Power Loss Definition in a Damped Wave Equation (Skin Depth Problem}

[Master1022]

Homework Statement

Find the maximum depth of a submarine below the surface of the sea which enables it to detect electromagnetic signals launched from the surface? (all the numbers given)

Relevant Equations

Damped Wave Equation
Definition of skin depth

Hi,

So the main question is: How to deal with power loss in E-M waves numerically when we are given power loss in dB's?

The context is that we are dealing with the damped wave equation: $\nabla ^ 2 \vec E = \mu \sigma \frac{\partial \vec E}{\partial t} + \mu \epsilon \frac{\partial ^ 2 \vec E}{\partial t^2}$. The problem is assuming a good conductor $\left( \sigma >> \omega \epsilon \right)$ and thus we get that the one-dimensional spatial solution is:
$$\vec E = \vec E_0 \exp \left( - \frac{(1 + j)z}{s} \right)$$ where $s = \sqrt \frac{2}{\omega \sigma \mu}$.

The data is given for all these constants, but am not concerned about that aspect of this problem. We are told that the maximum allowable power loss is 60 dB.

Here is my attempt to start the problem, and I was wondering whether I was starting the problem off correctly?
$$-60 dB = 10 \log_{10} \left( \frac{P_{sea}}{P_0} \right)$$ However, we know that $P_{EM} \propto | \vec E | ^2$ and thus we can write that $-60 dB = 20 \log_{10} \left( | \frac{E_{sea}}{E_{o}} | \right)$. Therefore, we have that $10^{-3} = | \exp \left( - \frac{(1 + j)z}{s} | \right)$.

This is where I am slightly confused on how to proceed - I am not fully sure how to evaluate this expression. If I just momentarily ignore the absolute value signs, then I get: $\ln \left( 10^{3} \right)s = |1 + j|z$. I feel as if I have cheated to get to this step. Anyways, that will lead me to the solution: $z = \frac{\ln \left( 10^{3} \right)s}{\sqrt 2}$.

When I substitute the numbers in, then I get the incorrect answer for some reason. I find that I need to have a 2 in the denominator rather than a $\sqrt 2$. Is my expression for the power loss in dB correct?

Thanks in advance for the help

 

[Master1022]

UPDATE: I realized that I didn't need to the 2 in the denominator to solve this problem and I need to evaluate the modulus of the exponential as follows:
$$| \exp \left( - \frac{(1 + j)z}{s} \right) | = | \exp \left( - \frac{z}{s} \right) | | \exp \left( - \frac{zj}{s} \right) |$$
Then the modulus of the complex exponential will be 1, so we can just consider the other exponent.

Therefore, we will get $ln(10^3) = \frac{z}{s}$ and thus that leads to the correct answer $z = s ln (10^3)$