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Making Waves in Multivariable Calculus
<http://blogs.ams.org/blogonmathblogs/2013/04/22/the-mathematics-of-planet-earth/>
J. B. ThooYuba College
2014 CMC3 Fall Conference, Monterey, Ca
December 10, 2014
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This presentation was produced using LATEX with C. Campani’sBeamer LATEX class and saved as a PDF file:<http://bitbucket.org/rivanvx/beamer>.
See Norm Matloff’s web page<http://heather.cs.ucdavis.edu/~matloff/beamer.html>for a quick tutorial.
Disclaimer: Our slides here won’t show off what Beamer can do.Sorry. :-)
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Are you sitting in the right room?
A common exercise in calculus textbooks is to verify that a givenfunction u = u(x , t) satisfies the heat equation, ut = Duxx , or thewave equation, utt = c2uxx . While this is a useful exercise in usingthe chain rule, it is not a very exciting one because it ends there.
The mathematical theory of waves is a rich source of partialdifferential equations. This talk is about introducing somemathematics of waves to multivariable calculus (vector calculus)students. We will show you some examples that we have presentedto our students that have given a context for what they are learning.
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Outline of the talk
Some examples of waves
Mathematical definition of a wave
Some wave equations
Using what we have learnt
Chain ruleIntegrating factorPartial fractions
Other examples (time permitting)
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References
Roger Knobel, An Introduction to the Mathematical Theory of Waves, StudentMathematics Library, IAS/Park City Mathematical Subseries, Volume 3, Ameri-can Mathematical Society, Providence (2000)
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Some examples of waves
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Typical
Pond Guitar Strings
(L) <http://astrobob.areavoices.com/2008/10/12/the-silence-of-crashing-waves/>
(R) <http://rekkerd.org/cinematique-instruments-releases-guitar-harmonics-for-kontakt/>
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Internal waves
Internal wave trains around Trinidad from space
Model of an estuary in a lab
(T) <http://en.wikipedia.org/wiki/Internal_wave>
(B) <http://www.ocean.washington.edu/research/gfd/hydraulics.html>
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Internal waves
Kelvin-Helmholtz instability
Clouds In a tank
(L) <http://www.documentingreality.com/forum/f241/amazing-clouds-89929/>
(R) <http://www.nwra.com/products/labservices/#tiltingtank>
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Water gravity waves
Deep-water waves
Bow waves or ship waves
(L) <http://wanderinweeta.blogspot.com/2011/12/bow-wave.html>
(R) <http://www.fluids.eng.vt.edu/msc/gallery/waves/jfkkub.jpg>
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Water gravity waves
Deep-water waves
Bow waves or ship waves
(L) <http://wanderinweeta.blogspot.com/2011/12/bow-wave.html>
(R) <http://www.fluids.eng.vt.edu/msc/gallery/waves/jfkkub.jpg>
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Water gravity waves
Shallow-water waves
Tsunami (2011 Tohoku, Japan, earthquake)
Iwanuma, Japan Crescent City, Ca Santa Cruz, Ca
(L) <http://www.telegraph.co.uk/news/picturegalleries/worldnews/8385237/Japan-disaster-30-powerful-images-of-the-earthquake-and-tsunami.html>
(C) <http://www.katu.com/news/local/117824673.html?tab=gallery&c=y&img=3>
(R) <http://www.conservation.ca.gov/cgs/geologic_hazards/Tsunami/Inundation_Maps/Pages/2011_tohoku.aspx>
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Solitary waves
Morning glory cloud Ocean wave
(L) <http://www.dropbears.com/m/morning_glory/rollclouds.htm>
(R) <http://www.math.upatras.gr/~weele/weelerecentresearch_SolitaryWaterWaves.htm>
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Solitary waves
Recreation of John Scott Russell’s soliton,Hariot-Watt University (1995)
<http://www.ma.hw.ac.uk/solitons/soliton1b.html>
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Shock waves
F-18 fighter jet Schlieren photograph
(L) <http://www.personal.psu.edu/pmd5102/blogs/its_only_rocket_science/about/>
(R) <http://www.neptunuslex.com/Wiki/2007/11/20/more-education/>
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Mathematical definition of a wave
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Definition
No single precise definition of what exactly constitutes a wave.Various restrictive definitions can be given, but to cover the wholerange of wave phenomena it seems preferable to be guided by theintuitive view that a wave is any recognizable signal that istransferred from one part of the medium to another with arecognizable velocity of propagation.
[Whitham]
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Definition
No single precise definition of what exactly constitutes a wave.Various restrictive definitions can be given, but to cover the wholerange of wave phenomena it seems preferable to be guided by theintuitive view that a wave is any recognizable signal that istransferred from one part of the medium to another with arecognizable velocity of propagation.
[Whitham]
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Some wave equations
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The wave equation
The wave equation: utt = c2uxx
Models a number of wavephenomena, e.g., vibrations ofa stretched string
Standing wave solution:
un(x , t) = [A cos(nπct/L) + B sin(nπct/L)] sin(nπx/L)
0 L
n = 3, A = B = 0.1, c = L = 1, t = 0 : 0.1 : 1, 0 ≤ x ≤ 1
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The Korteweg-de Vries (KdV) equation
The Korteweg-de Vries (KdV) equation: ut + uux + uxxx = 0
Models shallow water gravitywaves
x
u
speed c
Look for traveling wave solution u(x , t) = f (x − ct),
c > 0, f (z), f ′(z), f ′′(z)→ 0 as z → ±∞.
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The Sine-Gordon equation
The Sine-Gordon equation: utt = uxx − sin u
Models a mechanicaltransmission line such aspendula connected by a spring
u
Look for traveling wave solution: u(x , t) = f (x − ct),
c > 0, f (z), f ′(z)→ 0 as z →∞.
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Using what we have learnt
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Chain rule
h = g ◦ f =⇒ Dhm×n
= Dgm×p
Dfp×n
if f : Rn → Rp and g : Rp → Rm so that h : Rn → Rm
E.g., f : R → R2 : f (t) = (x , y), g : R2 → R2 : g(x , y) = (w , z),and
h = g ◦ f : R → R2 : h(t) = (w , z)
Then
Dh =
[∂w∂x
∂w∂y
∂z∂x
∂z∂y
]Dg
[dxdt
dydt
]Df
=⇒
[dwdt
dzdt
]=
[∂w∂x
dxdt +
∂w∂y
dydt
∂z∂x
dxdt +
∂z∂y
dydt
]
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Chain rule
h = g ◦ f =⇒ Dhm×n
= Dgm×p
Dfp×n
if f : Rn → Rp and g : Rp → Rm so that h : Rn → Rm
E.g., f : R → R2 : f (t) = (x , y), g : R2 → R2 : g(x , y) = (w , z),and
h = g ◦ f : R → R2 : h(t) = (w , z)
Then
Dh =
[∂w∂x
∂w∂y
∂z∂x
∂z∂y
]Dg
[dxdt
dydt
]Df
=⇒
[dwdt
dzdt
]=
[∂w∂x
dxdt +
∂w∂y
dydt
∂z∂x
dxdt +
∂z∂y
dydt
]
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Example 1
The wave equation:* utt = auxx , a > 0
Look for traveling wave solution: u(x , t) = f (x − ct)
i.e., look for a solution that advects at wave speed c withoutchanging its profile
E.g.,
u(x , t) = sin(x − ct),
u(x , t) = (x − ct)4,
u(x , t) = exp[−(x − ct)2
]*Models a number of wave phenomena, e.g., vibrations of a stretched string
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Typical exercise: Show that u(x , t) = exp[−(x − ct)2
]satisfies
utt = c2uxx .
Let z = x − ct. Then, u(x , t) = exp(−z2) = f (z) and, using the chainrule, we find that
ut =df
dz
∂z
∂t= 2cz exp(−z2),
utt =df ′
dz
∂z
∂t= 4c2z2 exp(−z2),
ux =df
dz
∂z
∂x= −2z exp(−z2),
uxx =df ′
dz
∂z
∂x= 4z2 exp(−z2)
utt = c2uxx =⇒ 4c2z2 exp(−z2) = c2 · 4z2 exp(−z2)
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Typical exercise: Show that u(x , t) = exp[−(x − ct)2
]satisfies
utt = c2uxx .
Let z = x − ct. Then, u(x , t) = exp(−z2) = f (z) and, using the chainrule, we find that
ut =df
dz
∂z
∂t= 2cz exp(−z2),
utt =df ′
dz
∂z
∂t= 4c2z2 exp(−z2),
ux =df
dz
∂z
∂x= −2z exp(−z2),
uxx =df ′
dz
∂z
∂x= 4z2 exp(−z2)
utt = c2uxx =⇒ 4c2z2 exp(−z2) = c2 · 4z2 exp(−z2)
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Let z = x − ct. Then u(x , t) = f (x − ct) = f (z) and, using thechain rule,
ut =df
dz
∂z
∂t= f ′(z)(−c) = −cf ′(z),
utt =df ′
dz
∂z
∂t= −cf ′′(z)(−c) = c2f ′′(z),
ux =df
dz
∂z
∂x= f ′(z)(1) = f ′(z),
uxx =df ′
dz
∂z
∂x= f ′′(z)(1) = f ′′(z)
utt = auxx =⇒ c2f ′′(z) = af ′′(z)
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Let z = x − ct. Then u(x , t) = f (x − ct) = f (z) and, using thechain rule,
ut =df
dz
∂z
∂t= f ′(z)(−c) = −cf ′(z),
utt =df ′
dz
∂z
∂t= −cf ′′(z)(−c) = c2f ′′(z),
ux =df
dz
∂z
∂x= f ′(z)(1) = f ′(z),
uxx =df ′
dz
∂z
∂x= f ′′(z)(1) = f ′′(z)
utt = auxx =⇒ c2f ′′(z) = af ′′(z)
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c2f ′′(z) = af ′′(z) =⇒ (c2 − a)f ′′(z) = 0
c2 − a = 0 =⇒ c = ±√
a:
u(x , t) = f (x ±√
at) provided f ′′ exists, otherwise f arbitrary
e.g., u(x , t) = sin(x +√
at) or u(x , t) = exp[−(x −
√at)2
]f ′′(z) = 0 =⇒ f (z) = A + Bz :
u(x , t) = A + B(x − ct) provided solution is not constant
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c2f ′′(z) = af ′′(z) =⇒ (c2 − a)f ′′(z) = 0
c2 − a = 0 =⇒ c = ±√
a:
u(x , t) = f (x ±√
at) provided f ′′ exists, otherwise f arbitrary
e.g., u(x , t) = sin(x +√
at) or u(x , t) = exp[−(x −
√at)2
]
f ′′(z) = 0 =⇒ f (z) = A + Bz :
u(x , t) = A + B(x − ct) provided solution is not constant
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c2f ′′(z) = af ′′(z) =⇒ (c2 − a)f ′′(z) = 0
c2 − a = 0 =⇒ c = ±√
a:
u(x , t) = f (x ±√
at) provided f ′′ exists, otherwise f arbitrary
e.g., u(x , t) = sin(x +√
at) or u(x , t) = exp[−(x −
√at)2
]f ′′(z) = 0 =⇒ f (z) = A + Bz :
u(x , t) = A + B(x − ct) provided solution is not constant
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Example 2
The linearized KdV* equation: ut + ux + uxxx = 0
Look for wave train solution: u(x , t) = A cos(kx − ωt) ,
where A 6= 0, k > 0, ω > 0
(a particular type of traveling wave solution, i.e., u(x , t) = f (x − ct))
Note: u(x , t) = A cos(k( x − (ω/k)t︸ ︷︷ ︸
x−ct
)advects at wave speed
c = ω/k
The number ω is the angular frequency and k is called thewavenumber. The wavelength is 2π/k .
*KdV = Korteweg-de Vries; the KdV equation models shallow-water gravitywaves
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Example 2
The linearized KdV* equation: ut + ux + uxxx = 0
Look for wave train solution: u(x , t) = A cos(kx − ωt) ,
where A 6= 0, k > 0, ω > 0
(a particular type of traveling wave solution, i.e., u(x , t) = f (x − ct))
Note: u(x , t) = A cos(k( x − (ω/k)t︸ ︷︷ ︸
x−ct
)advects at wave speed
c = ω/k
The number ω is the angular frequency and k is called thewavenumber. The wavelength is 2π/k .
*KdV = Korteweg-de Vries; the KdV equation models shallow-water gravitywaves
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Let z = kx − ωt and f (z) = A cos(z). Then
u(x , t) = A cos(kx − ωt) = f (z)
and, using the chain rule,
ut =df
dz
∂z
∂t= f ′(z)(−ω) = ωA sin(z),
ux =df
dz
∂z
∂x= f ′(z)(k) = −kA sin(z),
uxx =df ′
dz
∂z
∂x= f ′′(z)(k) = −k2A cos(z),
uxxx =df ′′
dz
∂z
∂x= f ′′′(z)(k) = k3A sin(z)
ut + ux + uxxx = 0 =⇒ (ω − k + k3)A sin(z) = 0
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Let z = kx − ωt and f (z) = A cos(z). Then
u(x , t) = A cos(kx − ωt) = f (z)
and, using the chain rule,
ut =df
dz
∂z
∂t= f ′(z)(−ω) = ωA sin(z),
ux =df
dz
∂z
∂x= f ′(z)(k) = −kA sin(z),
uxx =df ′
dz
∂z
∂x= f ′′(z)(k) = −k2A cos(z),
uxxx =df ′′
dz
∂z
∂x= f ′′′(z)(k) = k3A sin(z)
ut + ux + uxxx = 0 =⇒ (ω − k + k3)A sin(z) = 0
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(ω − k + k3)A sin(z) = 0 =⇒ ω − k + k3 = 0
Dispersion relation: ω = k − k3
Wave speed: c =ω
k= 1− k2
Note: That c depends on k means that wave trains of differentfrequencies travel at different speeds. Such a wave is called a dispersivewave. Here, smaller k or longer waves (λ = 2π/k) speed ahead, whilelarger k or shorter waves trail behind.
Group velocity: C = dωdk = 1− 3k2
The group velocity C is the velocity of the energy in the wave andis generally different from the wave speed c
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(ω − k + k3)A sin(z) = 0 =⇒ ω − k + k3 = 0
Dispersion relation: ω = k − k3
Wave speed: c =ω
k= 1− k2
Note: That c depends on k means that wave trains of differentfrequencies travel at different speeds. Such a wave is called a dispersivewave. Here, smaller k or longer waves (λ = 2π/k) speed ahead, whilelarger k or shorter waves trail behind.
Group velocity: C = dωdk = 1− 3k2
The group velocity C is the velocity of the energy in the wave andis generally different from the wave speed c
![Page 40: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/40.jpg)
(ω − k + k3)A sin(z) = 0 =⇒ ω − k + k3 = 0
Dispersion relation: ω = k − k3
Wave speed: c =ω
k= 1− k2
Note: That c depends on k means that wave trains of differentfrequencies travel at different speeds. Such a wave is called a dispersivewave. Here, smaller k or longer waves (λ = 2π/k) speed ahead, whilelarger k or shorter waves trail behind.
Group velocity: C = dωdk = 1− 3k2
The group velocity C is the velocity of the energy in the wave andis generally different from the wave speed c
![Page 41: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/41.jpg)
(ω − k + k3)A sin(z) = 0 =⇒ ω − k + k3 = 0
Dispersion relation: ω = k − k3
Wave speed: c =ω
k= 1− k2
Note: That c depends on k means that wave trains of differentfrequencies travel at different speeds. Such a wave is called a dispersivewave. Here, smaller k or longer waves (λ = 2π/k) speed ahead, whilelarger k or shorter waves trail behind.
Group velocity: C = dωdk = 1− 3k2
The group velocity C is the velocity of the energy in the wave andis generally different from the wave speed c
![Page 42: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/42.jpg)
In general, a wave train solution is u(x , t) = f (kx − ωt),
where k > 0, ω > 0, and f is periodic
(a particular type of traveling wave solution, i.e., u(x , t) = f (x − ct))
In general, not a solution for every possible k or ω
Note: u(x , t) = f(k(x − (ω/k)t
)advects at wave speed c = ω/k
![Page 43: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/43.jpg)
Integrating factor
To solve: y ′(x) + p(x)y(x) = q(x) for y = y(x)
Multiply through by integrating factor µ = µ(x)
µy ′ + µpy = µq
If µ′ = µp, then µy ′ + µpy = µy ′ + µ′y , so that
(µy)′ = µq =⇒ µy =
∫µq dx
and hence
y(x) =1
µ(x)
∫µ(x)q(x) dx where µ(x) = exp
[∫p(x) dx
]
![Page 44: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/44.jpg)
Integrating factor
To solve: y ′(x) + p(x)y(x) = q(x) for y = y(x)
Multiply through by integrating factor µ = µ(x)
µy ′ + µpy = µq
If µ′ = µp, then µy ′ + µpy = µy ′ + µ′y , so that
(µy)′ = µq =⇒ µy =
∫µq dx
and hence
y(x) =1
µ(x)
∫µ(x)q(x) dx where µ(x) = exp
[∫p(x) dx
]
![Page 45: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/45.jpg)
Integrating factor
To solve: y ′(x) + p(x)y(x) = q(x) for y = y(x)
Multiply through by integrating factor µ = µ(x)
µy ′ + µpy = µq
If µ′ = µp, then µy ′ + µpy = µy ′ + µ′y , so that
(µy)′ = µq =⇒ µy =
∫µq dx
and hence
y(x) =1
µ(x)
∫µ(x)q(x) dx where µ(x) = exp
[∫p(x) dx
]
![Page 46: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/46.jpg)
Example
The Sine-Gordon equation: utt = uxx − sin u
Models a mechanicaltransmission line such aspendula connected by a spring
u
Look for traveling wave solution: u(x , t) = f (x − ct),
c > 0, f (z), f ′(z)→ 0 as z →∞.
![Page 47: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/47.jpg)
Let z = x − ct. Then u(x , t) = f (x − ct) = f (z) and
utt = uxx − sin u =⇒ c2f ′′(z) = f ′′(z)− sin f
To solve the equation in f , we multiply through by f ′(z), anintegrating factor
c2f ′f ′′ = f ′f ′′ − f ′ sin f =⇒ c2(12 f ′ 2
)′=(1
2 f ′ 2)′+ (cos f )′
Now integrate w.r.t. z
12c2f ′ 2 = 1
2 f ′ 2 + cos f + a
To determine a, impose the conditions
f (z), f ′(z)→ 0 as z →∞
i.e., pendula ahead of the wave are undisturbed
![Page 48: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/48.jpg)
Let z = x − ct. Then u(x , t) = f (x − ct) = f (z) and
utt = uxx − sin u =⇒ c2f ′′(z) = f ′′(z)− sin f
To solve the equation in f , we multiply through by f ′(z), anintegrating factor
c2f ′f ′′ = f ′f ′′ − f ′ sin f =⇒ c2(12 f ′ 2
)′=(1
2 f ′ 2)′+ (cos f )′
Now integrate w.r.t. z
12c2f ′ 2 = 1
2 f ′ 2 + cos f + a
To determine a, impose the conditions
f (z), f ′(z)→ 0 as z →∞
i.e., pendula ahead of the wave are undisturbed
![Page 49: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/49.jpg)
Let z = x − ct. Then u(x , t) = f (x − ct) = f (z) and
utt = uxx − sin u =⇒ c2f ′′(z) = f ′′(z)− sin f
To solve the equation in f , we multiply through by f ′(z), anintegrating factor
c2f ′f ′′ = f ′f ′′ − f ′ sin f =⇒ c2(12 f ′ 2
)′=(1
2 f ′ 2)′+ (cos f )′
Now integrate w.r.t. z
12c2f ′ 2 = 1
2 f ′ 2 + cos f + a
To determine a, impose the conditions
f (z), f ′(z)→ 0 as z →∞
i.e., pendula ahead of the wave are undisturbed
![Page 50: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/50.jpg)
Let z = x − ct. Then u(x , t) = f (x − ct) = f (z) and
utt = uxx − sin u =⇒ c2f ′′(z) = f ′′(z)− sin f
To solve the equation in f , we multiply through by f ′(z), anintegrating factor
c2f ′f ′′ = f ′f ′′ − f ′ sin f =⇒ c2(12 f ′ 2
)′=(1
2 f ′ 2)′+ (cos f )′
Now integrate w.r.t. z
12c2f ′ 2 = 1
2 f ′ 2 + cos f + a
To determine a, impose the conditions
f (z), f ′(z)→ 0 as z →∞
i.e., pendula ahead of the wave are undisturbed
![Page 51: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/51.jpg)
Then, as z →∞,
12c2f ′ 2 = 1
2 f ′ 2 + cos f + a → 0 = 0+ cos 0+ a
so that a = −1,
i.e.
12c2f ′ 2 = 1
2 f ′ 2 + cos f − 1 =⇒ f ′ 2 =2
1− c2 (1− cos f )
Exercise:
1 Show that f (z) = 4 arctan[exp(− z√
1− c2
)]is a solution
2 Solve the equation to obtain the solution above(hint: 1− cos f = 2 sin2(f /2))
![Page 52: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/52.jpg)
Then, as z →∞,
12c2f ′ 2 = 1
2 f ′ 2 + cos f + a → 0 = 0+ cos 0+ a
so that a = −1, i.e.
12c2f ′ 2 = 1
2 f ′ 2 + cos f − 1 =⇒ f ′ 2 =2
1− c2 (1− cos f )
Exercise:
1 Show that f (z) = 4 arctan[exp(− z√
1− c2
)]is a solution
2 Solve the equation to obtain the solution above(hint: 1− cos f = 2 sin2(f /2))
![Page 53: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/53.jpg)
Then, as z →∞,
12c2f ′ 2 = 1
2 f ′ 2 + cos f + a → 0 = 0+ cos 0+ a
so that a = −1, i.e.
12c2f ′ 2 = 1
2 f ′ 2 + cos f − 1 =⇒ f ′ 2 =2
1− c2 (1− cos f )
Exercise:
1 Show that f (z) = 4 arctan[exp(− z√
1− c2
)]is a solution
2 Solve the equation to obtain the solution above(hint: 1− cos f = 2 sin2(f /2))
![Page 54: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/54.jpg)
Then, as z →∞,
12c2f ′ 2 = 1
2 f ′ 2 + cos f + a → 0 = 0+ cos 0+ a
so that a = −1, i.e.
12c2f ′ 2 = 1
2 f ′ 2 + cos f − 1 =⇒ f ′ 2 =2
1− c2 (1− cos f )
Exercise:
1 Show that f (z) = 4 arctan[exp(− z√
1− c2
)]is a solution
2 Solve the equation to obtain the solution above(hint: 1− cos f = 2 sin2(f /2))
![Page 55: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/55.jpg)
Wave front solution:
u(x , t) = 4 arctan[exp(− x − ct√
1− c2
)]
x
u
2π
speed cu
A wave front is a solution u(x , t) for which
limx→−∞
u(x , t) = k1 and limx→∞
u(x , t) = k2
![Page 56: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/56.jpg)
Partial fractions
Given a rational function p(x)/q(x)
p(x)
q(x)=
r1(x)
q1(x)+
r2(x)
q2(x)+ · · ·+ rn(x)
qn(x)
where qi (x) is a linear or an irreducible quadratic factor of q(x) and
ri (x) =
Bi (constant) if qi is linear,
Aix + Bi if qi is quadratic
![Page 57: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/57.jpg)
Example
The KdV equation: ut + uux + uxxx = 0
Look for traveling wave solution that is a pulse:
u(x , t) = f (x − ct),
f (z), f ′(z), f ′′(z)→ 0 as z → ±∞, where z = x − ct
x
u
speed c
![Page 58: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/58.jpg)
Then
ut + uux + uxxx = 0 =⇒ −cf ′ + ff ′ + f ′′′ = 0
Rewrite,
then integrate
−cf ′ +(1
2 f 2)′ + (f ′′)′ = 0
=⇒ −cf + 12 f 2 + f ′′ = a
To determine a, impose f (z), f ′′(z)→ 0 as z →∞. Then
−cf + 12 f 2 + f ′′ = a → 0+ 0+ 0 = a
so that−cf + 1
2 f 2 + f ′′ = 0
![Page 59: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/59.jpg)
Then
ut + uux + uxxx = 0 =⇒ −cf ′ + ff ′ + f ′′′ = 0
Rewrite, then integrate
−cf ′ +(1
2 f 2)′ + (f ′′)′ = 0 =⇒ −cf + 12 f 2 + f ′′ = a
To determine a, impose f (z), f ′′(z)→ 0 as z →∞. Then
−cf + 12 f 2 + f ′′ = a → 0+ 0+ 0 = a
so that−cf + 1
2 f 2 + f ′′ = 0
![Page 60: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/60.jpg)
Then
ut + uux + uxxx = 0 =⇒ −cf ′ + ff ′ + f ′′′ = 0
Rewrite, then integrate
−cf ′ +(1
2 f 2)′ + (f ′′)′ = 0 =⇒ −cf + 12 f 2 + f ′′ = a
To determine a, impose f (z), f ′′(z)→ 0 as z →∞.
Then
−cf + 12 f 2 + f ′′ = a → 0+ 0+ 0 = a
so that−cf + 1
2 f 2 + f ′′ = 0
![Page 61: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/61.jpg)
Then
ut + uux + uxxx = 0 =⇒ −cf ′ + ff ′ + f ′′′ = 0
Rewrite, then integrate
−cf ′ +(1
2 f 2)′ + (f ′′)′ = 0 =⇒ −cf + 12 f 2 + f ′′ = a
To determine a, impose f (z), f ′′(z)→ 0 as z →∞. Then
−cf + 12 f 2 + f ′′ = a → 0+ 0+ 0 = a
so that−cf + 1
2 f 2 + f ′′ = 0
![Page 62: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/62.jpg)
Now multiply through by integrating factor f ′, then integrate
− cff ′ + 12 f 2f ′ + f ′f ′′ = 0
=⇒ −c(1
2 f 2)′ + 12
(13 f 3)′ + (1
2 f ′ 2)′= 0
=⇒ −12cf 2 + 1
6 f 3 + 12 f ′ 2 = b
To determine b, impose f (z), f ′(z)→ 0 as z →∞.
Then
−12cf 2 + 1
6 f 3 + 12 f ′ 2 = b → 0+ 0+ 0 = b
so that−1
2cf 2 + 16 f 3 + 1
2 f ′ 2 = 0
![Page 63: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/63.jpg)
Now multiply through by integrating factor f ′, then integrate
− cff ′ + 12 f 2f ′ + f ′f ′′ = 0
=⇒ −c(1
2 f 2)′ + 12
(13 f 3)′ + (1
2 f ′ 2)′= 0
=⇒ −12cf 2 + 1
6 f 3 + 12 f ′ 2 = b
To determine b, impose f (z), f ′(z)→ 0 as z →∞. Then
−12cf 2 + 1
6 f 3 + 12 f ′ 2 = b → 0+ 0+ 0 = b
so that−1
2cf 2 + 16 f 3 + 1
2 f ′ 2 = 0
![Page 64: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/64.jpg)
Rewrite,
12 f ′ 2 = 1
2cf 2 − 16 f 3 =⇒
√3
f√3c − f
f ′ = 1
where we choose the positive√
and assume that 3c − f > 0.
Now let 3c − f = g2
√3
(3c − g2)g(−2gg ′) = 1 =⇒ 2
√3
3c − g2 g ′ = −1
To integrate, use partial fractions
13c − g2 =
A√3c − g
+B√
3c + g
![Page 65: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/65.jpg)
Rewrite,
12 f ′ 2 = 1
2cf 2 − 16 f 3 =⇒
√3
f√3c − f
f ′ = 1
where we choose the positive√
and assume that 3c − f > 0.
Now let 3c − f = g2
√3
(3c − g2)g(−2gg ′) = 1 =⇒ 2
√3
3c − g2 g ′ = −1
To integrate, use partial fractions
13c − g2 =
A√3c − g
+B√
3c + g
![Page 66: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/66.jpg)
Rewrite,
12 f ′ 2 = 1
2cf 2 − 16 f 3 =⇒
√3
f√3c − f
f ′ = 1
where we choose the positive√
and assume that 3c − f > 0.
Now let 3c − f = g2
√3
(3c − g2)g(−2gg ′) = 1 =⇒ 2
√3
3c − g2 g ′ = −1
To integrate, use partial fractions
13c − g2 =
A√3c − g
+B√
3c + g
![Page 67: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/67.jpg)
13c − g2 =
A√3c − g
+B√
3c + g
=⇒ 1 = A(√3c + g) + B(
√3c − g)
=⇒ A =1
2√3c, B =
12√3c
=⇒ 13c − g2 =
1/2√3c√
3c − g+
1/2√3c√
3c + g
=⇒ 2√3
3c − g2 g ′ =g ′
√c(√3c − g)
+g ′
√c(√3c + g)
![Page 68: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/68.jpg)
2√3
3c − g2 g ′ = −1
=⇒ g ′√
c(√3c − g)
+g ′
√c(√3c + g)
= −1
=⇒ g ′√3c − g
+g ′√
3c + g= −√
c
=⇒ − ln(√3c − g) + ln(
√3c + g) = −
√cz + d
=⇒ ln√3c + g√3c − g
= −√
cz + d
![Page 69: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/69.jpg)
Solve for g : g(z) =√3c
exp(−√
cz + d)− 1exp(−
√cz + d) + 1
Recall: f = 3c − g2
Use: tanh ζ =sinh ζcosh ζ
=12(e
ζ − e−ζ)12(e
ζ + e−ζ)= −exp(−2ζ)− 1
exp(−2ζ) + 1
Substitute −2ζ = −√
cz + d :
g(z) = −√3c tanh
[12(√
cz − d)]
Use f = 3c − g2 and choose d = 0:
f (z) = 3c sech2[12√
cz]
=⇒ u(x , t) = 3c sech2[√
c
2(x − ct)
]
![Page 70: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/70.jpg)
Solve for g : g(z) =√3c
exp(−√
cz + d)− 1exp(−
√cz + d) + 1
Recall: f = 3c − g2
Use: tanh ζ =sinh ζcosh ζ
=12(e
ζ − e−ζ)12(e
ζ + e−ζ)= −exp(−2ζ)− 1
exp(−2ζ) + 1
Substitute −2ζ = −√
cz + d :
g(z) = −√3c tanh
[12(√
cz − d)]
Use f = 3c − g2 and choose d = 0:
f (z) = 3c sech2[12√
cz]
=⇒ u(x , t) = 3c sech2[√
c
2(x − ct)
]
![Page 71: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/71.jpg)
Solve for g : g(z) =√3c
exp(−√
cz + d)− 1exp(−
√cz + d) + 1
Recall: f = 3c − g2
Use: tanh ζ =sinh ζcosh ζ
=12(e
ζ − e−ζ)12(e
ζ + e−ζ)= −exp(−2ζ)− 1
exp(−2ζ) + 1
Substitute −2ζ = −√
cz + d :
g(z) = −√3c tanh
[12(√
cz − d)]
Use f = 3c − g2 and choose d = 0:
f (z) = 3c sech2[12√
cz]
=⇒ u(x , t) = 3c sech2[√
c
2(x − ct)
]
![Page 72: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/72.jpg)
x
u
amplitude 3c
speed c
Soliton solution: u(x , t) = 3c sech2[√
c
2(x − ct)
]
Note: That amplitude is 3c means that taller waves move fasterthan shorter waves.
![Page 73: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/73.jpg)
Other examples
Water gravity waves
Ship waves
Tsunamis
Shock waves
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Water gravity waves
Consider water (inviscid incompressible fluid) in a constantgravitational field
Spatial coordinates (x , y , z), fluid velocity ~u = (u, v ,w)
Sinusoidal wave train solution, where the wave oscillates in~x = (x , y) and t, but not in z
Dispersion relation: ω2 = gk tanh(kd) , k = |~k | = 2π/λ
Here, ω is the frequency, ~k is the wavenumber vector, λ is thewavelength, g is gravity, and d is the depth of the water
![Page 75: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/75.jpg)
Water gravity waves
Consider water (inviscid incompressible fluid) in a constantgravitational field
Spatial coordinates (x , y , z), fluid velocity ~u = (u, v ,w)
Sinusoidal wave train solution, where the wave oscillates in~x = (x , y) and t, but not in z
Dispersion relation: ω2 = gk tanh(kd) , k = |~k | = 2π/λ
Here, ω is the frequency, ~k is the wavenumber vector, λ is thewavelength, g is gravity, and d is the depth of the water
![Page 76: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/76.jpg)
d
λ
“shallowness parameter” δ = dλ
deep water: δ > 0.28 shallow water: δ < 0.07
ω2 = gk tanh(kd)
=⇒ c =ω
k=
√g
ktanh(kd),
C =dω
dk=
12
√g
ktanh(kd) +
d√
gk sech2(kd)
2√
tanh(kd)
![Page 77: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/77.jpg)
d
λ
“shallowness parameter” δ = dλ
deep water: δ > 0.28 shallow water: δ < 0.07
ω2 = gk tanh(kd)
=⇒ c =ω
k=
√g
ktanh(kd),
C =dω
dk=
12
√g
ktanh(kd) +
d√
gk sech2(kd)
2√
tanh(kd)
![Page 78: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/78.jpg)
d
λ
“shallowness parameter” δ = dλ
deep water: δ > 0.28 shallow water: δ < 0.07
ω2 = gk tanh(kd)
=⇒ c =ω
k=
√g
ktanh(kd),
C =dω
dk=
12
√g
ktanh(kd) +
d√
gk sech2(kd)
2√
tanh(kd)
![Page 79: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/79.jpg)
Deep water: δ →∞ at fixed k
Using limθ→∞
tanh(θ) = 1
c =ω
k=
√g
ktanh(2πδ) →
√g
k
C =dω
dk=
12
√g
ktanh(2πδ) +
d√
gk sech2(2πδ)2√
tanh(2πδ)
→ 12
√g
k=
12
c
Energy in the wave propagates at half the wave speed
![Page 80: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/80.jpg)
travels at wave speed c = ω/k
travels at group velocity C = dω/dk
Ship waves: C = 12c
![Page 81: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/81.jpg)
travels at wave speed c = ω/k
travels at group velocity C = dω/dk
Ship waves: C = 12c
![Page 82: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/82.jpg)
(now) B A (earlier)
C
![Page 83: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/83.jpg)
(now) B A (earlier)
C
![Page 84: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/84.jpg)
(now) B A (earlier)
C
![Page 85: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/85.jpg)
B AD
C
E
OA′
D ′
C ′
O ′
Kelvin wedge
19.5◦ 35◦
![Page 86: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/86.jpg)
B AD
C
E
O
A′D ′
C ′
O ′
Kelvin wedge
19.5◦ 35◦
![Page 87: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/87.jpg)
B AD
C
E
OA′
D ′
C ′
O ′
Kelvin wedge
19.5◦ 35◦
![Page 88: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/88.jpg)
![Page 89: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/89.jpg)
Shallow water: δ → 0 at fixed d
Using limθ→0
tanh(θ)θ
= 1
c =ω
k=
√gd
tanh(2πδ)2πδ
→√
gd
C =dω
dk=
12
√gd
tanh(2πδ)2πδ
+d√
g · 2πδ/d sech2(2πδ)2√
tanh(2πδ)
→ 12
√gd +
d
2
√g
d=√
gd
Energy in the wave propagates at the wave speed
![Page 90: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/90.jpg)
Tsunamis
Typical wavelength of several hundred kilometers and deepest pointin the ocean in the Marianas Trench (Western Pacific Ocean) about11 kilometers makes a tsunami a shallow-water wave (long wave)
Wave speed c =√
gd
E.g., ocean depth 4 kilometers, gravity 9.8 m/s2 yields a wavespeed c =
√39 200 m/s ≈ 200 m/s or about 445 mph
Typical amplitude in the open ocean about 1 m, but rises up to10 m to 15 m as approaches shore
![Page 91: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/91.jpg)
d d
λ
a
Energy in the wave proportional to a2c ≈ a2√gd remains constant, so a
increases as d decreases
Hat Ray Leach beach, Thailand, December 2004
<http://geol105naturalhazards.voices.wooster.edu/page/32/>
![Page 92: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/92.jpg)
d d
λ
a
Energy in the wave proportional to a2c ≈ a2√gd remains constant, so a
increases as d decreases
Hat Ray Leach beach, Thailand, December 2004
<http://geol105naturalhazards.voices.wooster.edu/page/32/>
![Page 93: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/93.jpg)
What happens when a wave approaches the shore?
Typically the wave will break.*
*G. B. Witham, Linear and Nonlinear Waves, p. 22.
![Page 94: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/94.jpg)
What happens when a wave approaches the shore?
Typically the wave will break.*
*G. B. Witham, Linear and Nonlinear Waves, p. 22.
![Page 95: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/95.jpg)
Mathematically, to remove the multivalued part of the wave profile,we introduce a discontinuity or shock. We do this using the “equalarea rule” so that conservation is satisfied, i.e.,
∫ρ dx is the same
before and after a shock is introduced.*
*G. B. Witham, Linear and Nonlinear Waves, p. 42.
![Page 96: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/96.jpg)
Time series
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1shock forms
x
u
−1 1
1no longer a function
x
u
−1 1
1
![Page 97: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/97.jpg)
Time series
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1shock forms
x
u
−1 1
1no longer a function
x
u
−1 1
1
![Page 98: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/98.jpg)
Time series
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1shock forms
x
u
−1 1
1no longer a function
x
u
−1 1
1
![Page 99: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/99.jpg)
Time series
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1shock forms
x
u
−1 1
1no longer a function
x
u
−1 1
1
![Page 100: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/100.jpg)
Time series
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1shock forms
x
u
−1 1
1no longer a function
x
u
−1 1
1
![Page 101: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/101.jpg)
Time series
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1
x
u
−1 1
1shock forms
x
u
−1 1
1no longer a function
x
u
−1 1
1
![Page 102: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/102.jpg)
To define solution beyond shock formation: equal area rule
x
u
−1 1
1shock forms
x
u
−1 1
1shock propogates
x
u
−1 1
1
x
u
−1 1
1
Note: The amplitude diminishes as the shock propogates, i.e., thewave collapses after a shock forms
![Page 103: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/103.jpg)
To define solution beyond shock formation: equal area rule
x
u
−1 1
1shock forms
x
u
−1 1
1shock propogates
x
u
−1 1
1
x
u
−1 1
1
Note: The amplitude diminishes as the shock propogates, i.e., thewave collapses after a shock forms
![Page 104: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/104.jpg)
More
Can find time when shock forms (breaking time)
Can find the shock speed
But that would have to wait for another day.
![Page 105: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/105.jpg)
More
Can find time when shock forms (breaking time)
Can find the shock speed
But that would have to wait for another day.
![Page 106: Making Waves in Multivariable Calculus · James Lighthill, Waves in Fluids, Cambridge Mathematical Library, Cambridge](https://reader036.vdocuments.us/reader036/viewer/2022081505/5b90933109d3f21c788c3fd9/html5/thumbnails/106.jpg)
References
Adrian Constantin, Nonlinear Water Waves with Applications to Wave-CurrentInteractions and Tsunamis, CBMS-NSF Regional Conference Series in AppliedMathematics, Volume 81, Society for Industrial and Applied Mathematics,Philadelphia (2011).
Roger Knobel, An Introduction to the Mathematical Theory of Waves, StudentMathematics Library, IAS/Park City Mathematical Subseries, Volume 3,American Mathematical Society, Providence (2000).
James Lighthill, Waves in Fluids, Cambridge Mathematical Library, CambridgeUniversity Press, Cambridge (1978).
Bruce R. Sutherland, Internal Gravity Waves, Cambridge University Press,Cambridge (2010).
G. B. Whitham, Linear and Nonlinear Waves, A Wiley-Interscience Publication,John Wiley & Sons, Inc., New York (1999)