Plotting image with several functions

Question

i was trying to plot this image (for the values of B: 0, 1, 1/2 , 3/2):

Using this code:

ClearAll[x, y]
y[s_, B_] := (1 + B*B + 2*B*Sin[s])^(1/2)
x[s_?NumericQ, B_?NumericQ] := NIntegrate[(1 + B*Sin[t])/y[t, B], {t, 0, s}]
undulary = 
ParametricPlot[
{x[s, 0], y[s, 0]},
{x[s, 1], y[s, 1]},{x[s, 1/2], y[s, 1/2]},{x[s, 3/2], y[s, 3/2]},
{s, 0, 4*Pi}, 
AxesLabel -> {"x ", "y "}, AspectRatio -> 1/7]

But it's not working. does anyone know how to correct the code? Thank you! I really apreciate it.

Answer 1

Edit

Since the parametric curves travel different distances, the values for s have to be adjusted to fill the plot (if required). Here I have chosen some numbers by trial and error.

Show[
 MapThread[
  ParametricPlot[
    Evaluate[{x[s, #1], y[s, #1]}]
    , {s, 0, 2 (#1 + 3)*Pi}
    , AxesLabel -> {"x ", "y "}
    , AspectRatio -> 1/4
    , PlotStyle -> ColorData[97][#2]
    , PlotRange -> {{0, 12}, {-0.1, 2.5}}
    ] &
  , {{0, 1/2, 1, 3/2}, {1, 2, 3, 4}}
  ]
 ]

Original

Show[ParametricPlot[
  Evaluate[{x[s, #], y[s, #]} & /@ {0, 1/2, 1, 3/2}],
  {s, 0, 4*Pi},
  AxesLabel -> {"x ", "y "}, AspectRatio -> 1/4,
  PlotRange -> All]
 ]