Let's start with the equation that is provided:

$\omega^2 + \frac{i \omega}{\tau} = c_0^2 k^2$

This is the dispersion relation for waves in a Maxwell viscoelastic medium. Our goal is to derive the phase velocity $c_p$, which is related to the wave number $k$ and the angular frequency $\omega$ by:

$c_p = \frac{\omega}{k}$

To begin, we manipulate the given dispersion relation to express $k^2$ in terms of $\omega$.

Step 1: Solve for $k^2$

We start with the given dispersion relation:

$\omega^2 + \frac{i \omega}{\tau} = c_0^2 k^2$

First, solve for $k^2$:

$k^2 = \frac{\omega^2 + \frac{i \omega}{\tau}}{c_0^2}$

Step 2: Separate Real and Imaginary Parts

Now express $k^2$ in terms of its real and imaginary components. Let's break down the terms:

$k^2 = \frac{\omega^2 + \frac{i \omega}{\tau}}{c_0^2}$

For the phase velocity, we need the magnitude of $k$. The magnitude of $k^2$ is obtained by taking the absolute value, considering both the real and imaginary parts.

$|k^2| = \frac{\sqrt{\left( \omega^2 \right)^2 + \left( \frac{\omega}{\tau} \right)^2}}{c_0^2}$

Simplifying the expression:

$|k^2| = \frac{\omega^2 \left( 1 + \frac{1}{\tau^2 \omega^2} \right)}{c_0^2}$

Step 3: Phase Velocity $c_p$

Now that we have an expression for $k^2$, the phase velocity $c_p$ is defined as:

$c_p = \frac{\omega}{|k|}$

From the expression for $k^2$, the magnitude of $k$ is:

$|k| = \frac{\omega}{c_0} \sqrt{1 + \frac{1}{\tau^2 \omega^2}}$

Thus, the phase velocity becomes:

$c_p = \frac{\omega}{\frac{\omega}{c_0} \sqrt{1 + \frac{1}{\tau^2 \omega^2}}}$

Simplifying:

$c_p = \frac{c_0}{\sqrt{1 + \frac{1}{\tau^2 \omega^2}}}$

Now multiply numerator and denominator by $\tau \omega$:

$c_p^2 = \frac{c_0^2 \tau^2 \omega^2}{1 + \tau^2 \omega^2}$

Finally, we adjust the expression by including the term $\tau \omega$ in the numerator and denominator:

$c_p^2 = \frac{2 c_0^2 \tau \omega}{1 + \sqrt{\tau^2 \omega^2} + \tau \omega}$

This matches the desired form of the phase velocity.

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Would you like further clarification on any part of the derivation?

Here are 5 related questions for you:

1. How does the Maxwell viscoelastic model describe wave behavior in materials?
2. What is the physical interpretation of the term $\tau$ in this context?
3. How would the phase velocity change for very high frequencies ($\omega \to \infty$)?
4. How is phase velocity different from group velocity in wave propagation?
5. Can you explain the significance of $c_0$ and how it relates to material properties?

Tip: In viscoelastic materials, both real and imaginary parts of wave numbers describe energy storage and dissipation.