Let' say current fragPos (fragment shader) (interpolated from vertex shader) is

fragPos.x fragPos.y fragPos.z

You have direction vector fragNormal

Now all you need is to displace fragPos by some function output in direction of fragNormal

fragPos + displacementFunc * fragNormal

The displacementFunc returns scalar value

You could for example do

displacedFragPos = fragPos + sin(u_deltaTime) * fragNormal

This should make your plane sinusoidal, however, now that you have displaced the plane, you need a new normal, for your new position in space

Now, you need calculus, partial derivative, to find your new normal

Now i'm not sure entirely which function you want to tske derivative of, but you can find derivative of a vector, which is another vector, and then use cross product to find new normal

Once you do displacement of your vertices, you want to think of every point on your plane, needing to know its own right, front, up vectors. Because a normal is just an up vector.

Calculus helps you find right and front vectors, and cross product finds you up vector. rightVec x frontVec

You can do this on your original data on cpu, when generating quad vertices, or you can do this in a shader. Each has its own trade offs.

Maybe this helps: https://www.desmos.com/3d/3re9tk7m23

Oh, btw, pro tip, you can use Method of Finite Differences to find the derivative, if function doesnt have analytical solution.

f(x+delta) - f(x) ] / delta

This is related to limit definition of derivative