To solve this problem, we need to understand the concept of wave transmission at boundaries between two different materials. The fraction of the normal incident wave energy that is transmitted into the second layer is given by the transmission coefficient (T). The transmission coefficient can be calculated using the densities and wave velocities of the two layers. The formula for the transmission coefficient is:

T = \frac{4 \cdot \rho_1 \cdot v_1 \cdot \rho_2 \cdot v_2}{(\rho_1 \cdot v_1 + \rho_2 \cdot v_2)^2}

where:

• \rho_1 = 2 \, \text{g/cc} (Density of the first layer)
• v_1 = 3 \, \text{km/s} (Velocity of the first layer)
• \rho_2 = 3 \, \text{g/cc} (Density of the second layer)
• v_2 = 4 \, \text{km/s} (Velocity of the second layer)

Substituting the given values into the formula, we get:

T = \frac{4 \cdot 2 \cdot 3 \cdot 3 \cdot 4}{(2 \cdot 3 + 3 \cdot 4)^2}

Calculate the terms in the formula:

• Numerator: 4 \cdot 2 \cdot 3 \cdot 3 \cdot 4 = 288
• Denominator: (2 \cdot 3 + 3 \cdot 4)^2 = (6 + 12)^2 = 18^2 = 324

Finally, calculate the transmission coefficient:

T = \frac{288}{324} = 0.8889

Rounding this value gives approximately 0.89, which matches the correct answer option.

Thus, the fraction of the normal incident wave energy that will be transmitted into the second layer is 0.89. This aligns with option 0.89.