# What's the logic inside the formula for the calculation of the maximal envelope modulation amplitude of two temporal interfering electric fields?

Since my question is a bit elaborate, I will first write some details. Thank you all!

I refer to the study conducted by Nir Grossman et al. (2017) about the use of the temporal interfering stimulation to excite neural cells in a specific region of the brain. However, my question is concerning only for the electrical part. In the image above is represented a scheme of the addressed problem, in which two pairs of electrodes are applied on a spherical target (plane XY) (figure A). Figure B shows the time domain of the electric fields taken in two spatial point (see i and ii on figure A), in particular you can observe the generated envelope from the interaction between E1 and E2.

Considering the case of two interfering fields and , where is the position vector of interest, this study provides the following formula to calculate the module of the envelope modulation amplitude :

where is the direction of interest (details on page 20 in the link above). This formula is easy, and I have no problem to understand it. But, immediately after this formula, as you can see, is provided the formula for the calculation of the maximal envelope modulation amplitude. By referring to this last equation, I can understand the formula related to the first condition. In fact, looking for example figure B - case ii, I can observe that the envelope modulation amplitude is approximating at the amplitude of E2. On the other hand, is difficult to understand why otherwise it has to be applied the indicated formula. What's the logic about this formula?

Furthermore, I have another question. Why the maximal envelope modulation amplitude is computed with this formula? I mean, the first equation is sufficient to know the maximal, isn't it?

Reference to Grossman et al. (2017): https://www.cell.com/cell/pdf/S0092-8674(17)30584-6.pdf

• I wonder if this question would have a better chance on physics or math stack exchange. I think you'd be looking for a really specific engineer and possibly a less specific physicist.
– K H
Jan 21, 2021 at 5:46