An application of Mathematica to internal wave generation

David Vasholz

Johns Hopkins University
Applied Physics Laboratory

Work supported by Office of Naval Research
Grant No. N000140710113
Dr.  Ronald Joslin 

Approved for Public Release; Distribution is Unlimited

General comments

Submerged objects moving through a fluid can generate acoustic waves when vibrations are present

If the fluid is stratified, they will generate internal waves even when vibrations are absent

Internal waves generated because, in order to move forward, the object must push water out of the way

Internal waves and acoustic waves behave very differently

This work calculates internal wave wakes generated by a submerged object

Why Mathematica ?

Internal wave generation also generates an enormous amount of algebra when analyzed

Previously done by hand - with sketchy results

Use of Mathematica worthwhile just for keeping track of minus signs, factors of 2, etc.

Great documentation value

Benefits build up over a period of years

Have a "library" of  ~ 4000 notebooks dating from 1992

Geezer friendly

Ingredients

Geometry of the problem:

Coordinates:  x, y, z                       BV frequency profile: ((ℕ(z))^2)

Fluid variables:  u, v, w, p, ρ, ζ

"Maxwell's equations" of IW generation

μ ((∂u)/(∂x))+((∂v)/(∂y))+((∂w)/(∂z))==s                         ((∂ζ)/(∂x))-w==0

((∂p)/(∂y))+((∂v)/(∂x))==0                                         ((∂p)/(∂x))+((∂u)/(∂x))==0

γ ζ ((ℕ(z))^2)+((∂p)/(∂z))+((∂w)/(∂x))==0

w(x,y,0)==0                             w(x,y,-h)==0

The "hydrogen atom" of IW generation

Simplest possible situation for validation of theory

Uniform buoyancy frequency

Can make laboratory predictions for towing tank

Schematic of experiment:

Main result

Vertical displacement

a
(Subscript[ζ, f])== ((Subscript[𝕍, f])/(2 π)) (∑_(n=1)^∞)n cos(n π (Subscript[z, (Subscript[f, 0])])) sin(n π (Subscript[z, f])) ℍ(μ,(((π^2) γ)/(Φ^2)),n π,(Subscript[x, f]),(Subscript[y, f]))
b

Φ = (π (source speed))/(water depth)×(buoyancy frequency) 
                    =Froude number

ℍ = stratified wake form factor

All lengths in units of water depth

Definition of ℍ

ℍ(a,b,c,x,y)==(∫_(-∞)^∞)(E^(I y β)) (∫_(-∞)^∞)(((E^(I x α)) (α^2))/( (c^2) (α^2)+((α^2)-b) (a (α^2)+(β^2))))\[DifferentialD]α\[DifferentialD]β

ℍ==(Subscript[ℍ, wave])+(Subscript[ℍ, eva])

Coalescence

Pole locations move about as one varies a, b, c, and β

All four poles can meet at the origin, but if and only if  β=0 and

((n Φ)/(Sqrt[γ μ]))= 1

Various transitions occur as ((n Φ)/(Sqrt[γ μ])) passes through unity

Also have crucial μ-dependences apart from ((n Φ)/(Sqrt[γ μ]))

All transitions disappear in the HFNA

Upstream enhancement ratio

UER==(((∫_(-∞)^∞)(Subscript[ℍ, eva])(μ,(((π^2) γ)/(Φ^2)),n π,(Subscript[x, f]),(Subscript[y, f]))d(Subscript[y, f]))/((∫_(-∞)^∞)(Subscript[ℍ, eva])(μ,0,n π,(Subscript[x, f]),(Subscript[y, f]))d(Subscript[y, f])))==(((e^((n π (1-(Sqrt[1-((γ μ)/((n^2) (Φ^2)))])) |(Subscript[x, f])|)/(Sqrt[μ]))) θ(1-((γ μ)/((n^2) (Φ^2)))))/(Sqrt[1-((γ μ)/((n^2) (Φ^2)))]))

Plots of UER  at selected values of (Subscript[x, f]):

Second-order moments

Desired quantities:

(∫_(-∞)^∞)((Subscript[ℍ, wave])^2)d(Subscript[y, f])   and   (∫_(-∞)^∞)((Subscript[ℍ, eva])^2)d(Subscript[y, f])

Auxilliary quantities:

τ==(((((n Φ)/(Sqrt[γ μ]))))^(-2))-1        ρ==(((((((n Φ)/(Sqrt[γ μ]))))^(-2))+1)/(|τ|))     g(β)==(β^4)+2 ρ (β^2)+1

r==((|((γ μ)/((n^2) (Φ^2)))-1|)^2)(((((γ μ)/((n^2) (Φ^2)))+1))^(-2))

Analytic Results

(lim_((Subscript[x, f])->∞))(∫_(-∞)^∞)((Subscript[ℍ, wave])^2)d(Subscript[y, f])==((2 (π^2) (r^(1/4)))/(n μ (Sqrt[|τ|])))θ((Subscript[x, f])) (((2 K(2 (r^(-1)) (r-(Sqrt[1-r])-1)))/(Sqrt[1-(Sqrt[1-r])]))+((π sgn(τ)-π)/((Sqrt[1-(Sqrt[1-r])])+(Sqrt[(Sqrt[1-r])+1]))))

(lim_((Subscript[x, f])->0))(∫_(-∞)^∞)((Subscript[ℍ, eva])^2)d(Subscript[y, f])== (((π^2) (r^(1/4)))/(n μ (Sqrt[|τ|])))  (((2 K(2 (r^(-1)) (r-(Sqrt[1-r])-1)))/(Sqrt[1-(Sqrt[1-r])]))-((π sgn(τ)-π)/((Sqrt[1-(Sqrt[1-r])])+(Sqrt[(Sqrt[1-r])+1]))))

where K(x)  is a complete elliptic integral of the first kind

Can evaluate at finite (Subscript[x, f]) using numerical integration

Resonance plots

γ = μ = n =1

Above results are analytic; for other values of (Subscript[x, f]) must resort to numerical integration

Plots of  ℍ near Φ = 1

Notes and Derivations

Executable Notebook
 (not included)