Title:	Simulate Whale Ship Strikes
Version:	0.6.2
Maintainer:	Dan Kelley <dan.kelley@dal.ca>
Description:	Provides tools for simulating the biophysical effects of vessel-strikes on whales. The aim is to support the evaluation of marine policies limiting ship speeds through regions in which whales reside. This is important because ship strikes are a major source of lethality for several whale species, including the critically endangered North Atlantic right whale. In this analysis, whales are modelled with a four-layer system comprising skin, blubber, sub-layer (muscle or organ) and bone. Reasonable values for the material properties of these layers, along with other factors such as whale surface area and mass, are provided for a variety of whale species. Similarly, key values are provided for several ship types. The collision is modelled according to Newtonian dynamics, with stresses and strains within the whale layers being simulated over time. The simulation results are analyzed in the context of whale-strike data, to develop a Lethality Index for the whale in the modelled collision. For the underlying science, see Kelley and other "Assessing the Lethality of Ship Strikes on Whales Using Simple Biophysical Models." (2021) <doi:10.1111/mms.12745>. For more on the R code, see Kelley "'whalestrike': An R package for simulating ship strikes on whales" (2024) <doi:10.21105/joss.06473>.
Depends:	bslib, deSolve, R (≥ 3.5.0), shiny
Suggests:	knitr, rmarkdown, testthat
License:	GPL (≥ 3)
Encoding:	UTF-8
BugReports:	https://github.com/dankelley/whalestrike/issues
URL:	https://dankelley.github.io/whalestrike/
LazyData:	true
RoxygenNote:	7.3.2
BuildVignettes:	true
VignetteBuilder:	knitr
NeedsCompilation:	no
Packaged:	2026-03-03 13:25:51 UTC; kelley
Author:	Dan Kelley [aut, cre], James Vlasic [rtm], Sean Brilliant [rtm], Alexandra Mayette [rtm]
Repository:	CRAN
Date/Publication:	2026-03-06 18:20:02 UTC

whalestrike: A Package to Simulate Ship-Whale Collisions

Description

This package solves Newton's second law for a simple model of a ship colliding with a whale. This is a stripped-down model that does not attempt to simulate the biomechanical interactions that can be simulated in finite-element treatments such as that of Raymond (2007). For an in-depth discussion of the reason for writing the model, of the principles involved in its framing, and its use in developing a criterion for strike lethality, please see Kelley et al. (2021).

Details

The goal of the model is to establish a convenient framework for rapid computation of impacts in a wide variety of conditions. The model runs quickly enough to keep up with mouse movements to select conditions, in a companion R shiny app. That app lets the user see the effects of changing contact area, ship speed, etc., as a way to build intuition for scenarios ranging from collision with a slow-moving fishing boat to collision with a thin dagger-board of a much swifter racing sailboat. Another advantage of the simple formulation is that it makes it easy to modify various dynamical and biomechanical parameters, to add new forces, and to explore a range of criteria for whale damage.

The documentation for strike() provides a practical example of using the main functions of this package, while the package vignette provides a general overview. Kelley et al (2021) provide more detail about the mathematical framework of the package, along with a discussion of its purpose and application to real-world problems of ship strikes on whales.

Further reading

• Daoust, Pierre-Yves, Emilie L. Couture, Tonya Wimmer, and Laura Bourque. "Incident Report. North Atlantic Right Whale Mortality Event in the Gulf of St. Lawrence, 2017." Canadian Wildlife Health Cooperative, Marine Animal Response Society, and Fisheries and Oceans Canada, 2018. https://publications.gc.ca/site/eng/9.850838/publication.html.

• Fortune, Sarah M. E., Andrew W. Trites, Wayne L. Perryman, Michael J. Moore, Heather M. Pettis, and Morgan S. Lynn. "Growth and Rapid Early Development of North Atlantic Right Whales (Eubalaena Glacialis)." Journal of Mammalogy 93, no. 5 (2012): 1342-54. doi:10.1644/11-MAMM-A-297.1.

• Grear, Molly E., Michael R. Motley, Stephanie B. Crofts, Amanda E. Witt, Adam P. Summers, and Petra Ditsche. "Mechanical Properties of Harbor Seal Skin and Blubber–a Test of Anisotropy." Zoology 126 (2018): 137-44. doi:10.1016/j.zool.2017.11.002.

• Kelley, Dan E., James P. Vlasic, and Sean W. Brillant. "Assessing the Lethality of Ship Strikes on Whales Using Simple Biophysical Models." Marine Mammal Science 37, no. 1 (January 2021): 251–67. doi:10.1111/mms.12745.

• Kelley, Dan E. "Composite Spring," May 28, 2018. 20180528_composite_string. Dan Kelley's working notes.

• Kelley, Dan. "Whale Area," June 23, 2018. 20180623_whale_area. Dan Kelley's working notes.

• Kelley, Dan. "Ship Propulsion," July 1, 2018. 20180701_ship_propulsion. Dan Kelley's working notes.

• Kelley, Dan. "Whale Mass," July 7, 2018. 20180707_whale_mass. Dan Kelley's working notes.

• Kelley, Dan E."“Whalestrike: An R Package for Simulating Ship Strikes on Whales." Journal of Open Source Software 9, no. 97 (2024): 6473. doi:10.21105/joss.06473.

• MAN Diesel & Turbo. "Basic Principles of Propulsion." MAN Diesel & Turbo, 2011. https://spain.mandieselturbo.com/docs/librariesprovider10/sistemas-propulsivos-marinos/basic-principles-of-ship-propulsion.pdf?sfvrsn=2

• Manen, J. D. van, and P. van Oossanen. "Resistance." In Principles of Naval Architecture (Second Revision), Volume II - Resistance, Propulsion and Vibration, edited by Edward V Lewis, Second Edition, 1-125. Jersey City, NJ: Society of Naval Architects and Marine Engineers (U.S.), 1988.

• Mayette, Alexandra. "Whale Layer Thickness." December 15, 2025. (Personal communication of a 5-page document.)

• Mayette, Alexandra, and Sean W. Brillant. "A Regression-Based Method to Estimate Vessel Mass for Use in Whale-Ship Strike Risk Models." PloS One 21, no. 1 (2026): e0339760. doi:10.1371/journal.pone.0339760.

• Miller, Carolyn A., Desray Reeb, Peter B. Best, Amy R. Knowlton, Moira W. Brown, and Michael J. Moore. "Blubber Thickness in Right Whales Eubalaena Glacialis and Eubalaena Australis Related with Reproduction, Life History Status and Prey Abundance." Marine Ecology Progress Series 438 (2011): 267-83.

• Moore, M.J., A.R. Knowlton, S.D. Kraus, W.A. McLellan, and R.K. Bonde. "Morphometry, Gross Morphology and Available Histopathology in North Atlantic Right Whale (Eubalaena Glacialis) Mortalities (1970 to 2002)." Journal of Cetacean Research and Management 6, no. 3 (2005): 199-214.

• Ng, Laurel J., Vladislav Volman, Melissa M. Gibbons, Pi Phohomsiri, Jianxia Cui, Darrell J. Swenson, and James H. Stuhmiller. "A Mechanistic End-to-End Concussion Model That Translates Head Kinematics to Neurologic Injury." Frontiers in Neurology 8, no. JUN (2017): 1-18. doi:10.3389/fneur.2017.00269

• Raymond, J. J. "Development of a Numerical Model to Predict Impact Forces on a North Atlantic Right Whale during Collision with a Vessel." University of New Hampshire, 2007. https://scholars.unh.edu/thesis/309/.

• Soetaert, Karline, Thomas Petzoldt, and R. Woodrow Setzer. "Solving Differential Equations in R: Package DeSolve." Journal of Statistical Software; Vol 1, Issue 9, 2010. doi:10.18637/jss.v033.i09.

Author(s)

Maintainer: Dan Kelley dan.kelley@dal.ca (ORCID)

Other contributors:

• James Vlasic jvlasic@dal.ca (ORCID) [research team member]

• Sean Brilliant seanb@cwf-fcf.org (ORCID) [research team member]

• Alexandra Mayette alexandram@cwf-fcf.org (ORCID) [research team member]

See Also

Useful links:

• https://dankelley.github.io/whalestrike/

• Report bugs at https://github.com/dankelley/whalestrike/issues

GUI application for whale simulation

Description

Graphical-user-interface tool for exploring whale-strike simulations.

Usage

app(debug = FALSE)

Arguments

Details

Sliders, buttons, and choosers are grouped into panes that appear on the left of the view. When app() first opens, all of these panes are closed. To get acquainted with the app, try adjusting the controllers that are visible on the initial view. Then, open the "ship" pane and increase the ship mass. Do you find that the results make qualitative sense? Continue this process, exploring all the panes. A half-hour of such exploration should be enough to build enough confidence to start investigating practical applications. To learn more about how the simulations are carried out, and to read more about the underlying goals of this tool, please consult Kelley et al. (2021) and Kelley (2024). Extensive details on the calculations are provided in the help pages for the various functions of the whalestrike package, of which that for whalestrike() is a good starting point.

More information on app() in video form on youtube.

Note that an older version of a similar GUI application is still available as app_2025(), but it is not maintained and is slated for removal in the early months of 2026.

Value

A Shiny app object, created by a call to shinyApp().

Author(s)

Dan Kelley

References

Kelley, Dan E., James P. Vlasic, and Sean W. Brillant. "Assessing the Lethality of Ship Strikes on Whales Using Simple Biophysical Models." Marine Mammal Science 37, no. 1 (2021): 251–67. doi:10.1111/mms.12745.

Kelley, Dan E."“Whalestrike: An R Package for Simulating Ship Strikes on Whales." Journal of Open Source Software 9, no. 97 (2024): 6473. doi:10.21105/joss.06473.

Mayette, Alexandra. "Whale Layer Thickness." December 15, 2025. (Personal communication of a 5-page document.)

Mayette, Alexandra, and Sean W. Brillant. "A Regression-Based Method to Estimate Vessel Mass for Use in Whale-Ship Strike Risk Models." PloS One 21, no. 1 (2026): e0339760. doi:10.1371/journal.pone.0339760.

See Also

Other interactive apps: app_2025()

GUI app for interactive whale-strike simulations (OLD VERSION, no longer maintained)

Description

The app_2025() function starts a GUI application that makes it easy to run simple simulations and see the results in graphical form. Sliders and buttons permit a fair degree of customization. The application has some build-in documentation, which supplements what can be found in the ‘Details’ section of the present documentation.

Usage

app_2025(mode = "simple", options = list(height = 500))

Arguments

Details

When app_2025() is run, a window will appear within a few moments. At the top of that is a textual introduction to the system, with a button to hide that information. Below is a user-interaction area, with buttons and sliders that control the simulation and the plotted output. Below that is a plotting area, the contents of which depend on the configuration of the simulation as well as the user's selection of items to display.

The default setup, which is shown before the user alters any of the sliders, etc., is a simulation of a small fishing boat, of mass 45 tonnes, moving at speed 10 knots towards a whale of length 13.7m. (The whale length is used to compute its mass, using a formula that is described by the output of typing help("whaleMassFromLength","whalestrike") in an R console).

Sliders are provided for setting certain key properties of the ship and the whale, with italic labels for those properties that are deemed most likely to be adjusted during simulations. The details of these and the other parameters are revealed by typing help("parameters","whalestrike") and help("strike","whalestrike") in an R console.

To the right of the sliders is a column of checkboxes that control the plotted output. At startup, three of these boxes are ticked, yielding a display with three panels showing the time history of the simulation (help("plot.strike","whalestrike") provides details of the plots):

• The left-hand plot panel shows whale and boat location, the former with an indication of the interfaces between skin, blubber, sublayer, and bone.

• The middle panel shows the same information as the left one, but with a whale-centred coordinate system, and with labels for the components. This makes it easier to see the degree to which the layers are compressed during the impact.

• The right panel is an indication of the estimated threat to the four layers of the whale, with curves that are filled with grey for time intervals when the impact stress (force/area) is less than the strength of the material in the layer, and black for times when that threshold is exceeded.

Much can be learned by adjusting the sliders and examining the plotted output. As an exercise, try setting to a particular ship mass of interest, and then to slide the ship speed to higher and lower values, whilst monitoring the "threat" panel for black regions. This will reveal a critical speed for conditions that threaten the whale. Next, try altering the sublayer thickness, which is a surrogate for location along the whale body, because e.g. the sublayer is thinner near the mandible.

Advanced users are likely to want to alter the values of impact width and height. The default setting are intended to mimic a small fishing boat, such as a Cape Islander. Try lowering the width, to simulate a strike by a daggerboard or keel of a sailing boat.

Note that the pulldown menu for setting the whale species affects only whale mass. It does not affect the thicknesses of blubber or sublayer, the values of which have been set up to represent a midsection strike on a North Atlantic Right Whale.

Value

A Shiny app object, created by a call to shinyApp().

Author(s)

Dan Kelley

See Also

Other interactive apps: app()

Calculate derivative using first difference

Description

The derivative is estimated as the ratio of the first-difference of var divided by the first-difference of time. To make the results have the same length as time, the final result is appended at the end.

Usage

derivative(var, t)

Arguments

Value

Derivative estimated by using diff() on both var and time.

Author(s)

Dan Kelley

Dynamical law

Description

This function handles Newton's second law, which is the dynamical law that relates the accelerations of whale and ship to the forces upon each. It is used by strike(), as the latter integrates the acceleration equations to step forward in time through the simulation of a whale-strike event. Thus, dynamics() is a core function of this package. The code is very simple, because the forces are determined by other functions, as described in the “Details” section.

Usage

dynamics(t, y, parms)

Arguments

Details

Given a present state (defined by the positions and velocities of ship and whale) at the present time, apply Newton's second law to find the time derivatives of that state. Forces are determined with whaleCompressionForce(), whaleSkinForce(), shipWaterForce(), whaleWaterForce(), while engine force (assumed constant over the course of a collision) is computed from initial shipWaterForce(). Whale and ship masses are set by parameters(), which also sets up areas, drag coefficients, etc.

Value

An List contain items named dxsdt (time derivative of ship location), dvsdt (time derivative of ship speed), dxwdt (time derivative of whale location) and dvwdt (time derivative of whale speed). These are computed by solving the dynamical system using Newton's second law, based on the known masses of ship and whale, and the forces involved in the collision.

Author(s)

Dan Kelley

References

See whalestrike() for a list of references.

Draw polygon between two xy curves

Description

This adds to an existing plot by filling the area between the lower=lower(x) and upper=upper(x) curves. In most cases, as shown in “Examples”, it is helpful to use xaxs="i" in the preceding plot call, so that the polygon reaches to the edge of the plot area.

Usage

fillplot(x, lower, upper, ...)

Arguments

Value

None. This function is called to add to a plot.

Author(s)

Dan Kelley

Examples

# 1. CO2 record
plot(co2, xaxs = "i", yaxs = "i")
fillplot(time(co2), min(co2), co2, col = "pink")

# 2. stack (summed y) plot
x <- seq(0, 1, 0.01)
lower <- x
upper <- 0.5 * (1 + sin(2 * pi * x / 0.2))
plot(range(x), range(lower, lower + upper),
    type = "n",
    xlab = "x", ylab = "y1, y1+y2",
    xaxs = "i", yaxs = "i"
)
fillplot(x, min(lower), lower, col = "darkgray")
fillplot(x, lower, lower + upper, col = "lightgray")

Convert a speed in knots to a speed in m/s

Description

See also mps2knot(), which is the inverse of this function.

Usage

knot2mps(knot)

Arguments

Value

Speed in m/s.

Author(s)

Dan Kelley

See Also

Other functions dealing with units: mps2knot()

Examples

library(whalestrike)
knots <- seq(0, 20)
plot(knots, knot2mps(knots), xlab = "Speed [knots]", ylab = "Speed [m/s]", type = "l")

Compute lethality index, based on compression stress

Description

The model used for this is the logistic model, fitting observed injury/lethality statistics to the base-10 logarithm of the maximum compression stress during a simulated impact event.

Usage

lethalityIndexFromStress(stress)

Arguments

Value

threat of injury (in range 0 to 1)

Author(s)

Dan Kelley

See Also

Other functions dealing with Whale Lethality index: maximumLethalityIndex(), stressFromLethalityIndex()

Examples

lethalityIndexFromStress(parameters()$logistic$tau50) # approx. 0.5

Find maximum Lethality Index during a strike

Description

This works by finding the maximum Lethality Index encountered during a simulation created by calling strike(), and so it is important to use a detailed setting for the output times. In the example, the results are reported every 0.7/200 seconds (i.e. 3.5 milliseconds), which is likely sufficient (see the example, where a plot is used for this assessment).

Usage

maximumLethalityIndex(strike)

Arguments

Value

The maximum value of the Lethality Index that is involved in the simulation of the ship-whale collision event. This is a unitless number; see Kelley et al. (2021).

Author(s)

Dan Kelley, wrapping code provided by Alexandra Mayette

References

Kelley, Dan E., James P. Vlasic, and Sean W. Brillant. "Assessing the Lethality of Ship Strikes on Whales Using Simple Biophysical Models." Marine Mammal Science 37, no. 1 (January 2021): 251–67.

See Also

Other functions dealing with Whale Lethality index: lethalityIndexFromStress(), stressFromLethalityIndex()

Examples

library(whalestrike)
t <- seq(0, 0.7, length.out = 200)
state <- list(xs = -2, vs = knot2mps(10), xw = 0, vw = 0)
parms <- parameters()
s <- strike(t, state, parms)
# Compute the desired value and (for context) show it on a plot
maximumLethalityIndex(s)
# For context, this is how this can be done "by hand"
max(lethalityIndexFromStress(s[["WCF"]][["stress"]]))
# Show the maximum on a plot (see also the plot title)
plot(s, which = "lethality index")
abline(h=maximumLethalityIndex(s), col=2)

Convert a speed in m/s to a speed in knots

Description

This is done by dividing by the factor 1.852e3/3600, See also knot2mps(), which is the inverse of this function.

Usage

mps2knot(mps)

Arguments

Value

Speed in knots.

Author(s)

Dan Kelley

See Also

Other functions dealing with units: knot2mps()

Examples

library(whalestrike)
mps <- seq(0, 10)
plot(mps, mps2knot(mps), xlab = "Speed [m/s]", ylab = "Speed [knots]", type = "l")

Set parameters for a whale strike simulation

Description

Assembles control parameters into a list suitable for passing to strike() and the functions that it calls. If file is provided, then all the other arguments are read from that source. Note that updateParameters() may be used to modify the results of parameters, e.g. for use in sensitivity tests.

Usage

parameters(
  ms = 45000,
  Ss = NULL,
  Ly = 1.15,
  Lz = 1.15,
  species = "N. Atl. Right Whale",
  lw = 13.7,
  mw = NULL,
  Sw = NULL,
  l = NULL,
  a = NULL,
  b = NULL,
  s = NULL,
  theta = 55,
  Cs = 0.01,
  Cw = 0.0025,
  logistic = list(logStressCenter = 5.38, logStressWidth = 0.349, tau25 = 1e+05, tau50 =
    241000, tau75 = 581000),
  file = NULL
)

Arguments

Value

A named list holding the parameters, with defaults and alternatives reconciled according to the system described above, along with some items used internally, including lsum, which is the sum of the values in l, and stressFromStrain(), a function created by stressFromStrainFunction() that computes compression force from engineering strain.

Author(s)

Dan Kelley

References

Campbell-Malone, Regina. "Biomechanics of North Atlantic Right Whale Bone: Mandibular Fracture as a Fatal Endpoint for Blunt Vessel-Whale Collision Modeling." PhD Thesis, Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 2007. doi:10.1575/1912/1817.

Daoust, Pierre-Yves, Emilie L. Couture, Tonya Wimmer, and Laura Bourque. "Incident Report. North Atlantic Right Whale Mortality Event in the Gulf of St. Lawrence, 2017." Canadian Wildlife Health Cooperative, Marine Animal Response Society, and Fisheries and Oceans Canada, 2018. https://publications.gc.ca/site/eng/9.850838/publication.html.

Grear, Molly E., Michael R. Motley, Stephanie B. Crofts, Amanda E. Witt, Adam P. Summers, and Petra Ditsche. "Mechanical Properties of Harbor Seal Skin and Blubber - a Test of Anisotropy." Zoology 126 (2018): 137-44. doi:10.1016/j.zool.2017.11.002.

Raymond, J. J. "Development of a Numerical Model to Predict Impact Forces on a North Atlantic Right Whale during Collision with a Vessel." University of New Hampshire, 2007. https://scholars.unh.edu/thesis/309/.

Examples

parms <- parameters()
epsilon <- seq(0, 1, length.out = 100) # strain
sigma <- parms$stressFromStrain(epsilon) # stress
plot(epsilon, log10(sigma), xlab = "Strain", ylab = "log10(Stress [MPa])", type = "l")
mtext("Note sudden increase in stress, when bone compression starts")

Pin numerical values between stated limits

Description

Pin numerical values between stated limits

Usage

pin(x, lower = NULL, upper = NULL)

Arguments

Value

Copy of x, with any value that exceeds lim having been replaced by lim.

Author(s)

Dan Kelley

Plot a strike object

Description

Creates displays of various results of a simulation performed with strike().

Usage

## S3 method for class 'strike'
plot(
  x,
  which = "default",
  drawEvents = TRUE,
  colwcenter = "black",
  colwinterface = "black",
  colwskin = "black",
  cols = "black",
  colThreat = c("white", "lightgray", "darkgray", "black"),
  lwd = 1,
  D = 3,
  debug = 0,
  ...
)

Arguments

Value

None. This function is called to add to a plot.

Author(s)

Dan Kelley

References

See whalestrike() for a list of references.

Examples

# Plot lethality index
t <- seq(0, 0.7, length.out = 200)
state <- c(xs = -2, vs = knot2mps(12), xw = 0, vw = 0) # 12 knot ship
parms <- parameters() # default values
sol <- strike(t, state, parms)
plot(sol, which="lethality index")

Whale blubber stress-strain relationship

Description

This is a data frame with elements strain and stress, found by digitizing (accurate to perhaps 1 percent) the curve shown in Figure 2.13 of Raymond (2007). It is used to develop a stress-strain relationship used by parameters(), as shown in “Examples”.

References

Raymond, J. J. "Development of a Numerical Model to Predict Impact Forces on a North Atlantic Right Whale during Collision with a Vessel." University of New Hampshire, 2007. https://scholars.unh.edu/thesis/309/.

Examples

data(raymond2007)
attach(raymond2007)
# Next yields \code{a=1.64e5} Pa and \code{b=2.47}.
m <- nls(stress ~ a * (exp(b * strain) - 1), start = list(a = 1e5, b = 1))
plot(strain, stress, xaxs = "i", yaxs = "i")
x <- seq(0, max(strain), length.out = 100)
lines(x, predict(m, list(strain = x)))

Compute ship wetted area from mass

Description

Estimate the wetted area of a Cape Islander boat, given the vessel mass.

Usage

shipAreaFromMass(ms)

Arguments

Details

The method is based on scaling up the results for a single Cape Islander ship, of displacement 20.46 tonnes, length 11.73m, beam 4.63m, and draft 1.58m, on the assumption that the wetted area is proportional to length*(2*draft+beam). This reference area is scaled to the specified mass, ms, by multiplying by the 2/3 power of the mass ratio.

Note that this is a crude calculation meant as a stop-gap measure, for estimates values of the Ss argument to parameters(). It should not be used in preference to inferences made from architectural drawings of a given ship under study.

Value

Estimated area (m^2).

Author(s)

Dan Kelley

See Also

Other functions relating to ship characteristics: shipLength(), shipMassFromLength(), shipWaterForce()

Nominal ship length in m

Description

This is based on the "Average LOA in m" column in Table 1 of Mayette and Brillant (2026).

Usage

shipLength(type = NULL)

Arguments

Value

shipLength returns ship length in m, as defined in Mayette and Brillant (2026).

Author(s)

Dan Kelley, with help from Alexandra Mayette

References

Mayette, Alexandra, and Sean W. Brillant. "A Regression-Based Method to Estimate Vessel Mass for Use in Whale-Ship Strike Risk Models." PloS One 21, no. 1 (2026): e0339760. doi:10.1371/journal.pone.0339760.

See Also

Other functions relating to ship characteristics: shipAreaFromMass(), shipMassFromLength(), shipWaterForce()

Examples

library(whalestrike)
# An individual length
shipLength("Fishing")
# A table of lengths
shipLength()

Ship displacement in kg based on vessel type and length

Description

This is done using formulae in Table 3 of Mayette and Brillant (2026).

Usage

shipMassFromLength(type = NULL, L)

Arguments

Details

The formulae used are as follows.

Value

shipMassFromLength returns ship displacement mass (in kg), according to Mayette and Brillant (2026) Table 3.

Author(s)

Dan Kelley, with help from Alexandra Mayette

References

Mayette, Alexandra, and Sean W. Brillant. "A Regression-Based Method to Estimate Vessel Mass for Use in Whale-Ship Strike Risk Models." PloS One 21, no. 1 (2026): e0339760. doi:10.1371/journal.pone.0339760.

See Also

Other functions relating to ship characteristics: shipAreaFromMass(), shipLength(), shipWaterForce()

Examples

library(whalestrike)
shipMassFromLength("Tug", 50) / 1e3 # 1920.648

Ship Drag Force

Description

Compute the retarding force of water on the ship, based on a drag law (1/2)*rho*Cs*A*vs^2 where rho is water density taken to be 1024 (kg/m^3), Cs is drag coefficient stored in parms and A is area, also stored in ⁠parms, and ⁠vs' is the ship speed (m/s).

Usage

shipWaterForce(vs, parms)

Arguments

Value

Water drag force (N).

Author(s)

Dan Kelley

See Also

Other functions relating to ship characteristics: shipAreaFromMass(), shipLength(), shipMassFromLength()

Other functions relating to forces: stressFromStrainFunction(), whaleCompressionForce(), whaleSkinForce(), whaleWaterForce()

Reference strike() solution

Description

This was produced with the package as it existed on 2020-jul-8, prior to the publication of Kelley et al. (2021). It is used in testing, to ensure that the package does not inadvertently change in its predictions.

References

Kelley, Dan E., James P. Vlasic, and Sean W. Brillant. "Assessing the Lethality of Ship Strikes on Whales Using Simple Biophysical Models." Marine Mammal Science, 37(1), 2021, mms.12745. doi:10.1111/mms.12745.

Examples

# Three plots
data(sol20200708)
plot(sol20200708)

Compute stress, based on lethality index

Description

The model used for this is the logistic model, fitting observed injury/lethality statistics to the base-10 logarithm of the maximum compression stress during a simulated impact event.

Usage

stressFromLethalityIndex(injury)

Arguments

Value

whale compression stress, in Pascals.

Author(s)

Dan Kelley

See Also

Other functions dealing with Whale Lethality index: lethalityIndexFromStress(), maximumLethalityIndex()

Examples

stressFromLethalityIndex(0.5) # approx. 254000 Pa, i.e. parameters()$logistic$tau50

Create a function for stress in laminated layers

Description

Denoting unforced layer thickness in the i layer as l_i and strain there as \epsilon_i=\Delta l_i/l_i, we may write the stress-strain relationship as

\sigma = a_i*(exp(b_i*\epsilon_i)-1)

for each layer, where it is assumed that stress \sigma is equal across layers. Inverting this yields

\epsilon_i= ln(1 + \sigma/a_i)/b_i

where ln is the natural logarithm. Therefore, the change \Delta L in the total thickness L=\sum l_i may be written

0 = \Delta L - \sum((l_i/b_i) ln(1+\sigma/a_i))

. Note that zero-thickness layers are removed from the calculation, to avoid spurious forces.

Usage

stressFromStrainFunction(l, a, b, N = 1000)

Arguments

Details

This expression is not easily inverted to get \sigma in terms of \Delta L but it may be solved easily for particular numerical values, using uniroot().

This is done for a sequence of N values of strain \epsilon that range from 0 to 1. Then approxfun() is used to create a piecewise-linear representation of the relationship between \sigma and \Delta L, which becomes the return value of the present function. (The purpose of using a piecewise-linear representation to reduce computation time.)

Value

A piecewise-linear function, created with approxfun(), that returns stress as a function of total strain of the system of compressing layers. For the purposes of the whale-strike analysis, the strain should be between 0 and 1, i.e. there is no notion of compressing blubber, etc. to negative thickness.

Author(s)

Dan Kelley

See Also

Other functions relating to whale characteristics: whaleCompressionForce(), whaleLengthFromMass(), whaleMassFromLength(), whaleSkinForce(), whaleWaterForce()

Other functions relating to forces: shipWaterForce(), whaleCompressionForce(), whaleSkinForce(), whaleWaterForce()

Examples

library(whalestrike)
# Set blubber parameters for each layer, to see if
# we recover the raymond2007 data.
param <- parameters(a = rep(1.64e5, 4), b = rep(2.47, 4))
x <- seq(0, 0.5, length.out = 100)
y <- param$stressFromStrain(x)
plot(x, y, type = "l", lwd = 4, col = "gray")
data("raymond2007")
points(raymond2007$strain, raymond2007$stress, col = 2)

Simulate the collision of a ship and a whale

Description

Newtonian mechanics are used, taking the ship as non-deformable, and the whale as being cushioned by a skin layer and a blubber layer. The forces are calculated by shipWaterForce(), whaleSkinForce(), whaleCompressionForce(), and whaleWaterForce() and the integration is carried out with deSolve::lsoda().

Usage

strike(t, state, parms, debug = 0)

Arguments

Value

strike returns an object of class "strike", which is a list holding 13 items. Nine of these items are time-series vectors, namely time t, ship position xs, ship speed vs, whale position xw, whale speed vw, ship acceleration dvsdt, whale acceleration dvwdt, drag force on the ship SWF, and drag force on the whale WWF. In addition to these time-series items, there are 3 items that are lists: WSF holds time-series of surface forces on the whale; WCF holds time-series of compressive forces on the whale; and parameters holds parameters of the simulation (see parameters() for more information). Finally, there is a logical value named refinedGrid that indicates whether the simulation required a restart because the initial timestep proved inadequate to track the high forces that may arise quickly if bone starts to be compressed appreciably. All quantities are in SI units, s for time, m/s for speed, m/s^2 for acceleration, N for force, etc.

Author(s)

Dan Kelley

References

See whalestrike() for a list of references.

Examples

library(whalestrike)
# Example 1: three plots, as in the default three panels of app()
t <- seq(0, 0.7, length.out = 200)
state <- list(xs = -2, vs = knot2mps(10), xw = 0, vw = 0) # ship speed 10 knots
parms <- parameters()
sol <- strike(t, state, parms)
plot(sol)

# Example 2: time-series plots of blubber stress and stress/strength,
# for a 200 tonne ship moving at 10 knots
t <- seq(0, 0.7, length.out = 1000)
state <- list(xs = -2, vs = knot2mps(10), xw = 0, vw = 0) # ship 10 knots
parms <- parameters(ms = 200 * 1000) # 1 metric tonne is 1000 kg
sol <- strike(t, state, parms)
plot(t, sol$WCF$stress / 1e6,
    type = "l",
    xlab = "Time [s]", ylab = "Blubber stress [MPa]"
)
plot(t, sol$WCF$stress / sol$parms$s[2],
    type = "l",
    xlab = "Time [s]", ylab = "Blubber stress / strength"
)

# Example 3: max stress and stress/strength, for a 200 tonne ship
# moving at various speeds.
knots <- seq(0, 10, 1)
maxStress <- NULL
maxStressOverStrength <- NULL
for (speed in knot2mps(knots)) {
    t <- seq(0, 10, length.out = 1000)
    state <- list(xs = -2, vs = speed, xw = 0, vw = 0)
    parms <- parameters(ms = 200 * 1000) # 1 metric tonne is 1000 kg
    sol <- strike(t, state, parms)
    maxStress <- c(maxStress, max(sol$WCF$stress))
    maxStressOverStrength <- c(maxStressOverStrength, max(sol$WCF$stress / sol$parms$s[2]))
}
nonzero <- maxStress > 0
plot(knots[nonzero], log10(maxStress[nonzero]),
    type = "o", pch = 20, xaxs = "i", yaxs = "i",
    xlab = "Ship Speed [knots]", ylab = "log10 peak blubber stress"
)
abline(h = log10(sol$parms$s[2]), lty = 2)
plot(knots[nonzero], log10(maxStressOverStrength[nonzero]),
    type = "o", pch = 20, xaxs = "i", yaxs = "i",
    xlab = "Ship Speed [knots]", ylab = "log10 peak blubber stress / strength"
)
abline(h = 0, lty = 2)

Summarize a parameters object

Description

This provides an overview of the contents of an object created with parameters().

Usage

## S3 method for class 'parameters'
summary(object, ...)

Arguments

Value

summary.parameters returns nothing. It is called for its side effect of printing information about the parameters in a ship-whale collision simulation.

Author(s)

Dan Kelley

Examples

summary(parameters())

Summarize the parameters of a simulation, and its results

Description

Summarize the parameters of a simulation, and its results

Usage

## S3 method for class 'strike'
summary(object, ...)

Arguments

Value

None. This function is called for its side-effect of printing information about the ship-whale collision simulation.

Author(s)

Dan Kelley

Examples

library(whalestrike)
# Example 1: graphs, as in the shiny app
t <- seq(0, 0.7, length.out = 200)
state <- list(xs = -2, vs = knot2mps(10), xw = 0, vw = 0) # ship speed 10 knots
parms <- parameters()
sol <- strike(t, state, parms)
summary(sol)

Update parameters

Description

updateParameters() is used to alter one or more components of an existing object of type "parameters" that was created by parameters(). This can be useful for e.g. sensitivity tests (see “Details”).

Usage

updateParameters(
  original,
  ms,
  Ss,
  Ly,
  Lz,
  species,
  lw,
  mw,
  Sw,
  l,
  a,
  b,
  s,
  theta,
  Cs,
  Cw,
  logistic,
  debug = 0
)

Arguments

Details

Two important differences between argument handling in updateParameters() and parameters() should be kept in mind.

First, updateParameters() does not check its arguments for feasible values. This can lead to bad results when using strike(), which is e.g. expecting four layer thicknesses to be specified, and also that each thickness is positive.

Second, updateParameters() does not perform ancillary actions that parameters() performs, with regard to certain interlinking argument values. Such actions are set up for whale length and ship mass, which are easily-observed quantities from other quantities can be estimated using whalestrike functions. If lw (whale length) is supplied to parameters() without also supplying mw (whale mass), then parameters() uses whaleMassFromLength() to infer mw from lw. The same procedure is used to infer Sw if it is not given, using whaleAreaFromLength(). Similarly, parameters() uses shipAreaFromMass() to compute Ss (ship area) from ms (ship mass), if the Ss argument is not given. Importantly, these three inferences are not made by updateParameters(), which alters only those values that are supplied explicitly. It is easy to supply those values, however; for example,

parms <- updateParameters(PARMS, lw=1.01 * PARMS$lw))
parms <- updateParameters(parms, mw=whaleMassFromLength(parms$lw))
parms <- updateParameters(parms, Sw=whaleAreaFromLength(parms$lw))

modifies a base state stored in PARMS, increasing whale length by 1% and then increasing whale mass and area accordingly. This code block is excerpted from a sensitivity test of the model, in which

parms <- updateParameters(PARMS, ms=1.01 * PARMS$ms)
parms <- updateParameters(parms, Ss=shipAreaFromMass(parms$ms))

was also used to perturb ship mass (and inferred area).

Value

A named list holding the items of the same name as those in the list returned by parameters().

Author(s)

Dan Kelley

References

Daoust, Pierre-Yves, Emilie L. Couture, Tonya Wimmer, and Laura Bourque. "Incident Report. North Atlantic Right Whale Mortality Event in the Gulf of St. Lawrence, 2017." Canadian Wildlife Health Cooperative, Marine Animal Response Society, and Fisheries and Oceans Canada, 2018. https://publications.gc.ca/site/eng/9.850838/publication.html.

Whale projected area, as function of length

Description

This depends on calculations based on the digitized shape of a whale necropsy, which is provided by whaleShape(). The results are 0.143 * L^2 for the projected area (see reference 1) and 0.448 * (0.877 * L)^2 for the wetted area (see reference 2, but note that we use a correction related to whale mass).

Usage

whaleAreaFromLength(L, species = "N. Atl. Right Whale", type = "wetted")

Arguments

Details

Note that multiple digitizations were done, and that the coefficients used in the formulae agreed to under 0.7 percent percent between these digitizations.

Value

whaleAreaFromLength returns the surface area of the whale, in square metres.

Author(s)

Dan Kelley

References

1. Dan Kelley's internal document ⁠dek/20180623_whale_area.Rmd⁠, available upon request.

2. Dan Kelley's internal document ⁠dek/20180707_whale_mass.Rmd⁠, available upon request.

3. Daoust, Pierre-Yves, Emilie L. Couture, Tonya Wimmer, and Laura Bourque. "Incident Report. North Atlantic Right Whale Mortality Event in the Gulf of St. Lawrence, 2017." Canadian Wildlife Health Cooperative, Marine Animal Response Society, and Fisheries and Oceans Canada, 2018. https://publications.gc.ca/site/eng/9.850838/publication.html.

Examples

L <- 3:20
A <- whaleAreaFromLength(L)
plot(L, A, xlab = "Length [m]", ylab = "Area [m^2]", type = "l")

Whale compression force

Description

Calculate the total compression stress and force, along with the thicknesses of skin, blubber, sublayer, and bone. The stress is computed with the stressFromStrainFunction() function that is created by parameters() and stored in para. the force is computed by multiplying stess by area computed as the product of parms$Ly and parms$Lz. Any negative layer thicknesses are set to zero, as a way to avoid problems with aphysical engineering compression strains that exceed 1.

Usage

whaleCompressionForce(xs, xw, parms)

Arguments

Value

A list containing: force (N), the compression-resisting force; stress (Pa), the ratio of that force to the impact area; strain, the total strain, and compressed, a four-column matrix (m) with first column for skin compression, second for blubber compression, third for sublayer compression, and fourth for bone compression.

Author(s)

Dan Kelley

References

See whalestrike() for a list of references.

See Also

Other functions relating to whale characteristics: stressFromStrainFunction(), whaleLengthFromMass(), whaleMassFromLength(), whaleSkinForce(), whaleWaterForce()

Other functions relating to forces: shipWaterForce(), stressFromStrainFunction(), whaleSkinForce(), whaleWaterForce()

Compute whale length from mass

Description

This works by inverting whaleMassFromLength() using uniroot().

Usage

whaleLengthFromMass(M, species = "N. Atl. Right Whale", model = "fortune2012")

Arguments

Value

Whale length (m).

Author(s)

Dan Kelley

References

See whalestrike() for a list of references.

See Also

whaleMassFromLength() is the reverse of this.

Other functions relating to whale characteristics: stressFromStrainFunction(), whaleCompressionForce(), whaleMassFromLength(), whaleSkinForce(), whaleWaterForce()

Whale mass inferred from length

Description

Calculate an estimate of the mass of different species of whale, based on animal length, based on formulae as listed in “Details”.

Usage

whaleMassFromLength(L, species = "N. Atl. Right Whale", model = NULL)

Arguments

Details

The permitted values for model and species are as follows. Note that if model is not provided, then Fortune (2012) is used for "N. Atl. Right Whale", and Lockyer (1976) is used for all other species.

• "moore2005" (which only works if species is "N. Atl. Right Whale") yields 242.988 * exp(0.4 * length), which (apart from a unit change on L) is the regression equation shown above Figure 1d in Moore et al. (2005) for right whales. A difficult in the Moore et al. (2005) use of a single nonzero digit in the multiplier on L is illustrated in “Examples”.

• "fortune2012" with species="N. Atl. Right Whale" yields the formula exp(-10.095 + 2.825*log(100*L)) for North Atlantic right whales, according to a corrected version of the erroneous formula given in the caption of Figure 4 in Fortune et al (2012). (The error, an exchange of slope and intercept, was confirmed by S. Fortune in an email to D. Kelley dated June 22, 2018.)

• "fortune2012" with species="N. Pac. Right Whale" yields the formula exp(-12.286 + 3.158*log(100*L)) for North Pacific right whales, according to a corrected version of the erroneous formula given in the caption of Figure 4 in Fortune et al (2012). (The error, an exchange of slope and intercept, was confirmed by S. Fortune in an email to D. Kelley dated June 22, 2018.)

• "lockyer1976" uses formulae from Table 1 of Lockyer (1976). The permitted species and the formulae used are as follows (note that the "Gray Whale" formula is in the table's caption, not in the table itself).

  • "Blue Whale": 2.899 L^{3.25}

  • "Bryde Whale": 12.965 L^{2.74}

  • "Fin Whale": 7.996 L^{2.90}

  • "Gray Whale": 5.4 L^{3.28}

  • "Humpback Whale": 16.473 L^{2.95}

  • "Minke Whale": 49.574 L^{2.31}

  • "Pac. Right Whale": 13.200 L^{3.06}

  • "Sei Whale": 25.763 L^{2.43}

  • "Sperm Whale": 6.648 L^{3.18}

Value

Mass in kg.

Author(s)

Dan Kelley

References

• Lockyer, C. "Body Weights of Some Species of Large Whales." J. Cons. Int. Explor. Mer. 36, no. 3 (1976): 259-73.

• Moore, M.J., A.R. Knowlton, S.D. Kraus, W.A. McLellan, and R.K. Bonde. "Morphometry, Gross Morphology and Available Histopathology in North Atlantic Right Whale (Eubalaena Glacialis) Mortalities (1970 to 2002)." Journal of Cetacean Research and Management 6, no. 3 (2005): 199-214.

• Fortune, Sarah M. E., Andrew W. Trites, Wayne L. Perryman, Michael J. Moore, Heather M. Pettis, and Morgan S. Lynn. "Growth and Rapid Early Development of North Atlantic Right Whales (Eubalaena Glacialis)." Journal of Mammalogy 93, no. 5 (2012): 1342-54. doi:10.1644/11-MAMM-A-297.1.

See Also

whaleLengthFromMass() is the reverse of this.

Other functions relating to whale characteristics: stressFromStrainFunction(), whaleCompressionForce(), whaleLengthFromMass(), whaleSkinForce(), whaleWaterForce()

Examples

library(whalestrike)
L <- seq(5, 15, length.out = 100)
kpt <- 1000 # kg per tonne
# Demonstrate (with dashing) the sensitivity involved in the single-digit
# parameter in Moore's formula, and (with colour) the difference to the
# Fortune et al. (2012) formulae.
plot(L, whaleMassFromLength(L, model = "moore2005") / kpt,
    type = "l", lwd = 2,
    xlab = "Right-whale Length [m]", ylab = "Mass [tonne]"
)
lines(L, 242.988 * exp(0.35 * L) / kpt, lty = "dotted", lwd = 2)
lines(L, 242.988 * exp(0.45 * L) / kpt, lty = "dashed", lwd = 2)
lines(L, whaleMassFromLength(L,
    species = "N. Atl. Right Whale",
    model = "fortune2012"
) / kpt, col = 2, lwd = 2)
lines(L, whaleMassFromLength(L,
    species = "N. Pac. Right Whale",
    model = "fortune2012"
) / kpt, col = 3, lwd = 2)
grid()
legend("topleft",
    bg = "white", y.intersp = 1.4, lwd = 2, col = 1:3,
    legend = c("moore2005", "fortune2012 Atlantic", "fortune2012 Pacific")
)

# Emulate Figure 1 of Lockyer (1976), with roughly-chosen plot limits.
L <- seq(0, 18, 0.5)
m <- whaleMassFromLength(L, species = "Pac. Right Whale", model = "lockyer1976") / kpt
plot(L, m,
    col = 1, xlab = "Length [m]", ylab = "Mass [tonne]", type = "l", lwd = 2,
    xaxs = "i", yaxs = "i", xlim = c(3, 30), ylim = c(0, 180)
)
L <- seq(0, 28, 0.5)
m <- whaleMassFromLength(L, species = "Blue Whale", model = "lockyer1976") / kpt
lines(L, m, col = 2, lwd = 2)
L <- seq(0, 24, 0.5)
m <- whaleMassFromLength(L, species = "Fin Whale", model = "lockyer1976") / kpt
lines(L, m, col = 3, lwd = 2)
L <- seq(0, 18, 0.5)
m <- whaleMassFromLength(L, species = "Sei Whale", model = "lockyer1976") / kpt
lines(L, m, col = 1, lty = 2, lwd = 2)
L <- seq(0, 17, 0.5)
m <- whaleMassFromLength(L, species = "Bryde Whale", model = "lockyer1976") / kpt
lines(L, m, col = 2, lty = 2, lwd = 2)
L <- seq(0, 12, 0.5)
m <- whaleMassFromLength(L, species = "Minke Whale", model = "lockyer1976") / kpt
lines(L, m, col = 3, lty = 2, lwd = 2)
L <- seq(0, 17, 0.5)
m <- whaleMassFromLength(L, species = "Humpback Whale", model = "lockyer1976") / kpt
lines(L, m, col = 1, lty = 3, lwd = 2)
L <- seq(0, 18, 0.5)
m <- whaleMassFromLength(L, species = "Sperm Whale", model = "lockyer1976") / kpt
lines(L, m, col = 2, lty = 3, lwd = 2)
L <- seq(0, 15, 0.5)
m <- whaleMassFromLength(L, species = "Gray Whale", model = "lockyer1976") / kpt
lines(L, m, col = 3, lty = 3, lwd = 2)
grid()
legend("topleft",
    bg = "white", y.intersp = 1.4, col = c(1:3, 1:3, 1:2),
    lwd = 2, lty = c(rep(1, 3), rep(2, 3), rep(3, 3)),
    legend = c("Right", "Blue", "Fin", "Sei", "Bryde", "Minke", "Humpback", "Sperm", "Gray")
)

Get values for various whale measurements

Description

This uses a data frame containing information about several whale species, compiled by Alexandra Mayette (2026).

Usage

whaleMeasurements(species = NULL)

Arguments

Details

There are two species in the table that are not in Mayette's table. These are "Pac. Right Whale" and "Bryde Whale". For these, Mayette has suggesting using values for the "N. Atl. Right Whale" and "Sei Whale" cases, respectively, as conditional estimates for use in this package.

Value

The return value contains

• name species name, as used in e.g. whaleMassFromLength().

• Species proper species name. (This is not used in this package.)

• length whale length in metres.

• bone whale bone thickness in metres, measured from the centre to the sublayer.

• sublayer thickness of sublayer in meters; this was called muscle in Mayette's document.

• blubber whale blubber thickness in meters.

• skin whale skin thickness in metres.

Author(s)

Dan Kelley, using data and advice from Alexandra Mayette

References

Mayette, A. (2026). Measurements of large whale tissue thickness (Data set). Zenodo. doi:10.5281/zenodo.18764979.

Examples

library(whalestrike)
# All species in database
whaleMeasurements()
# A particular species
whaleMeasurements("N. Atl. Right Whale")

Whale side-view shape

Description

This is a data frame containing 45 points specifying the shape of a right whale, viewed from the side. It was created by digitizing the whale shape (ignoring fins) that is provided in the necropsy reports of Daoust et al. (2018). The data frame contains x and y, which are distances nondimensionalized by the range in x; that is, x ranges from 0 to 1. The point at the front of the whale is designated as x=y=0.

Usage

whaleShape()

Value

whaleShape returns a data frame with entries named x and y, which trace out the side-view shape of a Right Whale, using values digitized from a diagram in Daoust et al. (2017).

Author(s)

Dan Kelley

References

Daoust, Pierre-Yves, Emilie L. Couture, Tonya Wimmer, and Laura Bourque. "Incident Report. North Atlantic Right Whale Mortality Event in the Gulf of St. Lawrence, 2017." Canadian Wildlife Health Cooperative, Marine Animal Response Society, and Fisheries and Oceans Canada, 2018. https://publications.gc.ca/site/eng/9.850838/publication.html.

Examples

library(whalestrike)
shape <- whaleShape()
plot(shape$x, shape$y, asp = 1, type = "l")
polygon(shape$x, shape$y, col = "lightgray")
lw <- 13.7
Rmax <- 0.5 * lw * diff(range(shape$y))
mtext(sprintf("Max. radius %.2fm for %.1fm-long whale", Rmax, lw), side = 3)

Skin force

Description

The ship-whale separation is used to calculate the deformation of the skin. The parameters of the calculation are parms$Ly (impact area width, m), parms$Lz (impact area height, in m), parms$Ealpha (skin elastic modulus in Pa), parms$alpha (skin thickness in m), and parms$theta (skin bevel angle degrees, measured from a vector normal to undisturbed skin).

Usage

whaleSkinForce(xs, xw, parms)

Arguments

Value

A list containing force, the normal force (N), along with sigmay and sigmaz, which are stresses (Pa) in the y (beam) and z (draft) directions.

Author(s)

Dan Kelley

References

See whalestrike() for a list of references.

See Also

Other functions relating to whale characteristics: stressFromStrainFunction(), whaleCompressionForce(), whaleLengthFromMass(), whaleMassFromLength(), whaleWaterForce()

Other functions relating to forces: shipWaterForce(), stressFromStrainFunction(), whaleCompressionForce(), whaleWaterForce()

Whale Drag Force

Description

Compute the retarding force of water on the whale, based on a drag law (1/2)*rho*Cw*A*vw^2 where rho is 1024 (kg/m^3), Cw is parms$Cw and A is parms$Sw.

Usage

whaleWaterForce(vw, parms)

Arguments

Value

Water drag force (N).

Author(s)

Dan Kelley

See Also

Other functions relating to whale characteristics: stressFromStrainFunction(), whaleCompressionForce(), whaleLengthFromMass(), whaleMassFromLength(), whaleSkinForce()

Other functions relating to forces: shipWaterForce(), stressFromStrainFunction(), whaleCompressionForce(), whaleSkinForce()