The Homogeneous Wave Equation: IVP

Manipulate[Plot[0.5*(Sin[x+t]+Sin[x-t]),{x,-15,15},PlotRange->{-1.25,1.25},PlotLabel->"Standing Wave Solution to Homogeneous Wave Equation"],{t,0,10*Pi}]

The Inhomogeneous Wave Equation: IVP

Manipulate[Plot[0.75*Sin[x+t]+0.25*Sin[x-t]-0.25*(Cos[x+t]+Cos[x-t])+0.5*Exp[-t]*Cos[x],{x,-15,15},PlotRange->{-1.25,1.25},PlotLabel->"Effect of Slight Perturbation to Homogeneous Wave Equation"],{t,0,10*Pi}]

Manipulate[Plot[{0.75*Sin[x+t]+0.25*Sin[x-t]-0.25*(Cos[x+t]+Cos[x-t])+0.5*Exp[-t]*Cos[x],0.5*Exp[-t]*Cos[x],0.75*Sin[x+t]-0.25*Cos[x+t],0.25*(Sin[x-t]-Cos[x-t])},{x,-15,15},PlotRange->{-1.25,1.25},PlotLabel->"Decomposition into left/right-movers and forcing effect",PlotLegends->{"superposition","external force","left mover","right mover"}],{t,0,10*Pi}]

The Inhomogeneous Wave Equation: Turning on and off a force

Manipulate[Plot[
UnitStep[t-1]*(
(1-UnitStep[t-2])*(
Cos[x]/(c^2)-(Cos[x+c*(t-1)]+Cos[x-c*(t-1)])/(2*c^2)
)
+UnitStep[t-2]*(
(Cos[x+c*(t-2)]-Cos[x+c*(t-1)]+Cos[x-c*(t-2)]-Cos[x-c*(t-1)])/(2*c^2))
),{x,-10,10},PlotRange->{-1,1},PlotLabel->"Application of finite duration force to stationary flat string"],{t,0,20,0.1},{c,1,10}]