To determine the values for the Lamé parameters, Young's modulus, and the bulk modulus for the given sample of granite, we can use the following relationships and given parameters:

Given:

• P-wave velocity (Vp) = 5.5 km/s = $5500 \, \text{m/s}$
• Density (ρ) = 2.6 Mg/m³ = $2.6 \times 10^3 \, \text{kg/m}^3$
• The material is a Poisson solid (so, it is isotropic and elastic).

Equations:

1. Bulk modulus (K): The bulk modulus can be obtained from the P-wave velocity using the formula: $V_p = \sqrt{\frac{K + \frac{4}{3}G}{\rho}}$ where $V_p$ is the P-wave velocity, $K$ is the bulk modulus, $G$ is the shear modulus, and $\rho$ is the density.

2. Shear modulus (G): We can express $G$ using the relationship between $V_p$ and $V_s$ (S-wave velocity). However, to proceed, we need the value of $V_s$ (S-wave velocity), which can be derived from the relationship between the longitudinal and shear velocities in a Poisson solid.

3. Lamé parameters ($\lambda$ and $\mu$): The Lamé parameters can be obtained from the relations between them and the elastic constants (shear modulus $G$ and bulk modulus $K$):

   • $\lambda = \frac{3K - 2G}{3}$
   • $\mu = G$

4. Young’s Modulus (E): Young’s modulus is related to the Lamé parameters by the formula: $E = \frac{3K(1 - 2\nu)}{2(1 + \nu)}$ where $\nu$ is Poisson's ratio.

Step 1: Calculate the Bulk Modulus (K)

We can use the formula for the P-wave velocity to solve for the bulk modulus $K$:

$V_p = \sqrt{\frac{K + \frac{4}{3}G}{\rho}}$

Rearranging for $K$, we get:

$K = \left( V_p^2 \rho - \frac{4}{3}G \right)$

To fully solve for $K$, we need $G$, the shear modulus, which is typically derived from the shear wave velocity. If this information isn't provided, we can't solve the system completely. However, assuming a typical relationship for granite (which usually has a shear modulus of around 30 GPa), we could proceed with that assumption. Would you like me to proceed with this assumption, or do you have the shear wave velocity or any additional data?