Propagation of Reflected and Refracted Waves at an Interface

Manipulate[
θ1=θ Degree;
sn1=Sin[θ1];
sn2=(v2/v1)sn1;
θ2=ArcSin[sn2];
kx1=sn1*f/v1;
ky1=cs1*f/v1;
kx2=sn2*f/v2;
ky2=cs2*f/v2;
If[Im[θ2] ≠0, RC = 1.0; TC = 0.0,cs1=Sqrt[1-sn1*sn1];cs2=Sqrt[1-sn2*sn2];del = v1*cs1 + v2*cs2;RC = (v2*cs2 - v1*cs1)/del;TC = 2v2*cs2/del];
p1=DensityPlot[0.5+(Cos[kx1*x - ky1*y - 2π*f*t]+RC*Cos[kx1*x + ky1*y - 2π*f*t])/4,{x,0,50000},{y,0,25000},PlotPoints->{101,51},Frame->False,ColorFunction->"GrayTones",ColorFunctionScaling->False];
p2=DensityPlot[0.5 + (TC*Cos[kx2*x - ky2*y - 2π*f*t])/4,{x,0,50000},{y,-25000,0},PlotPoints->{101,51},Frame->False,ColorFunction->"GrayTones",ColorFunctionScaling->False];
p3 = Graphics[Line[{{0,0},{50000,0}}]];
Show[p1,p2,p3,PlotRange->{{0,50000},{-25000,25000}},AspectRatio->1,ImageSize->{400,400}],
{{t,0,"time"},0,10,1,Appearance->"Labeled",ImageSize->Tiny},
{{θ,30.,"incident angle"},0,90.,.01,Appearance->"Labeled",ImageSize->Tiny},
{{v1,300,"upper velocity"},50,1000,50,Appearance->"Labeled",ImageSize->Tiny},
{{v2,500,"lower velocity"},50,1000,50,Appearance->"Labeled",ImageSize->Tiny},
{{f,0.3,"frequency"},0.1,1.0,0.1,Appearance->"Labeled",ImageSize->Tiny},
ControlPlacement->Left,TrackedSymbols->True
]

CAPTION

A propagating wave of given frequency (in Hz) is incident on an interface between two acoustic media, with varying propagation velocities. The incident angle is measured from the vertical, running from 0 to 90 degrees. The amplitudes of the reflected and refracted waves are computed for the specified incident angle. The reflected wave is added to the incident wave in the upper medium and the refracted wave is shown in the lower medium. At a certain incident angle that depends on the relative velocities, the incident wave will be totally reflected and the bottom half of the plot will become solid gray, indicating zero amplitude.

DETAILS

This Demonstration shows how a propagating plane wave interacts with an interface between two acoustic media. The incident wave impinges on the boundary from the top and causes a reflected wave that interferes with the incident wave. The incident wave is transmitted into the bottom layer as a refracted wave with its own propagation velocity. The incident and refracted wave angles θ (angle of the wave normal relative to vertical) are related by the well-known equation

((sin((Subscript[θ, 1])))/(sin((Subscript[θ, 2]))))= ((Subscript[v, 1])/(Subscript[v, 2])),

where v is velocity and the subscripts 1 and 2 denote the top and bottom media. The incident wave is taken to have amplitude 1. If RC denotes the reflection coefficient for the scattered wave amplitudes, then the wave motion in the top layer is described by (Towne, 1988)

cos((Subscript[k, (Subscript[x, 1])])x- (Subscript[k, (Subscript[y, 1])])y-ωt)+RC cos((Subscript[k, (Subscript[x, 1])])x+ (Subscript[k, (Subscript[y, 1])])y-ω t),

while, with TC denoting the transmission coefficient, the motion in the bottom layer is described by

TC cos[(Subscript[k, (Subscript[x, 2])])x- (Subscript[k, (Subscript[y, 2])])y-ω t],

where (Subscript[k, x]) and (Subscript[k, y]) are the horizontal and vertical wavenumbers, with the subscripts 1 and 2 for the top and bottom layers, respectively, and ω is the frequency in radians. Here all lengths are in meters.

The coefficients RC and TC are computed for variable angles of incidence and velocities with

RC=(((Subscript[v, 2])cos((Subscript[θ, 2])) - (Subscript[v, 1])cos((Subscript[θ, 1])) )/((Subscript[v, 1])cos((Subscript[θ, 1])) + (Subscript[v, 2])cos((Subscript[θ, 2])))) ,

TC=((2(Subscript[v, 2])cos((Subscript[θ, 2])))/((Subscript[v, 1])cos((Subscript[θ, 1])) + (Subscript[v, 2])cos((Subscript[θ, 2])))),

assuming, with no loss of generality, that the densities of the two media are equal. This definition puts RC in the range [-1,+1], while TC has the range [0,2]. Because of wave interference, the top media can have amplitudes in the range [-2,+2]. The overall amplitudes are scaled in the plots such that black is -2, white is +2, and medium gray is 0. Note that amplitudes appear to be continuous across the boundary. This physical fact is the basis for the equations for computation of RC and TC.

Reference:

D. H. Towne, Wave Phenomena, New York: Dover Publications, 1988.

THIS NOTEBOOK IS THE SOURCE CODE FROM

"Propagation of Reflected and Refracted Waves at an Interface" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/PropagationOfReflectedAndRefractedWavesAtAnInterface/

Contributed by: David von Seggern (University of Nevada)

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