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I am trying to graph the partials of two equations with respect to t, with both equations being implicit. I know I should be getting ellipses with the x-axis as the major axis.

k = 5;
h = 1/4;
w = 5;
A = 100;
g = 9.81;

u[t_, x_, y_] := g*k*A/w*(Tanh[k*h]*Sinh[k*y] + Cosh[k*y]) Cos[k*x - w*t]
v[t_, x_, y_] := g*A*k/w*(Tanh[k*h]*Cosh[k*y] + Sinh[k*y]) Sin[k*x - w*t]
uu[t_, x_, y_] := D[u[t, x, y], t]
vv[t_, x_, y_] := D[v[t, x, y], t]

Print[uu[t, x, y]]
Print[vv[t, x, y]]

ImplicitPlot[{uu[t, x, y], vv[t, x, y]}, {x, -1, 1}, {y, -1, 1}, {t == 1}]

I am just getting back:

ImplicitPlot[{-4905. Sin[
5 t - 5 x] (Cosh[5 y] + Sinh[5 y] Tanh[5/4]), -4905. Cos[
5 t - 5 x] (Sinh[5 y] + Cosh[5 y] Tanh[5/4])}, {x, -1, 1}, {y, -1,1}, {t == 1}]

I believe that I am correct in fixing a t and graphing different x and y's for each t, but I am not sure if this is how to approach this. Thank you

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    $\begingroup$ is this close to what you expect: ContourPlot[{uu[t, x, y], vv[t, x, y]} /. t -> 1, {x, -1, 1}, {y, -1, 1}, Contours->5]? $\endgroup$ – kglr Oct 19 '18 at 22:26
  • $\begingroup$ I know to expect ellipses with the horizontal axis being the major axis and the vertical axis being the minor axis of the ellipse, i know that the minor axis diminishes to zero, so the ellipses collapse to a horizontal segment. Thank you for your comment, I will think on it. $\endgroup$ – elcharlosmaster Oct 19 '18 at 22:33
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 ContourPlot[{uu[t, x, y], vv[t, x, y]} /. t -> 1, {x, -1,  1}, {y, -1,   1}, 
   Contours -> 5, ImageSize -> 500]  

enter image description here

Or with Contours -> 30:

 ContourPlot[{uu[t, x, y], vv[t, x, y]} /. t -> 1, {x, -1, 1}, {y, -1,   1}, 
   Contours -> 5, ImageSize -> 500] 

enter image description here

Alternatively, plot the two functions separately:

Row[ContourPlot[#[t, x, y] /. t -> 1, {x, -1, 1}, {y, -1, 1}, 
  Contours -> 5, ImageSize -> 300, PlotLabel -> #] & /@ {uu, vv}]

enter image description here

and with Contours-> 30`:

enter image description here

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