Initialization

SetOptions[Plot,PlotStyle->{Blue,AbsoluteThickness[2]},
ImageSize->500,AxesStyle->Directive[FontFamily->"Arial",FontSize->18,Black,AbsoluteThickness[0.5],Arrowheads[0.04]]];

Work

PH424 Basics: Traveling and Standing Waves

Traveling waves

Any wave form "travels" in space if the space and time variables x and t appear in the argument of an oscillatory function in the combination kx ± ωt.

Clear[x,t,A,k,ω]

ψ[x_,t_]:=A*Sin[k*x-ω*t]

For easy plotting choose wavelength = 1 space unit, period = 1 time unit, A = 1.

A=1;
k=2*π/2;
ω=2*π/2;

Plot & animate the function.

Animate[Plot[ψ[x,t],{x,0,5},AxesLabel->{x,ψ},PlotLabel->t],{t,0,5},
AnimationRate->0.25,
AnimationRunning->TRUE]

Standing Waves

Any wave form "stands" in space if it is the product of two functions, one a function of space, and one an oscillatory function of time.  The space and time variables x and t thus appear separately. From a mathematical point of view, why is this characteristic of a standing wave function?

ϕ[x_,t_]:=A*Sin[k*x]*Sin[ω*t]

ϕ[x,t]

Sin[((3 π t)/2)] Sin[((3 π x)/2)]

Sin[((3 π t)/2)] Sin[((3 π x)/2)]

Plot[ϕ[x,0],{x,0,5},AxesLabel->{x,ϕ}]

Animate[Plot[ϕ[x,t],{x,0,5},AxesLabel->{x,ϕ},PlotRange->1.2,PlotLabel->t],{t,0,5},
AnimationRate->0.25,
AnimationRunning->TRUE]

Predict the behavior of the functions Pulse1[x,t] and Pulse2[x,t]

Pulse1[x_,t_]:=A*Exp[-(k*x-ω*t)^2]
Pulse2[x_,t_]:=A*Exp[-k*(x-2.5)^2]*Cos[ω*t]

Animate[Plot[Pulse1[x,t],{x,0,5},AxesLabel->{x,ϕ},PlotRange->1.2,PlotLabel->t],{t,0,5},
AnimationRate->0.25,
AnimationRunning->TRUE]

Animate[Plot[Pulse2[x,t],{x,0,5},AxesLabel->{x,ϕ},PlotRange->1.2,PlotLabel->t],{t,0,5},
AnimationRate->0.25,
AnimationRunning->TRUE]

Copyright 2017 Oregon State University
Version February 2, 2017