Rayleigh Surface Wave

Example from the PscFunctions package

With the help of routines of the PscFunctions package one can visualize some types of waves .
Rayleigh has shown that in elastic half-space there can be special waves called as surface.
At the beginning we create the equation of the undeformed half-plane x(u,v), y(u,v) . Then we create equation of the Rayleigh surface wave U(x,y,t), V(x,y,t) . Coordinates of points after deforming are determined by functions X(u,v,t),Y(u,v,t) . The following example shows a graphical representation of such wave in some point of time.

>    restart;
with(PscFunctions):
x:=(u,v)->u:
y:=(u,v)->QR(v,0):
a:=0.8475: b:=0.3933: c:=0.5753: d:=1.732: p:=1: C:=1:
# Rayleigh surface wave equation
U:=(x,y,t)->C*(-exp(-a*y)+c*exp(-b*y))*sin(x+p*t):
V:=(x,y,t)->C*a*(exp(-a*y)-d*exp(-b*y))*cos(x+p*t):
# deformed half-plane equation
X:=(u,v,t)->x(u,v)+U(x(u,v),y(u,v),t):
Y:=(u,v,t)->y(u,v)+V(x(u,v),y(u,v),t):
plot3d([X(u,v,0.5),Y(u,v,0.5),0],u=-5..5,v=-2..5,grid=[31,26],style=HIDDEN,color=BLACK,labels=["X","Y","Z"],axes=BOXED,view=[-4..4,-2..4,-1..1],orientation=[90,0],scaling=CONSTRAINED);

[Maple Plot]

The following animation shows visually this wave.

>    plots[animate](plot3d,[[X(u,v,t),Y(u,v,t),0],u=-5..5,v=-2..5,grid=[31,26],style=HIDDEN,color=BLACK,labels=["X","Y","Z"],axes=BOXED,view=[-4..4,-2..4,-1..1]],t=0..2*Pi,orientation=[90,0],scaling=CONSTRAINED);

[Maple Plot]