Underwater acoustics

Applied Ocean Research 111 (2021) 102649
0141-1187/© 2021 Elsevier Ltd. All rights reserved.
A contribution to underwater acoustic problem in water entry of
wedge-shaped bodies
Mojtaba Barjasteh, Hamid Zeraatgar *
Amirkabir Laboratory of Hydrodynamics (ALH), Faculty of Maritime Engineering, Amirkabir University of Technology, Tehran 15875-4413, Iran
A R T I C L E I N F O
Keywords:
Water-entry
Wedge
Underwater Acoustics
Pressure wave
Impact pressure
A B S T R A C T
Generation, propagation, reflection, and interference of pressure waves in the water entry of wedge-shaped
bodies are investigated in this study using a modified 2D acoustic equation. Impact pressure generated by
symmetric and asymmetric entry of wedges is considered as excitation for the initiation of an acoustic wave.
Available impact pressure is discretized and applied as multiple nodal source functions. The acoustic equation is
numerically solved using finite difference method. Pattern of propagating pressure waves is comparedfor the
symmetric and asymmetric entries with different deadrise angles. The comparison shows that asymmetric entry
can significantly change the propagation pattern. The problem is also inspected for the shallow water entry by
considering reflection from boundaries. The shallow water results reveal that special cares should be considered
in specifying tank dimensions in experimental measurement of the impact pressure to avoid undesired pressure
oscillation.
1. Introduction
The water entry problem is a well-known phenomenon in naval ar
chitecture and ocean engineering. Hydrodynamic performance evalua
tion of marine craft can be one of the most important application of the
water entry problem. Analytical studyof the problem is initiated by
technical reports of Karman (1929) and Wagner (1366) in designing of
sea-plane structures. They investigated water entry of different 2D rigid
bodies particularly wedges due to its wide application in their topic.
Although both reports employed the same concepts to derive their
governing equations, the Wagner model could simulate the problem in
more details. Most of later analytical models such as Korobkin (1992),
Toyama (1993), Mei et al. (1998) and Oliver (2002)) are also based on
Wagner theory to develop different application of this modeling. Wag
ner predicted to water entry pressure and its distribution over the sur
face of a penetrating rigid wedge. The pile-up effect was considered for
the first time in his research. Fig. 1 illustrates the entry of a wedge at
penetration depth of h. This figure shows free surface pile-ups on the
both sides of the wedge as well as distribution of the hydrodynamic
impact pressure and its peak, Pmax, on the wet chord of the wedge.
Wagner showed that the controlling parameter in peak impact
pressure is the wedge deadrise angle which is shown by β. Once the
wedge touches the water and then starts penetrating, the free surface
rises right close to the chords which is called pile-up. These regions are
hatched in Fig. 1. A jet-flow spray is then formed at vicinity of these pile-
up regions. It can be easily shown that the inner boundary of pile-up
regions experiences the hydrodynamic impact pressure while as the
outer one faces to the ambient pressure, say Patm, (Wagner, 1366, Fal
tinsen, 2005). Additionally, the pile-up region is localized just around
the intersection of the chords and the free surface. Thus, this region is
usually neglected in comparison to the main water domain beneath the
wedge, for sake of simplicity, specially at initial stage of penetration.
Furthermore, the rising of the free surface profile due to immersed
volume of the wedge, h2
/tanβ, is very small and can be ignored in real
practice because of large width of the domain. Wagner theory can
generally evaluate the impact pressure in water entry of wedge-shaped
bodies with reasonable engineering accuracy. Accordingly, several
later researches have focused in extension of this theory notably
experimental studies of Chuang (1973), Greenhow (1988), Whelan
(2004), Yettou et al. (2006), and Barjasteh et al. (2016). Fig. 2 shows a
typical measured impact pressure in water entry of a wedge with β = 20∘
reported by Barjasteh et al. (2016). They showed that, Wagner theory
overestimates the impact pressure for small deadrise angles.
The generated hydrodynamic impact pressure causes formation of a
pressure gradient as a source of a disturbance in the water right close to
the wedge. The disturbance turns to the source of a sound wave and
* Corresponding author.
E-mail address: hamidz@aut.ac.ir (H. Zeraatgar).
Contents lists available at ScienceDirect
Applied Ocean Research
journal homepage: www.elsevier.com/locate/apor
https://doi.org/10.1016/j.apor.2021.102649
Received 1 December 2020; Received in revised form 2 March 2021; Accepted 5 April 2021

2
starts propagating in surrounding water with the speed of sound. The
propagation continues until the waves are fully dissipated or confronted
with the boundaries such as interfaces, obstacles, or waterbed. When the
sound waves encounter to different boundaries, they may be trans
mitted, reflected, absorbed and so on. The propagation of the sound
waves can generally affect the underwater pressure distribution on the
bodies and consequently its hydrodynamic performances. Additionally,
its effects may be more pronounced in shallow waters due to reflection
from beds.
An interesting application of water entry of wedge sections is hy
drodynamic evaluation of planing craft. Various types of symmetric and
asymmetric wedge sections are employed in designing of these boats.
Newmann (1977) developed aEulerian strip theory for marine vessels to
calculate a longitudinal quantity of a slender body. In this theory, the
slender marine vessel is initially divided into multiple transvers 2D
sections and the desired quantity is investigated for each section.
Finally, the obtained results are integrated over the vessel length to
compute the main longitudinal quantity for the vessel. Fig. 3 illustrates a
planing craft with steady trim angle of τ and surge speed of V. The
deadriseangle of boat section are increased from the aft toward fore.
Using the Eulerian representation of strip theory, the plane ABCD is
fixed in time and space and the boat gradually passes through this plane.
The intersections of the boat geometry and this plane are 2D symmetric
wedges which are penetrating to the water with constant steady speed of
Vtanτ. Similar analysis is also available when the boat rolls, but sym
metric wedges convert to asymmetric ones as described by Barjasteh
et al. (2016).
Fig. 4 shows a general wedge section of boat the with deadrise angle
of β which is approaching the water surface with pure vertical velocity of
V. Two types of entry including symmetric and asymmetric is investi
gated for different boat dynamics. Asymmetric entry is defined by
inclining the symmetric wedge by predefined angle, ϕ, in respect to
apex, O, resulting the left and the right deadrise angles of βL = β − ϕ and
βR = β + ϕ, respectively. Initially, the water surface is at rest and
assumed to be undisturbed even after penetration as discussed in pre
vious section. A rectangular domain is chosen to study the problem. The
water depth is controlled by parameter h, which can have any arbitrary
values. The horizontal axis of coordinate system coincides with the
water surface and located at the first touch point between the wedge and
the water. Domain width is set to 2a, which is reasonably higher than
wet beam of the wedge to vanish undesired interaction of the wedge and
domain sides.
After penetration, two high-pressure regions are generated on wet
chords of the wedge as depicted in Fig. 5 according to conservation of
momentum. These regions are indicated by PL and PR to address the left
and the right sides of the wedge, respectively. Magnitudes of the pres
sure are highly dependent to the left and right deadrise angles of the
wedge as shown by Barjasteh et al. (2016). These high-pressure regions
are sources of pressure disturbance in the water. According to continuity
and conservation of momentum, these disturbances start propagating in
the water and generating their corresponding leading waves called WL
and WR. Leading waves gradually turn to a singlestanding wave which is
Fig. 2. (a) History of impact Pressure measured for a wedge with β = 20∘
recorded by three pressure transducers mounted on shown positions P1, P2, and
P3(Barjasteh et al., 2016).
Fig. 3. Eulerian representation of strip theory in hydrodynamic evaluation of
planing crafts for its different three sections shown by 1, 2 and 3 (aftward).
Draft of all sections are illustrated at the fourth figure.
Fig. 4. Schematic view of impacting body.
Fig. 1. Symmetric water entry of a wedge.
M. Barjasteh and H. Zeraatgar

3
named pulse wave, PW, as displayed in Fig. 5.
Entry of each section of a craft generates and propagates different
impact pressure fields in the supporting water. Resultant of all pressure
fields determine the total propagating underwater pressure due to a craft
motion. The present research focuses on the first step of strip theory
which is generation, propagation, and interference of a 2D impact
pressure. The second step of establishing a strip theory, requires several
other mathematical techniques which will be considered in future.
The study of propagating the underwater pressure due to impact or
other sources of pressure is well known as hydroacoustics or underwater
acoustics. This topic has many applications such as in hydrography,
SONAR, oceanography, measurement of underwater sea currents, un
derwater noise detection, water entry of projectiles, and propeller
optimization. Many maritime researches have been conducted on the
underwater acoustics. Honghui and Makoto (2004) carried out an
experimental study for measurement of underwater acoustic and
supercavitation in high speed entry of different bodies using PVDF
needle hydrophones. Averbuch et al. (2011) developed an algorithm to
detect the river boat via their acoustic signature. Similar research was
also performed by Barlett and Wilson (2002) for planinghulls in which
underwater acoustic of power boats are measured using hydrophones
mounted on bed. Abrahamsen (2012) introduced a ship as a source of
underwater acoustic noise and made a review on sources of noises in a
typical cargo ship. Ianniello et al. (2014) simulated the underwater
acoustic noise generated by ship motion and rotation of its propeller.
They used Ffowcs Williams–Hawkings equation for evaluation of the
acoustics and coupled RANS and level-set method for flow estimation
and updated the free surface and flow. Since the main source of un
derwater acoustic noises is the ship propellers, Lloyd et al. (2015)
investigate the under-water acoustic problem for a submerged propeller
and compared CFD results with acoustic analogy method. Belibassakis
(2018) carried out a similar research for evaluation of acoustic noise of a
propeller using a coupled panel method for hydrodynamic problem and
Farassat acoustic formulation.
Propagation of pressure waves in water entry of different types of
high-speed projectiles are also well studied by various researches mainly
those focused on experimental investigation of the problem. This special
class of water entry problem considers formation, propagation and
rarefaction of front shock waves and ballistic waves, particularly for
supersonic entries. These shock waves result in generating a highly
concentrated physical discontinuity right before and after font waves
(Lee et al. (1997), Ji et al. (2020), Korobkin (1992), Korobkin and
Pukhnach (1988), Laverty (2004), Truscott et al. (2014), Bodily et al.
(2014)). Most of these studies utilized various concept of aerodynamics
analogy to manipulate the problem in the water, specially capturing the
discontinuities. However, the entry speed in the present study is limited
to a couple of meters per second and essentially the above type of
discontinuity cannot be occurred. Therefore, the common methods used
in high-speed entry of projectiles are not applicable to the present
problem.
Regardless of application of this study on underwater acoustic of
planing craft, the authors reported some oscillation behavior in their
previous research in the water entry of wedges, (Barjasteh et al., 2016).
Similar oscillations are also distinguished and reported by Yettou et al.
(2006), Whelan (2004), and Zeraatgar et al. (2019) in their experimental
study of the water entry problem. It seems that source of such
oscillations is reflection of the acoustic waves from the tank bed. Illu
mination of sources of such oscillations is another motivation of the
authors to investigate this problem.
The reviewed literature shows that a few papers studied the under
water acoustic with maritime application and their solution method are
very complicated and have many limitations. This research simulates
generation, formation, reflection, and interference of pressure waves
due to the water entry of wedge-shaped 2D bodies. The method benefits
a simplified form of wave equation and some straightforward FDM
techniques to provide a stable and convergent solution applicable to
other desired geometries, too.
2. Formulation
The problem is schematically defined in Fig. 5. The regions PL and
PR are high-pressure sources of disturbance in the domain. These
sources are very similar to the acoustic sources of disturbance, say
noises, which produce sounds in the air due to an initial pressure
gradient. It is well known that propagation of a mechanical noise in a
physical medium can be accurately evaluated using classical wave
equation derived by Stokes dating back to 1845 (Stokes, 1845).
∇2
P −
1
c2
∂2
P
∂t2
+ Γ = O(x, y, t) (1)
where∇2
is the Laplace operator, and P, c, Γ, and O are pressure field,
local sound speed, dissipation, and source of excitation, respectively.
The wave equation which is also known as acoustic equation is appli
cable to the underwater acoustics in prediction of wave propagation.
The mechanism of generation, propagation, and dissipation of noises
both in the air and the water are very similar.The pressure field in wave
equation is derived for perturbation pressure. The total pressure is
regarded as linear summation of uniform field and the perturbation.
Pt = Pu + P (2)
where Pt, Pu and P are total, uniform, and perturbation pressures,
respectively. In this study, hydrostatic pressure is a uniform field
whereas the impact pressure due to water entry of the wedge is the
perturbation. Thus, all undesired effects of vertical variation of hydro
static pressure, ρgh, in solution of wave equation is vanished.
According to the wave equation, the pattern of propagation of the
noise in different mediums is very similar but with different propagation
speeds which is governed by c, the local speed of sound in the mediums.
Thus, this parameter plays a crucial role in the wave equation. The
sound speed in the water is the thermodynamic property of the water
and defined; c = √(B /ρ) where B = ρ0(∂P /∂ρ) is bulk modulus. ρ0, and
ρ are initial and instantaneous water density, respectively. Here, the
medium is sea water. The sound speed in the sea water can be expressed
as a known function of temperature, salinity, and depth (Mackenzie,
1981).
c = 1448.96 + 4.591T − 0.05304T2
+ 0.0002.374T3
+ 1.340(S − 35)
+ 0.01630h + 1.675*10− 7
h2
− 0.01025T(S − 35) − 7.139*10− 13
Th3
(3)
whereT, S, and h are water temperature in◦
C, salinity in ppt and sea
depth in meters, respectively. Assuming the fresh water of 20◦
C, salinity
of 20 ppt and the water depth less than 50 meters, the sound speed is
calculated about 1505 m/s. The sound speed is supposed to be constant
due to almost unchanged effective parameters. The depth of sea water is
limited to 50 meters for real application of the problem and simplifies its
solution, simultaneously.
Another important term in the wave equation is dissipation or Γ.
Dissipation or attenuation determines the energy loss of pressure wave
due to propagation in a physical medium. There are three mechanisms of
energy dissipation during propagation of pressure waves in sea water
Fig. 5. Generation and propagation of pressure waves for symmetric entry.

4
including absorption, divergence, and scattering (Kinsler and Frey,
1962, Etter, 2003, Bjørnø, 2017). Divergence and scattering represent
those parts of energy loss due to presence of the water surface, seabed,
suspended sediments, any inhomogeneities, undercurrents, organisms,
etc. These types of energy loss are out of scope this study and they are
disregarded. The absorption is energy loss as a results of water viscosity,
heat conduction, and relaxation. Relaxation is caused by presence of
magnesium sulfate (MgSO4) and boric acid B(OH)3 (Schulkin and
Marsh, 1962, Yeager et al., 1973), which is not considered in this study
because of the specified water properties. It is shown that viscosity and
heat conduction losses cannot be studied separately, and they are usu
ally investigated as thermo-viscous attenuation. However, heat con
duction may be neglected when dimension of problem domain is large
enough to assume the propagation is adiabatic and there are not any
underwater thermal flows [28, 30, 31], which are met in the present
study. Since the only remaining mechanism of dissipation is the viscous
attenuation, it is simply referred as dissipation. Omitting all other types
of attenuation, the dissipation is considered which is represented by
following equation as proposed by Stokes (1845).
Γ = γ
∂
∂t
(
∇2
P
)
, γ =
4μ
3ρ0c2
0
(4)
where γ is dissipation coefficient. For sea water at the prespecified
condition, μ=0.00109 Pa.s, ρ0=1025 Kg/m3
, and c0=1505 m/s, the
dissipation coefficient is very small, say γ = 6.26 × 10− 13
. Additionally,
the dissipation term is a third order derivative of time and space dif
ferentials. So, even for a very large time differential, order of magnitude
of the dissipation term is at least one order lower than other terms of
wave equation. Furthermore, it is common, in solution of wave equa
tion, to convert time domain of the problem into frequency domain for
easier analytical solution of the wave equation. If it is assumed that the
source is a mono-frequency excitation as it meets in this study, the
classical wave equation converts to Helmholtz equation in frequency
domain. After some mathematical manipulation, the dissipated pressure
can be expressed as follows (Kinsler and Frey, 1962).
P = P0e− αx
ej(ωt− kx)
(5)
where α, ω, and k are absorption coefficient, wave frequency and wave
number, respectively.The dissipated pressure can be rearranged with
some assumptions to relate the absorption coefficient with acoustic in
tensity, I, and damped pressure due to dissipation (Bjørnø, 2017).
α =
10
x
log
(
I1
I2
)
=
20
x
log
(
P0
P
)
(6)
where unit of α is in dB/m but it is more common to express it as dB/Km
due to very small contribution of viscous dissipation on pressure
damping in regions close to the source.The absorption coefficient is
proportional to square of the source frequency (Kinsler and Frey, 1962,
Etter, 2003). Thorp (1965) showed that the total sound absorption co
efficient related to viscosity dissipation and all other terms of attenua
tion are very small for lower frequency as it is depicted in Fig. 6. He also
made a comparison between the freshwater and the sea water absorp
tion and concluded that the absorption in the sea water is much higher
than fresh water, especially for lower frequency and mainly due to
relaxation. However, for a very high supersonic excitation frequency,
say more than 1MHz, absorptions in both mediums are almost the same.
It should be noted that absorption coefficient which is plotted in
Fig. 6 includes all terms of attenuation while it was discussed earlier that
the viscous dissipation is the only available term of attenuation in the
present study. Thus, the desired absorption coefficient is lower than that
shown in Thorp results. To address an appropriate value of absorption
coefficient, the frequency of excitation of pressure source should be
determined. The hydrodynamic impact pressure in the present study is a
pulse wave, very similar to explosion pressure wave. Generally, this type
of waves is low frequency high level noises. Referring to Fig. 2, one can
find three measured pressure pulses during wedge entry. The frequency
of these pulses can be easily estimated using Fourier transformation of
the measured pressure as shown in Fig. 7. Three peak frequencies are
readily distinguished including 0, 73.3, and 305.3 Hz. These are the
main frequencies of the impact pressure propagating into water. The
first frequency of 0 Hz is omitted for avoiding numerical errors in
Fourier transformation and the main frequency of impact pressure is
evaluated as 73.3 Hz.
Having known the wave frequency, Equations (5) and (6) result in an
initial pressure damping of less than 0.1 % after traveling 100 meters
and about 1 % after 1 Km in the sea water. Based on this discussion, all
attenuation and viscous dissipation mechanism is ignored in this study
and the governing equation is reduced as follows.
∇2
P −
1
c2
∂2
P
∂t2
= O(x, y, t) (7)
Eliminating the dissipation term in classical wave equation yields to
a 2D acoustic equation which fairly simulates the propagation of pres
sure wave as well as its interaction with boundaries. A rectangle domain,
ABCD, is defined as shown in Fig. 4 which is two dimensional. As dis
cussed earlier, the wave equation is modified for perturbation impact
pressure. Thus, all uniform pressure such as those caused by the gravity
and all other body forces are neglected, for the sake of simplicity.
Therefore, the air-water interface remains undisturbed including pile-up
effect as explained in Introduction. The acoustic wave equation is a
linear homogeneous hyperbolic partial differential equation. The num
ber of required boundary conditions areequal to the sum of derivative
orders of spatial and temporal variables. So, four boundary conditions
and two initial conditions should be specified.The boundaries are
definedfar enough from the pressure sources and remain undisturbed
during solution. This configuration is carried out using Dirichlet
boundary condition.
Fig. 6. Absorption of sound in seawater (Thorp, 1965).
Fig. 7. Spectrum of Fourier transformation of experimental data shown
in Figure 2.

5
{
P(− a, y, t) = P(a, y, t) = 0, − h ≤ y ≤ 0, t > 0
P(x, 0, t) = P(x, − h, t) = 0, |x| ≤ a, t > 0
(8)
This type of boundary condition is also known as reflecting boundary
condition. Applying the above boundary conditions result in reflecting
waves immediately as the sound waves meet the boundaries. Another
type of boundary condition is symmetric boundary condition which is a
special case of Neumann boundary condition. This boundary condition
is usually utilized for reduction of computation time in numerical so
lution. If line x = 0 is assumed to be the symmetric line of the problem,
this boundary condition can be presented as follows and applied on the
symmetry line.
∂P(0, y, t)
∂x
= 0 (9)
Depending on the application of the problem, another type of
boundary condition may be employed which is called absorbing
boundary condition (Cohen and Pernet, 2016).
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∂P(− a, y, t)
∂x
= ±c
∂P(− a, y, t)
∂t
= 0,
∂P(a, y, t)
∂x
= ∓c
∂P(a, y, t)
∂t
= 0
∂P(x, 0, t)
∂x
= ±c
∂P(x, 0, t)
∂t
= 0,
∂P(x, − h, t)
∂x
= ∓c
∂P(0, − h, t)
∂t
= 0
(10)
The above boundary condition simply states that the spatial gradient
of pressure is equal to its temporal derivative. This readily yields to the
fact that all waves can artificially pass the boundaries without any
induced perturbation. In other words, this boundary condition simulates
the situation where the boundaries physically do not exist.
In addition to spatial boundary condition, two temporal conditions,
say initial conditions, are also required for completing the solution. It is
common in solution of the acoustic equation, to employ mixed initial
conditions to achieve more precise results as follows:
⎧
⎨
⎩
P(x, y, 0) = f(x, y), (x, y) ∈ R
∂P
∂t
(x, y, 0) = g(x, y), (x, y) ∈ R
(11)
Where R = [− a, a] × [0, − h] defines the domain region and f and g are
any arbitrary spatial functions.
Analytical solution of 2D wave equation is available in different
mathematical forms depending on the boundary conditions and solution
techniques. Some of the most famous techniques are d’Alembert solu
tion, separation of variables and Fourier transformation. Since the sep
aration of variable method gives the most general solution, this method
is utilized as the analytical solution of the problem as given in Equation
(7) ignoring source term temporary, for sake of simplicity. One can
define pressure field as a product of three independent functions as
follows.
P(x, y, t) = F(x, y)T(t), F(x, y) = X(x)Y(y) (12)
where F, T, X, and Y are any arbitrary functions of their corresponding
variables. Substituting Equation (8) in Equation (1) yields to three ODEs
as follows:
⎧
⎨
⎩
T̈ + ω2
T = 0, ω = cλ
Fxx + Fyy = − λ2
F →
{
X′′
− K1X = 0
Y′′
− K2Y = 0
(13)
where ω, c, λ, K1, and K2 are all positive constants. The first application
of the separation of variables method is deriving 2D Helmholtz equation,
(Fxx + Fyy = − λ2
F), which is well-known in the acoustic problem. By
two consecutive application of theseparation of variables method and
assuming reflecting boundary condition, a general form of an analytical
solution can be found as follows.
Pmn(x, y, t) = Xm(x)Yn(y)Tmn(t)
=
∑
∞
n=1
∑
∞
m=1
(sinμmx)(sinνny)
(
Bmncosλmnt + B*
mnsinλmnt
)
Bmn =
8
ah
∫a
− a
∫0
− h
f(x, y)(sinμmx)(sinνny)dydx
B*
mn =
8
ahλmn
∫a
− a
∫0
− h
g(x, y)(sinμmx)(sinνny)dydx (14)
wherem and n are counter indices in x and y directions and Bmn and
B*mnare harmonic coefficients, respectively. This solution is a harmonic
solution of 2D wave equation which is the summation of different solo
frequency harmonic answers. Three distinctive frequencies are intro
duced i.e., oscillating in time by λmn, in x direction by μm and in y di
rection by νn. These frequencies are called characteristics frequencies
and mathematically related to each other as follows:
λmn = c
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
̅
μ2
m + ν2
n
√
= cπ
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
m2
a2
+
4n2
h2
√
(15)
Pressure waves like other waves can be also distinguished by their
wave number,λ = ω/c, and wavelength, L = 2πc/ω. Analytical solution
of the acoustic equation is not always available mainly due to compli
cated mathematical form of initial conditions f(x,y) and g(x,y). If the
integral of harmonic coefficients cannot be analytically calculated, nu
merical integration methods may be utilized. However, the initial con
ditions, f(x,y) and g(x,y), cannot not be simply evaluated in practical
applications. Once initial conditions defined with complicated func
tions, the concept of source function may solve the problem as shown in
Numerical Implementation. Including source function in acoustic
equation can be effectively employed in numerical implementation of
the equation. Many types of complicated initial conditions can be
reproduced using the concept of source function in the acoustic problem.
Fig. 8 illustrates the stepwise symmetric and asymmetric water entry
of wedges into the water. Initially, a wedge touches the water surface at
t0. It pierces more at the next time increments with velocity of V(t).
Immediately after chords of the wedge get wet, a hydrodynamic impact
pressure is induced just on the wet sides of the wedge. A typical distri
bution of the hydrodynamic impact pressure is schematically shown in
Fig. 8 with hatched dark gray regions.
Fig. 8. Wedge entry in different time steps for (a) symmetric entry and (b)
asymmetric entry. Pressure distribution on the wedges’ chords are shown in
hatched dark gray regions.

6
The profile of such pressure distribution is already available by
analytical, experimental, and numerical methods. A symmetric entry
yields to a symmetric pressure distribution on both sides of the wedge.
The pressure distribution can be expressed as a function of the length of
the chords which is indicated by ζ and η for the left and the right sides,
respectively. Thus, P(ζ) and P(η) are identical for the symmetric entry.
As far as an asymmetry entry is concerned as shown in Fig. 8, P(ζ) is
higher and generated earlier than P(η) because the left side pierces the
water sooner with lower deadrise angle. It is worth noting that as wedge
body gradually penetrates the water, the wet breadth of the wedge in
creases, and the pressure amplitude amplifies.
The solution of the acoustic problem requires precise capturing of the
impact pressure distribution, P(ζ) and P(η) as time marches. For correct
application of the impact pressure on the wedge boundaries, one can
modify the upper rectangular geometry of the domain by subtracting the
instantaneous immersed geometry of the wedge from the rectangular
domain as shown in Fig. 9. Then the initial conditions are applied on the
upper boundary of domain by imposing P(ζ) on line EF, and P(η) on FG.
However, its mathematical implementation is not straightforward and
makes the solution very complicated.
Alternatively, the domain can be unchanged where the pressure
distribution is applied using source function technique. Now, the
domain is rectangular and pressure distribution, P(ζ) and P(η), are
applied on artificial lines of EF and FG as source functions. Since the
wave equation is a linear PDE, its solution as well as its sources can be
sum of other solutions or sources. So, the total source function is simply
defined as follows:
O(x, y) = P(ζ(x, y)) + P(η(x, y))
Although analytical treatment of the source function is not easy, its
numerical implementation is straightforward. Similar procedure is also
available for asymmetric entry but with different lengths of lines EF and
FG. The appropriate distribution of the impact pressure, P(ζ) and P(η),
are evaluated using Wagner theory (Wagner, 1366) for symmetric entry,
Toyama modeling (Toyama, 1993) for asymmetric entry, and experi
mental results conducted by Barjasteh et al. (2016). The experimental
results are used to correct impact pressure predicted by analytical the
ories for small deadrise angles. Now, the governing equation, boundary
conditions, initial conditions, and excitation impact pressure are all
known and numerical solution of the problem can be started.
3. Numerical implementation
The main idea behind the numerical solution of the problem is to
substitute distribution of impact pressure with multiple nodal source
functions in the acoustic equation. Wave equation as a hyperbolic PDE
has a straightforward numerical solution especially using Finite Differ
ence Method (FDM). As it is common in FDM, the first step is to check
whether a FDM scheme is available for the governing equation. To do
such a task, all terms of the governing equation should be expressed in
their well-known differencing schemes and substituting in the Equation
(7) yields the following differencing scheme:
Pk+1
i,j − 2Pk
i,j + Pk− 1
i,j
Δt2
= c2
Pk
i+1,j − 2Pk
i,j + Pk
i− 1,j
Δx2
+ c2
Pk
i,j+1 − 2Pk
i,j + Pk
i,j− 1
Δy2
+ Ok
i,j
(16)
where here i, j, and k are counter indices in horizontal, vertical, and
temporal directions. Recalling the governing equation as presented by
Equation (7) and substituting the corresponding differencing de
rivatives, the following explicit differencing scheme is driven assuming
Δx = Δy.
Pk+1
i,j = 2Pk
i,j − Pk− 1
i,j + [CFL]2
(
Pk
i+1,j + Pk
i− 1,j − 4Pk
i,j + Pk
i,j+1 + Pk
i,j− 1
)
+ Δt2
Ok
i,j
(17)
where CFL = (cΔt)/Δx is Courant–Friedrichs–Lewy parameter to ensure
convergence of numerical solution. The pressure field at time step k+1 is
a function of known parameters in the previous time step, k. The
boundary conditions should also be discretized in numerical solution of
Fig. 9. Modified domain of the problem by removing instantaneous immersed
part of the wedge.
Fig. 10. Application of the impact pressure as multiple nodal source functions.
Fig. 11. Density of nodal distribution.
Table 1
Variation of nodal sources for a typical symmetric entry.
Time step Nodes (ni,nj)
t1 (1,2, 2,2, 3,1)
t2 (1,2, 2,2, 3,1)
t3 (1,3, 2,2, 3,2, 4,1)
t4 (1,3, 2,3, 3,2, 4,1)

7
the problem. Depending on specified boundary conditions, the general
forms of their discretization are varied. The absorbing boundary con
ditions discretized as followings.
Pk+1
1,j = Pk
2,j +
CFL − 1
CFL + 1
[
Pk+1
2,j − Pk
1,j
]
Pk+1
nx,j = Pk
nx− 1,j +
CFL − 1
CFL + 1
[
Pk+1
nx− 1,j − Pk
nx,j
]
Pk+1
i,1 = Pk
i,2 +
CFL − 1
CFL + 1
[
Pk+1
i,2 − Pk
i,1
]
Pk+1
i,ny
= Pk
i,ny− 1
+
CFL − 1
CFL + 1
[
Pk+1
i,ny− 1
− Pk
i,ny
]
(18)
where nx and ny indicate the last domain nodes on boundaries in xand y
directions, respectively. After discretization of the governing equation
and its boundary conditions, the application of pressure source function
is started. Fig. 10 indicates a discretized half-domain for entry of a
symmetric wedge. The wedge is penetrating the water and the impact
pressure is increasing on the wet chords of the wedge. p(η) is a function
of chord length, η, and evaluated from analytical impact modeling of
Wagner (1366), Toyama (1993) and experimental results reported by
Barjasteh et al. (2016).
The source function is modeled by summation of multiple nodal
sources. initially, the solver computes intersection points between the
chord and vertical lines x = xni
and calculates their corresponding ver
tical positions, say yni
= y0 + η(xni
)tanβ, where y0 is vertical position of
the wedge apex. Additionally, it finds upper and lower vertical nodes for
each intersection point using following inequality at x = xni
.
y
(
ni, nj
)〈
yni < y
(
ni, nj+1
)
(19)
Furthermore, the solver checks which adjacent vertical node is closer
to the intersection point at x = xni
by comparing their vertical distances,
[ywi
− y(ni, nj)] and [y(ni,nj+1) − ywi
]. Once this node is found, the related
impact pressure, P(η(ni)), is applied on it. To ensure adequate applica
tion of the impact pressure, density of mesh distribution is increased at
wedge location. Fig. 11illustrates the mesh grid for a typical symmetric
entry.
The wedge location and related position of nodal sources are
continuously updated at each time step.Table 1 shows the variation of
nodal sources for four successive time steps which is illustrated in
Fig. 10.
Once the nodal sources are determined, the numerical solution of the
problem is started using the known values of nodal pressure sources.
Depending on the symmetry of the impact and deadrise angle of the
wedge, different pressure equations are employed for prediction of
nodal pressures. For instance, the Wagner equation determines the
Fig. 13. Propagation of the impact pressure for symmetric entry of a wedge with β0 = 5∘
at four different timesteps.
Fig. 12. Validation of numerical results using percentage of absolute error.

8
impact pressure for symmetric entry with deadrise angles of higher than
15◦
at intersection points, η(xni
).
P(η)
ρ
= V
L
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
L2 − η2
√
dL
dt
+
dV
dt
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
L2 − η2
√
(20)
whereρ is the water density, V is instantaneous vertical component of the
body speed and L is wet beam of the wedge. Toyama relation is also used
for asymmetric entries.
P(η)
ρ
= V
(1 + ξσ)
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 − ξ2
√
dL
dt
−
V2
2
⋅
ξ2
1 − ξ2
+
(
L
̅̅̅̅̅̅̅̅̅̅̅̅̅
1 − ξ2
√ ) dV
dt
(21)
where ξ = (η − σL)/L and σ is eccentric values of the asymmetric wedge
at any instantaneous draft. Both equations are overestimate the impact
pressure for small deadrise angles lower than 15◦
. These overestimated
pressures are corrected using measured impact pressure reported by
Barjasteh et al. (2016).
4. Results and discussions
4.1.
To validate the solver convergence and its stability, the numerical
results are compared with an analytical solution of a typical 2D wave
equation in solid shells. Suppose that a rectangular shell with 2a=3 and
h=2 is deformed to have the given shape of f(x, y) = xy(2 − x)(3 − y) and
an initial velocity of g(x,y) = 0. One can employ 2D wave equation to
find the shell shape form for t>0. All edges are kept fixed in space before
solution and then released at t=0. The analytical solution of the problem
can be found using the separation techniques as follows (Soedel, 2004).
P(x, y, t) = Pf + Pg =
576
π6
∑
∞
n=1
∑
∞
m=1
[(
1 + (− 1)m+1)(
1 + (− 1)n+1)
m3n3
sin
mπ
2
xsin
nπ
3
ycosπ
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
9m2 + 4n2
√
t
]
(22)
The solver results are shown as percentage of absolute error
compared to the analytical solution in Fig. 12 . The error is less than one
percent with 2000 increments. This type of error presentation is also
known as mesh independency analysis in numerical methods. The re
sults show that the solver properly solves a 2D wave equation.
Fig. 13 illustrates the propagation of the impact pressure for the
symmetric entry of a wedgewith β0 = 5∘
. The domain is a rectangle of 10
by 5 meters where half of domain, 5 by 5, is calculated taking to account
the symmetry quality for the less calculation time. Discretization of
domain boundaries is implemented by Δxmin = Δymin = 0.1 mm at the
wedge location and Δxmax = Δymax = 13 mm at far field. The full rect
angle of 10 by 5 meters as calculation domain,has approximately
4,100,000 elements and the half rectangle has almost half elements
number.
Time increment is a crucial parameter which guarantees the
convergence of the numerical solution. The time increment is controlled
by CFL numbers. The disturbance is propagating with local sound speed.
So, the CFL number is chosen 0.5 or lower to ensure accurate capturing
of propagation in each time step. All wedges are freely falling from
height of 0.5 meters with initial impact speed of v0 =
̅̅̅̅̅̅̅̅
2gh
√
= 3.13 m/s.
The impact pressure distribution is calculated by Wagner method while
the peak pressure is corrected by Barjasteh et al. (2016). The numerical
results are presented for four time-steps with increment of about one
Fig. 14. Propagation of the impact pressure for symmetric entry of a wedge with β = 30∘
at four different timesteps.

9
Fig. 15. Propagation of the impact pressure for asymmetric entry of a wedge of βL = 5∘
and βR = 35∘
.

10
millisecond. The impact pressure reaches its maximum value in less than
two milliseconds. The solution continues for 4 milliseconds after the
impact.
Recalling Fig. 13, the wave-front carries higher pressure as can be
seen beneath the wedge and shown by different colors. One can connect
the wavefronts and draw a straight line which deviates from the left
boundary by about 5◦
. This angle is called propagating angle and is very
close to the deadrise angle of the wedge.
Upon detaching the pressure wave from the wedge, a low vacuum
region is generated. The low pressure due to the vacuum propagates
right behind the pressure waves as illustrated with dark blue color in
Fig. 10. Although the propagation speed assumed constant in the present
study, the wavefronts carry different value of the wave energy. So, the
distribution of propagating front pressures is not symmetric. The pres
sure disturbance travels with the speed of sound and meets the bottom
boundary of the domain att = 5/1505 = 0.0033 seconds. This time is
checked during numerical manipulation and it can also be verified by
referring to Fig. 13.
Similar simulation is carried out for symmetric entry of a wedge with
β0 = 30∘
depicted in Fig. 14. Since the propagation speed remains con
stant, propagation pattern of the impact pressure is very similar to the
previous case of β=5o
. However, the higher deadrise angle causes the
lower impact pressure as well as different propagating angle. Similar
results of symmetric entry of two wedges may lead to a generalized
conclusion that is the symmetric wedges have the same propagating
pattern.
The asymmetric entry is investigated as shown in Fig. 15 for a wedge
with β0 = 20∘
and inclination angle of ϕ = 15∘
. At this orientation, the
left deadrise angle, βL = 5∘
, is lower than the right deadrise angle, βR =
35∘
. This asymmetric water entry readily results in much higher impact
pressure at the left side in respect to the right side.The simulation of the
asymmetric problem is conducted on the full rectangle of 10 by 5 meters.
The impact pressure at the left side is higher than the right side about
one order of magnitude. Thus, hydrodynamic pressure at the left side
governs the problem. This can be immediately seen in the numerical
results shown by different colors. Both the left and the right sides of the
wedge, produces impact pressures as multiple nodal source functions.
The propagation angle of the problem is generally determined by the
deadrise angle of the left side as seen in Fig. 15. This pattern is an
important result of asymmetric problem and recommends that the lower
deadrise angle is mainly controls the impact pressure and the right
impact pressure can be neglected for large inclination angles.
It is common to analyze underwater acoustics using special types of
pressure transducers called hydrophones. These sensors can adequately
measure the acoustic pressure with high resolutions. One can study the
propagation of underwater acoustic pressure by an array of hydrophones
which are mounted in different desired locations. Phase and amplitude
differences between all measured pressure can effectively evaluate the
propagation of the waves especially when multiple arrays are used. This
can also be carried out in the present investigation by virtue of numer
ical hydrophones as shown in Fig. 16. This figure illustrates an array of
hydrophones mounted symmetrically about the symmetry line of the
domain, say x = 0.
Fig. 17 depicts the recorded acoustic pressure in symmetric entry of
wedges corresponding to Fig. 13 and Fig. 14. These pressures are
recorded by hydrophones H3, H4, and H5, respectively. The problem is
symmetric and thus the pressures reported by H1 and H5 as well as H2
and H4 are the same. The numerical measurement of the propagated
pressure is made for t = 5/1505 = 0.0033 seconds, the time period that
the wave requires to travel from its source to the bed of the domain. This
ensures full capturing of the propagation behavior during this period.
The boundary conditions are set to absorbing conditions and waves can
freely pass through the boundaries without any interactions.
The first issue which is instantly found from Fig. 17 is ascending
values of recorded pressures. After initiating the solution and commence
of the water entry, the wave starts propagating and takes about 0.1
milliseconds to reach the hydrophones and nothing is recorded by sen
sors. Then the pressures are rapidly rising to their peaks at the endpoints.
The peak impact pressure in entry of wedges with β=5◦
and β=30◦
are
418 KPa and 41 KPa as reported by Barjasteh et al. (2016). These peaks,
marginally lower, can be traced at the end of the measurement time. As
time marches, the acoustic pressures asymptotically tend to their peak
values. Although dissipation mechanisms are ignored, there is a kind of
mathematical attenuation for cylindrical and spherical waves (Kinsler
and Frey, 1962, Etter, 2003, Bjørnø, 2017). The deviation between the
experimental pressure peak and those recorded by hydrophones is
referred to this type of attenuation.
Recalling Fig. 17-a, although both hydrophones H3 and H5 are
placed at the same distance from the origin, hydrophone H3 records a
higher pressure and earlier than H5. Barjasteh et al. (2016) showed that
the peak pressure in this deadrise angle appears close to the apex. Thus,
the first wave front is seen by hydrophone H3 that is mounted right
beneath the wedge apex. All recorded pressures show an oscillating
behavior with different frequencies and amplitudes. This oscillation
refers to alternating compression waves due to vacuum regions behind
the strong pressure waves as discussed in the previous section. Measured
pressures carry oscillation with a phase difference mainly because of
Fig. 16. Location of numerical hydrophones.
Fig. 17. Acoustic pressure in symmetric entry of wedges recorded by hydro
phones (a) β=5◦
, (b) β=30◦
.
Fig. 18. Acoustic pressure in asymmetric entry of wedges recorded by hydro
phones,β=20◦
and ϕ=15◦
.

11
wave interactions. Interestingly, hydrophone H4 reports a higher value
than that recorded by H5. This phenomenon and pressure difference
between H3 and H5 are totally controlled by deadrise angle of the wedge
and related wave propagation.
Similar behavior is also observed in recorded pressures by the hy
drophones for entry of wedge with β=30◦
as illustrated in Fig. 17-b.
There are two distinctive differences between the results of these two
wedges. The first one refers to lower amplitude of pressure oscillation in
wedge entry of β=30◦
. The impact pressure is lower than wedge of β=5◦
and results in weaker wave fronts and subsequently weaker compression
vacuum waves. The second difference is higher pressure of H5 rather
than H4, in contrast to wedge of β=5◦
. As discussed before, this subject is
controlled by wedge deadrise angle.
Fig. 18 illustrates the underwater acoustic pressure for water entry of
an asymmetric wedge with β = 20∘
and ϕ = 15∘
. This entry is also
studied in experiments conducted by Barjasteh et al. (2016). This
asymmetric entry generates a propagation pattern which is totally
different with symmetric entry. In asymmetric entry, the deadrise angle
of inclined side is smaller than the other side. This smaller deadrise angle
causes higher impact pressure. The pressure difference can be well
distinguished in Fig. 18. Highest peak pressure logged by hydrophone
H3 beneath the wedge apex as previously seen in symmetric entry of
wedge with β = 5∘
while the lowest pressure measured by sensor H5
located at the free surface on the right side. Similar oscillations and
phase differences are also visible in this figure with the same reasons as
analyzed before.
Some boundary reflects the sound waves. Fig. 19 shows the propa
gation of the impact pressure for a symmetric entry of a wedge of β0 =
30∘
, and reflecting boundary conditions on the right and bottom,
namely x = 5 and y = − 5. It clearly shows the reflection waves and
their interference with the incoming waves. Propagation angle of the
problem is about 30◦
. Thus, the reflected wave from the bottom
Fig. 19. Reflection of pressure wave from bottom and the right boundaries for symmetric entry of a wedge with β0 = 30∘
in a range of time-steps.

12
boundary is stronger than those reflected from the right side. The
complicated pattern of the wave reflection and interference show that
the numerical solver is stable and appropriately captures the problem.
Modeling the reflection of pressure wave has a wide application in
simulation of shallow water impact. Additionally, the capability of the
numerical simulation can be effectively employed to assess a suitable
tank dimensions as well as to disregard sound wave effects on pressure
record in water impact experiments.
5. Conclusion
The propagation of the acoustic wave in the water entry of wedge-
shaped bodies is investigated in this study. It is shown that the impact
pressure changes to the acoustic waves and can be effectively modeled
by the wave equation. Thus, the propagation is modeled by the 2D
acoustic equation. Both the symmetric and asymmetric water entries are
studied. Symmetric entry of wedges shows that the patterns of the
propagating waves are similar for different deadriseangles having
different peak pressures and propagation angles. Furthermore, the
asymmetric entry results in different propagating pattern which its
characteristics is dominated by lower deadrise angle of the wedge sides.
Reflection of pressure waves is also modeled using reflecting bound
aries. Finally, the study of propagation of the impact pressure in the
water entry problem is accomplished using modified 2D acoustic
equation with appropriate boundary and initial conditions. The devel
oped numerical solver can provide an approximate numerical module
for the engineering design of boats and ships.
Authorship statement
All persons who meet authorship criteria are listed as authors, and all
authors certify that they have participated sufficiently in the work to
take public responsibility for the content, including participation in the
concept, design, analysis, writing, or revision of the manuscript.
Furthermore, each author certifies that this material or similar material
has not been and will not be submitted to or published in any other
publication before its appearance in the Journal of Applied Ocean
Research.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
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