Chapter 15      Problem 54

f[x_,t_]:=((a^3)/((a^2)+(((x-v t))^2)))

Animate[Plot[f[x,t]/.{a->.01,v->20.},{x,-0.2,0.2},PlotRange->{0.,0.015}],{t,0.,0.010, 0.001}]

Plot[f[x,0.00]/.{a->.01,v->20.},{x,-0.1,0.1},PlotRange->{0.,0.015}]

Does f[x,t] satisfy the Wave Equation?

(∂_(x,x))f[x,t]//Simplify

(∂_(t,t))f[x,t]//Simplify

(∂_(x,x))f[x,t]==(1/(v^2)) (∂_(t,t))f[x,t]//Simplify

Yes !!     It does satisfy the Wave Equation.

Assuming[n∈Integers,(∫_0^ℓ)((Sin[((2 π x)/(2 ℓ/n))])^2)\[DifferentialD]x]

x=.
λ=.