Transverse and Longitudinal Directional Planar Waves

Visualize the propagation of transverse and longitudinal directional planar waves through a square lattice

We'll need to define how particles, or parcels of the medium through which our waves travel, are displaced by the waves.

Define a function to compute the position of a particle subject to displacement by a directional two-dimensional transverse wave:

In[1]:=

transverseWave2D[{x_,y_},A_,λ_,θ_,ϕ_,t_:0]:=Module[{(*Wavenumber:*)k=2π/λ},{x,y}+(A*π*Sin[k*(x*Cos[θ]+y*Sin[θ])-2π*t-ϕ]/4)*AngleVector[θ+π/2]]

Define a function to compute the position of a particle subject to displacement by a directional two-dimensional longitudinal wave:

In[2]:=

longitudinalWave2D[{x_,y_},A_,ϕ_,λ_,θ_,t_:0]:=Module[{(*Wavenumber:*)k=2π/λ},{x,y}+(A*λ/7*Sin[k*(x*Cos[θ]+y*Sin[θ])-2π*t-ϕ])*{Cos[θ],Sin[θ]}]

We'll also need to define the rectangular point lattice through which to propagate the waves.

Define a square point lattice:

In[3]:=

lattice=Flatten[Outer[List,Subdivide[-π,π,50],Subdivide[-π,π,50]],1];

Now, by mapping the wave displacement functions over the lattice points, we can visualize how waves displace them. Let's make these visualizations interactive using

Manipulate

Interactive transverse planar wave visualization:

In[4]:=

ManipulateWith{lattice=Outer[List,Subdivide[0,6,50],Subdivide[0,6,50]]},RowGraphics#,

&@Map[Point@transverseWave2D[#,amplitude,wavelength,directionAngle,phase,time]&,lattice,{2}],

Graphics[

]

,{{amplitude,1},-4,4},{{phase,π},0,2π},{{wavelength,2},1,10},{{directionAngle,0},-π,π},{time,0,1},SaveDefinitionsTrue

Out[4]=

amplitude

phase

wavelength

directionAngle

time

Interactive longitudinal planar wave visualization:

In[5]:=

ManipulateWith{lattice=Outer[List,Subdivide[-π,π,50],Subdivide[-π,π,50]]},RowGraphics#,

&@Map[Point@longitudinalWave2D[#,amplitudeFactor,phase,wavelength,directionAngle,t]&,lattice,{2}],

Graphics[

]

,{{amplitudeFactor,1},0,1},{{phase,π},0,2π},{{wavelength,1},1,10},{{directionAngle,π/4},-π,π},{t,0,1},SaveDefinitionsTrue

Out[5]=

amplitudeFactor

phase

wavelength

directionAngle

Finally, we can combine both visualizations into a single hybrid interactive demonstration to highlight the differences between both wave types.

Make a single interactive visualization of particle displacements from both wave types:

In[6]:=

ManipulateWith{latticePoints=Flatten[Outer[List,Subdivide[-π,π,50],Subdivide[-π,π,50]],1]},LegendedGraphics

,Point[Select[N[#1],RegionMember[Rotate[Rectangle[{-2π,0},{2π,2π}],directionAngle,{0,0}]]]]&@Map[transverseWave2D[#,amplitudeFactor/2,wavelength,directionAngle,phase+π/2,0]&,latticePoints],

,Point[Select[N[#1],RegionMember[Rotate[Rectangle[{-2π,-2π},{2π,0}],directionAngle,{0,0}]]]]&@Map[longitudinalWave2D[#,amplitudeFactor,phase,wavelength,directionAngle,0]&,latticePoints],