Looking to create a standing wave, specifically a third harmonic–using expressions

If you click in the comp window with the pen tool and then add this path expression it might get you headed in the right direction:

freq = .5;
amp = 100;
t = time - inPoint;
a = amp*Math.sin(t*freq*Math.PI*2);
w = 500; // width
cycles = 3;
cycleW = w/cycles;

p = [[-w/2,0]]; // points
iT = [[0,0]]; // in tangents
oT = [[0,0]]; // out tangents
for (i = 0; i < cycles*2; i++){
  n = Math.floor(i/2)
  if (i%2){
    p.push([-w/2 + cycleW*n + 3*cycleW/4,-a]);
  }else{
    p.push([-w/2 + cycleW*n + cycleW/4,a]);
  }
  iT.push([-.1821*cycleW,0]);
  oT.push([.1821*cycleW,0]);
}
p.push([w/2,0]);
iT.push([0,0]);
oT.push([0,0]);
createPath(p,iT,oT,false);

If you own iExpressions, you could create a shape path with the desired animation using the Sine Wave iExpression.

If you want to attach objects to the path, you can first convert the expression into keyframes and then connect null layers to the vertices of the path using the "create nulls from path" script that is included in After Effects.

However, if you don't need a path at all, and only want to oscillate some layers up an down, then you could probably also write an expression along those lines:
[value[0],value[1]+Math.sin(time*multiplier)*amplitude]

This expression should make the layer oscilate up/down from its current position where "multiplier" controls the speed and "amplitude" defines how much it moves up and down.