# Implicit and Parametric Plot

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

• 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]?
– kglr
Oct 19, 2018 at 22:26
• 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. Oct 19, 2018 at 22:33

 ContourPlot[{uu[t, x, y], vv[t, x, y]} /. t -> 1, {x, -1,  1}, {y, -1,   1},
Contours -> 5, ImageSize -> 500] 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] 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}] and with Contours-> 30`: 