ≡ ∂ 3 ψ (q0 , a0 )/∂ q3 = 4.4516 σ 5/2 and ψ0 ≡ ∂ 4 ψ (q0 , a0 )/∂ q4 = −28.7012 σ 7/2 .

(23b)

If a < a0 , the two roots q− and q+ become a complex conjugate pair, expressed in [41] as

q± = qR ± i qI ,

where

(24a)

 qR =

a

1

σ 3/2 3 5/4



1 a a + √ +O √ 9 σ σ

 qI =

1

1

σ 3 1/2

−

1 3 5/4



a a √ +O √

σ

σ

2 ,

(24b)

.

(24c)

2

The top half of Fig. 5 depicts the function F (δ = 0.01, h, τ ), determined via accurate numerical evaluation of the integral (19) or via the corresponding CFU asymptotic approximation or the KHP approximations KHP1 and KHP2, for 16 ≤ τ ≤ 19 at a radial distance h = 1 away from the point source (on the left side) and for distances 0.5 ≤ h ≤ 0.65 at a time τ = 10 (on the right side). The related errors  ≡ | Fapp − Fnum | where Fapp denotes the values of the function F (δ = 0.01, h, τ ) given by the CFU, KHP1 or KHP2 approximations and Fnum is determined via accurate numerical evaluation, are depicted in the bottom half of Fig. 5. This half of the figure shows that the errors associated with the refined KHP2 approximation are significantly smaller than the errors associated with the KHP1 approximation, as was also found in Fig. 3 for ship waves. 7. Conclusions Asymptotic approximations to the classical generic wave integral (4) are considered in this study. In particular, two alternative Kelvin–Havelock–Peters (KHP) approximations are compared to the Chester, Friedman and Ursell (CFU) approximation [10]. The KHP approximations considered in the study consist of the KHP1 approximation given in [3] and the modification KHP2, given by (18), of the KHP1 approximation that is obtained in this study. The KHP2 and KHP1 approximations differ in that (i) the first-order Havelock approximation (8a) used in KHP1 at the cusps is replaced by the second-order Havelock approximation (8b) in KHP2, (ii) the approximation (11a) to the Airy function used in KHP1 is replaced by the more accurate approximation (13) in KHP2, and (iii) the approximations (16) used in KHP1 are replaced by the somewhat simpler (and, in the case of transient capillary-gravity waves, more accurate) approximations (17) in KHP2.

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The alternative KHP approximations KHP1 and KHP2 combine the three classical complementary asymptotic approximations, associated with ship waves, given by Kelvin [6], Havelock [7] and Peters [8]. The Havelock approximation is only valid at a cusp, whereas Kelvin’s and Peters’ approximations are singular at a cusp and are only valid in regions strictly inside or outside cusps. The KHP approximations KHP1 and KHP2 are valid at the cusps, where they agree with the classical Havelock (first-order) approximation or a refined (second-order) Havelock approximation. The KHP1 and KHP2 approximations are also valid in regions inside and outside cusps, where they are nearly identical to Kelvin’s or Peters’ approximations. The errors associated with the KHP1 and KHP2 approximations are then comparable to the O(h−1 ) errors associated with Kelvin’s and Peters’ approximations inside or outside the cusps, and the O(h−2/3 ) or O(h−4/3 ) errors associated with Havelock’s first- or secondorder approximations at the cusps. Thus, the errors related to the KHP2 approximation are O(h−4/3 ) at the cusps, O(h−1 ) inside or outside the cusps, and between O(h−4/3 ) and O(h−1 ) in the vicinity of the cusps. Kelvin’s, Havelock’s and Peters’ approximations, and the related KHP1 and KHP2 approximations, only involve elementary functions and are then fully analytical, whereas the CFU approximation involves the Airy function and its derivative. Another notable difference between the KHP approximations and the CFU approximation is that the KHP approximations – like Kelvin’s classical approximation but unlike the CFU approximation – provide a formal decomposition into the two distinct systems of waves (called transverse and divergent waves in the special case of ship waves) that exist inside the cusps of the wave pattern. This formal decomposition provides essential physical insight because it explicitly defines the phases and the amplitudes of the waves associated with the two systems of waves inside the cusps. Such physical insight is important, as is illustrated for ship waves in [1,2,9,13,30–34]. Indeed, the analysis of farfield ship waves reported in these studies is based on Kelvin’s classical approximation – except for [9], based on the KHP1 approximation [3] – and could not be performed via the CFU approximation. Moreover, the analysis of farfield ship waves given in [9] requires an analytical approximation valid everywhere (i.e. inside, at, and outside the cusps of the Kelvin wake) far behind the ship and could not be performed via Kelvin’s, Havelock’s and Peters’ approximations. The illustrative applications to farfield ship waves (created by a ship traveling at a constant speed in calm water) and transient capillary-gravity waves considered in the study show that the KHP1 and KHP2 approximations both provide realistic practical alternatives to the more accurate but more complicated CFU approximation. Indeed, these illustrative applications, specifically Figs. 3–5, show that the simple analytical approximations KHP1 and KHP2 are sufficiently accurate and well suited for practical purposes, as is also illustrated in [9] where the KHP1 approximation is applied to study the influence of the Froude number and, for a fully submerged body, of the submergence depth on farfield ship waves. The applications to farfield ship waves and transient capillary-gravity waves considered in the study also show that the KHP2 approximation given here is appreciably more accurate than the KHP1 approximation given in [3] in the vicinity of a cusp, although differences between the KHP1 and the KHP2 approximations, as well as differences among these approximations and the CFU approximation, may be inconsequential for some practical applications. E.g., the three asymptotic approximations CFU, KHP1 and KHP2 can hardly be distinguished, and cannot even be distinguished from accurate numerical computations, in Fig. 4. Indeed, the CFU and KHP2 approximations are asymptotic approximations that evidently become more accurate as the parameter 1  h in the generic wave integral (4) increases. Appendix A. The KHP2 approximation with the approximations (16) or (17) The KHP2 approximations — where the approximations (16) or (17) are used — are compared in Fig. 6 for the wave component GW (x, y, z) in the Green function for steady ship waves (considered in Section 5) and the function F (δ = 0.01, h, τ ) related to transient capillary-gravity waves (considered in Section 6). The left side of Fig. 6 shows that the errors  associated with the KHP2 approximations with (16) or (17) are nearly indistinguishable for the function GW (x, y, z), whereas the errors for the function F (δ = 0.01, h, τ ) are significantly smaller if the approximations (17) are used instead of the approximations (16). The differences between the KHP2 approximations with (16) or (17) are related to the behaviors of the function (identified as ‘Exact’ in Fig. 6) defined by the left side of equations (16) and (17) and the corresponding approximations (denoted as M+ , M− , M ± or M0 ) defined by the right sides of (16) or (17). Specifically, the top right corner of Fig. 6 shows that the functions M+ , M− , M ± and the corresponding exact function vary slowly and do not differ much from the value M0 at the cusp for the wave component GW (x, y, z) in the Green function for steady ship waves. However, for the function F (δ = 0.01, h, τ ) associated with transient capillary-gravity waves, the two figures in the center and bottom rows of Fig. 6 show that the approximations M+ or M− vary very rapidly in the vicinity of the cusps, and are significantly different from the corresponding exact function and the cusp-value M0 . It should be noted that A− and A+ are real functions inside the region where two distinct points of stationary phase exist whereas A ± is complex outside that region, which yields the complex functions M ± depicted in Fig. 6.

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Fig. 6. The left side of the figure compares the errors  associated with the KHP2 approximations, with equations (16) or (17), for the wave component GW ( x = −50, 15 ≤ y ≤ 20, z = 0 ) in the Green function for steady ship waves (considered in Fig. 3) in the top row, and the functions F ( δ = 0.01, h = 1, 16 ≤ τ ≤ 19 ) and F ( δ = 0.01, 0.5 ≤ h ≤ 0.65, τ = 10 ) related to transient capillary-gravity waves (considered in Fig. 5) in the center and bottom rows. The right side of the figure depicts the exact function (Exact) and the related approximations defined by (16) and (17) and denoted as M+ ,M− , M ± or M0 .

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