Transmission Line, TEM wave, transverse Laplacian operator (electromagnetic) Time-Harmonic Electromagnetic Fields (1961) by R. F. Harrington, ISBN 07-026745-6, p. 63, he discusses TEM mode in a transmission line.  The wave travels in the z direction.  He defines the transverse Laplacian operator as the standard Laplacian though without the z partial derivatives.  He then says that the transverse Laplacian for both E and H is zero for a TEM wave, i.e., where neither E nor H components in the z direction.  This is equation (2-71) on p. 63.
I cannot see this for the life of me.  Any thoughts would be appreciated.
BTW, I am not a student.  I did EM graduate work about 25 years ago, but then my employer, who paid for that education, assigned me to other duties, and I stopped doing EM work.  I am doing this now for fun.
Thanks.
 A: I'd like to flesh out dmckee's comment:

It has been a long time since I did that kind of thing, but at least some of the time you can make these claims as an ansatz and then show that they are consistent later on. And you are talking about the "transverse electric and magnetic" mode

I'll show you the "ansatz" (German for a "setting-into" i.e. a substitution). The grounding idea of a transverse electromagnetic mode is one that, in each transverse plane and at each instant in time, behaves exactly like static electromagnetic field defined wholly by scalar electric and magnetic potentials. As such, the conductor cross sections must be electric equipotential lines, and the only way this can be true without violating the principle of charge conservation is if there are two "there" and "back again" electrical paths as in a DC circuit. So TEM transmission lines are always at least two conductor lines, unlike general waveguides which can be single conductors, as in a Goubau line (see here as well
.  
To understand what this means: let's by assumption (i.e. an "ansatz") assume electro- / magentostatic fields are gradients of scalar potentials i.e. $\mathbf{E} = \nabla_\perp \psi_E = \partial_x \psi_E \hat{\mathbf{x}} + \partial_y \psi_E \hat{\mathbf{y}} =\mathbf{E}_\perp$,  $\mathbf{H} = \nabla_\perp \psi_H= \partial_x \psi_H \hat{\mathbf{x}} + \partial_y \psi_H \hat{\mathbf{y}} =\mathbf{H}_\perp$: to see that TEM modes exist, we substitute fields of the form $\mathbf{E}_\perp E_z(z, t)$ and $\mathbf{H}_\perp H_z(z, t)$ into Faraday's and Ampère's laws, thus proving that they do fulfill Maxwell's equations as long as:
$$\hat{\mathbf{z}}\wedge \mathbf{E}_\perp = \mathbf{H}_\perp\qquad(1)$$
$$\partial_z E_z = -\mu_0 \partial_t H_z\qquad(2)$$
$$\partial_z H_z = -\epsilon_0 E_z\qquad(3)$$
so that both $E_z$ and $H_z$ fulfill the wave equation $\partial_t^2 E_z = c^2 \partial_z^2 E_z$ (where $c$ is the freespace lightspeed and $c^2 \epsilon_0 \mu_0 = 1$), so that their solutions are dispersionless waves of the general form:
$$E_z(z, t) = f_+(z - c\,t) + f_-(z + c\,t)\qquad(4)$$
$$H_z(z, t) = \sqrt{\frac{\epsilon_0}{\mu_0}}\left(f_+(z - c\,t) - f_-(z + c\,t)\right)\qquad(5)$$
Here $f_+$ and $f_-$ are arbitrary twice differentiable functions. . Equation (1) means $\partial_x \psi_E = \partial_y \psi_H$ and $\partial_y \psi_E = -\partial_x \psi_H$, i.e. the Cauchy Riemann relationships and so there is a really neat and compact way of doing TEM waveguide analysis wherein we can define a complex potential $\Psi(\zeta) = \Psi(x + i\,y) = \psi_E(x, y) + i\,\psi_H(x, y)$ which is a holomorphic function of the complex variable $\zeta = x + i\,y$ and the electric and magnetic vector fields  can be interpreted as the complex numbers:
$$\mathbf{E}_\perp = \left(\frac{\mathrm{d} \Psi}{\mathrm{d} \zeta}\right)^*\qquad(6)$$
$$\mathbf{H}_\perp = -i \left(\frac{\mathrm{d} \Psi}{\mathrm{d} \zeta}\right)^*\qquad(7)$$
so that the whole field variation as a function of $\zeta$ (encoding the transverse co-ordinate as a complex number) $z$ (axial position) and $t$ (time) is:
$$\mathbf{E}(\zeta,\,z,\,t)= \left(\frac{\mathrm{d} \Psi}{\mathrm{d} \zeta}\right)^* \left(f_+(z - c\,t) + f_-(z + c\,t)\right)\quad\quad(8)$$
$$\mathbf{H}(\zeta,\,z,\,t) = -i \sqrt{\frac{\epsilon_0}{\mu_0}} \left(\frac{\mathrm{d} \Psi}{\mathrm{d} \zeta}\right)^*\left(f_+(z - c\,t) - f_-(z + c\,t)\right)\quad\quad(9)$$
One of the follow-ons from a function's being holomorphic is that both its real and imaginary parts fulfill the two dimensional Laplace equation. You can readily prove this from the Cauchy-Riemann relationships which, from our "ansatz", we know the scalar potentials and Cartesian components of the electric and magnetic fields must fulfil.
A: I am not sure if I understood your question, but I will try my best to answer what I got from it.
If a TEM wave travels in the Z direction, then neither the E field nor the H field will have a component in that direction (this is the very definition of a TEM wave). Now the transverse Laplacian is simply the standard Laplacian - minus all the partial derivatives with respect to the direction of propagation. This can be understood easily - just take the standard Laplacian and put Ez and Hz as 0 (assuming the direction of propagation is Z). 
Now if you are trying to understand how you got those equations in the first place, then its equally simple - just use Maxwell's equations to find the relationship between the transverse field components and then make them satisfy Helmholtz equation. 
Hope this answers your query. If not, feel free to ask more.
