# How would one calculate the electric field produced by a piezoelectric device in response to a sound wave?

So the governing equations of piezoelectricity are:

Where E is the electric field, D is the displacement field, S is the strain, and T is the stress. Say I have a piezoelectric device where the mechanical force on the device is given by a sound wave (such as in an ultrasound-powered device). Pressure and stress share the same units, so given that there is no existing electric field across a piezoelectric device, would I be correct to assume that the equation simply is D = dT and plug in the pressure for the value of T?

• Judging from the description of your question, this seems like a homework problem. Commented Jan 13, 2022 at 5:25
• It isn't, it's a question for a simulation program that I'm trying to do for a research project. @Newbie
– NaP
Commented Jan 13, 2022 at 17:21

The equations that you are presenting in your questions are not "the equations" for piezoelectricity. These are termed constitutive relations. You are missing the "definition" of the electric field in terms of the potential, the definition of the strain in terms of displacement, and the conservation equations.

Following, I will rewrite the equations.

The constitutive equations are (the inverse of yours):

\begin{align} T_{ij} = c_{ijkl} S_{kl} - e_{kij} E_k\, \\ D_i = e_{ijk} S_{kl} + \epsilon_{ij} E_j\, . \end{align}

The electric field and strain are written as:

$$E_k = -\phi_{,k}\, ,\quad S_{kl} = \frac{1}{2}(u_{k,l} + u_{l, k})\, ,$$

being $$\phi$$ the electric potential and $$u$$ the displacement.

You can write the (conservation) equations using the above information to obtain

\begin{align} c_{ijkl} u_{k, lk} + e_{kij} \phi_{,kj} + f_i = \rho \ddot{u}_i\, ,\\ e_{ikl} u_{k, li} - \epsilon_{ij} \phi_{,ij} = \rho_e\, , \end{align}

where $$f_i$$ is the body force, $$\rho$$ is the mass density, and $$\rho_e$$ is the charge density.

You could say that the last equations are the equations "of motion" for the material.

• What exactly is phi and u supposed to represent? Is it potential and particle velocity respectively?
– NaP
Commented Jan 14, 2022 at 19:36
• @PrithviNathan, I have updated the answer. Commented Jan 14, 2022 at 19:41