###### 2-D Wave Index of PDE's ######

All animations presented in this section represent finite-difference solutions to some variation
of the standard the 2+1 wave equation

               Utt=a**2(Uxx+Uyy)

On some domain or other.

The specific PDE's solved for each video are listed below.

1  ##### 2d Wave, Square Domain #####

	Utt=Uxx+Uyy
	U(x,y,0)=exp(-(x-5)**2-(y-5)**2)
	Ut(x,y,0)=0
	U(Boundary,t)=0

	Domain={(x,y) | 0<x<10,0<y<10}

2  ##### 2d Wave, Circle Domain #####

	Utt=Uxx+Uyy
	U(x,y,0)=exp(-(x-7.5)**2-(y-7.5)**2)
	Ut(x,y,0)=0
	U(Boundary,t)=0

	Domain={(x,y,t) | (x-7.5)**2+(y-7.5)**2<7.5**2,0<t<20}

3  ##### 2d Wave, Periodic Square Domain #####

	Utt=Uxx+Uyy
	U(x,y,0)=exp(-(x-15)**2-(y-15)**2)
	Ut(x,y,0)=0
	U(0,y,t)=U(20,y,t)
	U(x,0,t)=U(x,20,t)

	Domain={(x,y) | 0<x<20,0<y<20}

4  ##### 2d Wave, Forcing, Square Domain #####

	Utt=Uxx+Uyy+exp(-(x-5)**2-(y-5)**2)*sin(5t)
	U(x,y,0)=0
	Ut(x,y,0)=0
	U(Boundary,t)=0

	Domain={(x,y) | 0<x<10,0<y<10}

5  ##### 2d Wave, Interference, Square Domain #####

	Utt=Uxx+Uyy+(exp(-(x-5)**2-(y-5)**2)+exp(-(x-15)**2-(y-15)**2))*sin(5t)
	U(x,y,0)=0
	Ut(x,y,0)=0
	U(Boundary,t)=0

	Domain={(x,y) | 0<x<20,0<y<20}

5  ##### 2d Wave, ShockWave, Square Domain #####

	Utt=Uxx+Uyy+exp(-20*((x-10)**2+(y-3(1+t))**2)*sin(5t)
	U(x,y,0)=0
	Ut(x,y,0)=0
	U(Boundary,t)=0

	Domain={(x,y) | 0<x<20,0<y<20}

6  ##### 2d Wave, Diffraction #####

	Utt=Uxx+Uyy
	U(x,y,0)=Sum(n=1,4) of exp(-10(y-n)**2)
	Ut(x,y,0)=20*Sum(n=1,4) of (y-n)*exp(-10(y-n)**2)
	U(0,y,t)=U(10,y,t)
	U(Boundary Wall_1,t)=U(Boundary Wall_2,t)=0
	U(x,0,t)=U(x,10,t)=0

	Domain=Main-Wall_1-Wall_2

		Main={(x,y) | 0<x<10,0<y<10}
		Wall_1={(x,y) | x<4.7 && 5.8<y<6.2}
		Wall_2={(x,y) | x>5.3 && 5.8<y<6.2}

7  ##### 2d Wave, DoubleSlit #####

	Utt=Uxx+Uyy 0<t<7
	U(x,y,0)=Sum(n=2,8) of exp(-20(y-n/2)**2)
	Ut(x,y,0)=40*Sum(n=2,8) of (y-n/2)*exp(-20(y-n/2)**2)
	U(0,y,t)=U(10,y,t)
	U(Boundary Wall_1)=U(Boundary Middle)=U(Boundary Wall_2)=0
	U(x,0,t)=U(x,10,t)=0

	Domain=Main-Wall_1-Middle-Wall_2

		Main={(x,y)|0<x<10,0<y<10}
		Wall_1={(x,y)|x<4.1 && 5.8<y<6.2}
		Middle={(x,y)|4.7<x<5.3 && 5.8<y<6.2}
		Wall_2={(x,y)|x>5.9 && 5.8<y<6.2}

7  ##### 2d Wave, Refraction #####

	Utt=v(x,y)*(Uxx+Uyy)
	U(x,y,0)=exp(-10*(x-15)**2)
	Ut(x,y,0)=-10(x-15)*exp(-10*(x-15)**2)
	U(0,y,t)=U(20,y,t)=0
	U(x,0,t)=U(x,10,t)

	Domain={(x,y) | 0<x<20,0<y<10}

	v(x,y)=5 if y>2x-10,
	       1 otherwise

8  ##### 2d Wave, Parabolic Domain, Circular Pulse Centered at Focus #####

	Utt=Uxx+Uyy
	U(x,y,0)=exp(-20((x-5)**2+(y-9)**2))
	Ut(x,y,0)=0
	U(Boundary,t)=0

	Domain={(x,y,t) | 0<x<10,-10<y<-(1/4)*(x-5)**2+10,0<t<10}

8  ##### 2d Wave, Parabolic Domain, Flat Wave Front Moving Into Parabola #####

	Utt=Uxx+Uyy
	U(x,y,0)=exp(-10(y-5)**2)
	Ut(x,y,0)=20(y-5)exp(-10(y-5)**2)
	U(x,-(1/4)*(x-5)**2+5,t)=0
	U(x,-10,t)=0
	U(0,y,t)=U(10,y,t)

	Domain={(x,y,t) | 0<x<10,-10<y<-(1/4)*(x-5)**2+5,0<t<14}

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