Calculations to find the solution

Clear[λ,L,H,U,a]

λ[n_]=(2n +1)Pi/2/H

Integrate[Sin[λ[n]y]^2,{y,0,H},Assumptions->Element[n,Integers]]

Simplify[%,Assumptions->Element[n,Integers]]

Integrate[100Sin[λ[n]y],{y,0,H},Assumptions->Element[n,Integers]]

Simplify[%,Assumptions->Element[n,Integers]]

a[n_]=400/(2n+1)/Pi/Cosh[λ[n]L]

u[x_,y_,nmax_]:=Sum[a[n]Cosh[λ[n]x]Sin[λ[n]y],{n,0,nmax}]

u[x,y,1]

Plots to check the boundary conditions are correct

L=1;
H=1;
Plot[Evaluate[u[L,y,200]],{y,0,H},PlotRange->{0,110}]

L=1;
H=1;
Plot[Evaluate[u[0,y,200]],{y,0,H},PlotRange->All]

L=1;
H=1;
Plot[Evaluate[u[x,H,200]],{y,0,H},PlotRange->{0,110}]

Surface and level curves plots of the solution

L=1;
H=1;
Plot3D[Evaluate[u[x,y,20]],{x,0,L},{y,0,H},PlotRange->All,
AxesLabel->{x,y,u}]

L=1;
H=1;
ContourPlot[Evaluate[u[x,y,200]],{x,0,L},{y,0,H},ContourShading->False,Contours->Range[0,100,5],
AxesLabel->{x,y}]

L=1;
H=2;
ContourPlot[Evaluate[u[x,y,200]],{x,0,L},{y,0,H},ContourShading->False,Contours->Range[0,100,5],
AxesLabel->{x,y},AspectRatio->Automatic]

L=2;
H=1;
ContourPlot[Evaluate[u[x,y,200]],{x,0,L},{y,0,H},ContourShading->False,Contours->Range[0,100,5],
AxesLabel->{x,y},AspectRatio->Automatic]