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. From a mathematical point of view, why is this so?

Clear[x,t,A,k,ω]

ψ[x_,t_]:=A*Cos[k*x-ω*t]

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

A=1;
k=2*π/1;
ω=2*π/1;

Plot & animate the function.

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

Change the relative signs of the space and time terms.  Which way does the wave travel?
Change the Cos function to a non-oscillatory function.
Add a constant phase angle δ to the argument of the Cos.

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*Cos[k*x]*Cos[ω*t]

ϕ[x,t]

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->False]

Change the relative signs of the space and time terms.
Change the Cos functions to Sin functions. Try combinations of both.
Add a constant phase angle δ to the argument of either or both functions.
Change the space function to a non-oscillatory function.

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[YourFunction[x,t],{x,0,5},AxesLabel->{x,ϕ},PlotRange->1.2,PlotLabel->t],{t,0,5},
AnimationRate->0.25,
AnimationRunning->False]

Copyright 2011 Oregon State University
Version February 3, 2012