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Here is your script (a) doing what I think you want it to do. It makes a mesh object at each frame, and adds a location keyframe at the frame it was created in script.

If you are using an operator, need to be in, or set the context it requires. Make a new object obj context with context.scene.objects.active = obj. If it works on selected objects, like say join, need to make sure other objects are (de)selected correctly by setting obj.select accordingly on each.

Going to use bpy_struct.keyframe_insert(...) rather than the operator. This will be quicker, especially for high numbers.

import bpy
from math import sin, cos
scn = bpy.context.scene

numY = 10
freq = 1
amp = 1
scale = 1

for numX in range (2, 20):
    verts = []
    faces = []

    for i in range (0, numX):
        x = scale * i
        for j in range (0, numY):                
            y = scale * j
            z = scale * amp * (cos(i * freq) + sin(j * freq))
            vert = (x, y, z)
            verts.append(vert)

    count = 0
    for i in range (0, numY * (numX - 1)):
        if count < numY - 1:
            A = i
            B = i + 1
            C = (i + numY) + 1
            D = (i + numY)

            face = (A, B, C, D)
            faces.append(face)
            count = count + 1
        else:
            count = 0

    mesh = bpy.data.meshes.new("Wave")

    mesh.from_pydata(verts, [], faces)
    mesh.update(calc_edges = True)

    obj = bpy.data.objects.new("Wave", mesh)

    obj.location = (0, 0, 0)
    scn.objects.link(obj)
    obj.keyframe_insert("location", frame=numX, group="Location")