I Harmonically forcing a drum membrane -- are the waves isotropic?

[Spinnor]

Suppose I apply a pair of equal and opposite harmonically varying forces perpendicular to an infinite drum membrane. Consider the following forcing functions at two nearby points,(x=0,y=a) and (x=0,y=-a), separated by a distance 2a,

F(t,0,a) = Acos(ωt), F(t,0,-a) = -Acos(ωt)

Let the forcing functions act not at a point but over some very small area so we don't puncture the membrane. Let ω be small enough so that 2a<<c/ω

I suspect that far from the origin, (0,0), that the waves will not propagate equally in all directions? Is there a simple argument one can make to show that my suspicion is correct or incorrect?

Thanks!

 

[Baluncore]

For small amplitude signals, if the membrane is isotropic then the propagation will be the same in all directions.

A standing wave pattern will be created once signals reflect at the membrane boundary. That will change the apparent impedance of the membrane where the forcing functions are applied.

 

[Spinnor]

For small amplitude signals, if the membrane is isotropic then the propagation will be the same in all directions.

I think the above is true only if the two forcing functions are in phase and very close together in relation to the wavelength? In my problem the forcing functions are close in relation to the wavelength but the sources are out of phase by one half cycle, so that far away and equal distant from the "sources" the waves will cancel?

By process of elimination it seems waves are most strong along the line that contains the sources?

Because the sources are close and are out of phase little energy is radiated? If we move the sources apart more energy is radiated?

This membrane was allowed to be infinite so there are no reflections.
Thanks!