Nonlinear Shallow Water Waves - Revision history

Levi: /* Speed of the shock */ fix

2025-08-24T21:41:01Z

Speed of the shock: fix

← Older revision		Revision as of 21:41, 24 August 2025
Line 468:		Line 468:
	</math>		</math>
	</center>		</center>
	where <math>u^+</math> is the ~~height~~ on the right (positive) side of		where <math>u^+</math> is the flow speed on the right (positive) side of
	the jump and <math>u^-</math> is the ~~height~~ on the left (negative) side.		the jump and <math>u^-</math> is the flow speed on the left (negative) side.

	We now need to consider the equation for conservation of momentum. In integral form		We now need to consider the equation for conservation of momentum. In integral form

2025-08-23T23:12:11Z

Speed of the shock: fix signs

← Older revision		Revision as of 23:12, 23 August 2025
Line 443:		Line 443:
	<math>		<math>
	\partial_t \int_{x_1}^{x_2} \rho h \mathrm{d}x		\partial_t \int_{x_1}^{x_2} \rho h \mathrm{d}x
	+ \left. \rho u h \right\|_{~~x_2~~}^{~~x_1~~} =0		+ \left. \rho u h \right\|_{x_1}^{x_2} =0
	</math>		</math>
	</center>		</center>
Line 455:		Line 455:
	\int_{s(t)}^{x_2} \right) h \mathrm{d}x		\int_{s(t)}^{x_2} \right) h \mathrm{d}x
	= \left( \int_{x_1}^{s(t)} +		= \left( \int_{x_1}^{s(t)} +
	\int_{s(t)}^{x_2} \right) \partial_t h \mathrm{d}x + h^{+} \partial_t s(t) - h^{-}\partial_t s(t),		\int_{s(t)}^{x_2} \right) \partial_t h \mathrm{d}x - h^{+} \partial_t s(t) + h^{-}\partial_t s(t),
	</math>		</math>
	</center>		</center>
Line 476:		Line 476:
	<math>		<math>
	\partial_t \int_{x_1}^{x_2} \rho h u \mathrm{d}x		\partial_t \int_{x_1}^{x_2} \rho h u \mathrm{d}x
	= \left. \rho u^2 h \right\|_{~~x_1~~}^{~~x_2~~}		= \left. \rho u^2 h \right\|_{x_2}^{x_1}
	+ \int_0^{h(x_1)} P(x_1,z,t) \mathrm{d}z -		+ \int_0^{h(x_1)} P(x_1,z,t) \mathrm{d}z -
	\int_0^{h(x_2)} P(x_2,z,t) \mathrm{d}z		\int_0^{h(x_2)} P(x_2,z,t) \mathrm{d}z

2025-08-06T23:42:27Z

Equations of Motion

← Older revision		Revision as of 23:42, 6 August 2025
Line 40:		Line 40:
	<math>		<math>
	\partial_t \int_{x}^{x + \Delta x} \rho h u \mathrm{d}x		\partial_t \int_{x}^{x + \Delta x} \rho h u \mathrm{d}x
	= \left. \rho u^2 h \right\|_{x}^{~~x + \Delta~~ x}		= \left. \rho u^2 h \right\|_{x + \Delta x}^{x}
	+ \int_0^{h(x)} P(x,z,t) \mathrm{d}z -		+ \int_0^{h(x)} P(x,z,t) \mathrm{d}z -
	\int_0^{h(x + \Delta x)} P(x+\Delta x,z,t) \mathrm{d}z		\int_0^{h(x + \Delta x)} P(x+\Delta x,z,t) \mathrm{d}z

2020-08-17T04:31:16Z

Lecture Videos

← Older revision		Revision as of 04:31, 17 August 2020
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	=== Part 1 ===		=== Part 1 ===

	{{#ev:youtube\|~~SWIXL97H0fw~~}}		{{#ev:youtube\|Yi8fytUszrQ}}

	=== Part 2 ===		=== Part 2 ===

	{{#ev:youtube\|~~j652tvrgm4c~~}}		{{#ev:youtube\|_668xmJ9DAQ}}

	=== Part 3 ===		=== Part 3 ===

	{{#ev:youtube\|~~QVtYsUGtb7I~~}}		{{#ev:youtube\|yTat11ERJMg}}

	=== Part 4 ===		=== Part 4 ===

	{{#ev:youtube\|~~Ik2Q7H_aIV8~~}}		{{#ev:youtube\|qG8bPJwX96k}}

	=== Part 5 ===		=== Part 5 ===

	{{#ev:youtube\|~~lNeFNHAqgEA~~}}		{{#ev:youtube\|38ZWW2dM1Qs}}

	=== Part 6 ===		=== Part 6 ===

	{{#ev:youtube\|~~Yo0sW3Nh2FI~~}}		{{#ev:youtube\|AZHlhA5pQBg}}

	=== Part 7 ===		=== Part 7 ===

	{{#ev:youtube\|~~gOEMfOQZAAM}}~~		{{#ev:youtube\|VhlY5lQcTlM}}

	~~=== Part 8 ===~~

	~~{{#ev:youtube\|ZwvTZcWLZKM~~}}

	[[Category:Simple Nonlinear Waves]]		[[Category:Simple Nonlinear Waves]]

2020-08-17T04:26:37Z

Shallow Water Bore

← Older revision		Revision as of 04:26, 17 August 2020
Line 568:		Line 568:
	Below is a video of surfing on the [http://en.wikipedia.org/wiki/Severn Severn] bore, do not believe everything they		Below is a video of surfing on the [http://en.wikipedia.org/wiki/Severn Severn] bore, do not believe everything they
	say. You might also want to check out the [http://en.wikipedia.org/wiki/Pororoca Pororoca]		say. You might also want to check out the [http://en.wikipedia.org/wiki/Pororoca Pororoca]
	a tidal bore on the Amazon.		a tidal bore on the Amazon.
	~~{{#ev:youtube\|0jtkyuuMgVQ}}~~

	== Lecture Videos ==		== Lecture Videos ==

2020-08-17T04:26:11Z

Shallow Water Bore

@@ Line 570: / Line 570: @@
 a tidal bore on the Amazon.
 {{#ev:youtube|0jtkyuuMgVQ}}
 [[Category:Simple Nonlinear Waves]]
 [[Category:Nonlinear Water-Wave Theory]]
 [[Category:Shallow Depth]]

2020-08-17T03:47:39Z

Accelerating Piston

← Older revision		Revision as of 03:47, 17 August 2020
Line 382:		Line 382:
	<center>		<center>
	<math>		<math>
	X_+(t,t_0 + \Delta t) = X_+(t,t_0) + \Delta t \frac{\partial X_+}{\partial t_0} (t,t_0)		X_+(t,t_0 + \Delta t) = X_+(t,t_0) + \Delta t \frac{\partial X_+}{\partial t_0} (t,t_0) = X_+(t,t_0)
	</math>		</math>
	</center>		</center>

2015-01-09T02:46:15Z

Introduction

← Older revision		Revision as of 02:46, 9 January 2015
Line 14:		Line 14:
	This allows us to assume [[:Category:Shallow Depth\|Shallow Depth]]. We assume that the problem has not variation		This allows us to assume [[:Category:Shallow Depth\|Shallow Depth]]. We assume that the problem has not variation
	in either the <math>y</math> or <math>z</math> direction. The fluid is governed by two parameters,		in either the <math>y</math> or <math>z</math> direction. The fluid is governed by two parameters,
	<math>u</math>, the velocity of the water, and <math>h</math> the water depth (note that this is not the still water depth since the problem		<math>u(x,t)</math>, the velocity of the water, and <math>h(x,t)</math> the water depth (note that this is not the still water depth since the problem
	is nonlinear).		is nonlinear).

2010-11-06T09:51:24Z

Accelerating Piston

← Older revision		Revision as of 09:51, 6 November 2010
Line 350:		Line 350:
	<center>		<center>
	<math>		<math>
	u+2c = a t_0 + 2c_{\~~textrm~~{plate}} \,		u+2c = a t_0 + 2c_{\text{plate}} \,
	</math>		</math>
	</center>		</center>
	where <math> a t_0</math> is the velocity of the piston at time <math>t=t_0</math>		where <math> a t_0</math> is the velocity of the piston at time <math>t=t_0</math>
	and <math>c_{\~~textrm~~{plate}}</math> is the speed (related to height) at the plate.		and <math>c_{\text{plate}}</math> is the speed (related to height) at the plate.
	We know that <math> R_- = u - 2 c =-2c_0</math> through out the fluid, so that if		We know that <math> R_- = u - 2 c =-2c_0</math> through out the fluid, so that if
	we solve this at the plate (where <math>u=at_0</math> and <math>c=c_{\~~textrm~~{plate}}</math>)		we solve this at the plate (where <math>u=at_0</math> and <math>c=c_{\text{plate}}</math>)
	then we get		then we get
	<center><math>		<center><math>
	c_{\~~textrm~~{plate}} = at_0/2 + c_0\,		c_{\text{plate}} = at_0/2 + c_0\,
	</math></center>		</math></center>

2010-10-17T23:59:36Z

Accelerating Piston

← Older revision		Revision as of 23:59, 17 October 2010
Line 357:		Line 357:
	We know that <math> R_- = u - 2 c =-2c_0</math> through out the fluid, so that if		We know that <math> R_- = u - 2 c =-2c_0</math> through out the fluid, so that if
	we solve this at the plate (where <math>u=at_0</math> and <math>c=c_{\textrm{plate}}</math>)		we solve this at the plate (where <math>u=at_0</math> and <math>c=c_{\textrm{plate}}</math>)
	then we get <math>c_{\textrm{plate}} = at_0/2 + c_0</math>		then we get
			<center><math>
			c_{\textrm{plate}} = at_0/2 + c_0\,
			</math></center>

	On the <math>C_+</math> characteristics <math>u</math> and <math>c</math>		On the <math>C_+</math> characteristics <math>u</math> and <math>c</math>