A longitudinal earthquake wave strikes a boundary between two types of rock at a 41° angle. As the wave crosses the boundary, the specific gravity of the rock changes from 3.6 to 2.8. Assuming that the elastic modulus (Section 15–2)is the same for both types of rock, determine the angle of refraction.

Verified step by step guidance

1. Identify the relevant physics principle: This problem involves the refraction of waves at a boundary, which is governed by Snell's Law. Snell's Law is expressed as: $n$₁⁢sin⁢θ₁=n₂⁢sin⁢θ₂, where $n$ is the refractive index, and $\theta$ is the angle of incidence or refraction.
2. Relate the refractive index to the specific gravity: The refractive index for longitudinal waves in a medium is proportional to the square root of the specific gravity (density) divided by the elastic modulus. Since the elastic modulus is constant, the refractive index is proportional to the square root of the specific gravity. Thus, $n\propto \sqrt{\rho }$, where $\rho$ is the specific gravity.
3. Calculate the ratio of refractive indices: Using the proportionality, the ratio of refractive indices is given by $\frac{\sqrt{3.6}}{\sqrt{2.8}}$. This ratio will be used in Snell's Law.
4. Apply Snell's Law: Substitute the known values into Snell's Law. The angle of incidence is $41°$, and the ratio of refractive indices is as calculated in the previous step. The equation becomes: $\sin 41°=\frac{\sqrt{2.8}}{\sqrt{3.6}}\sin \theta$₂, where $\theta$₂ is the angle of refraction.
5. Solve for the angle of refraction: Rearrange the equation to isolate $\theta$₂. Use the inverse sine function: $\theta$₂=sin⁢−1⁢3.62.8⁢sin⁢41°. This will give the angle of refraction.

Key Concepts

Snell's Law

Snell's Law describes how waves change direction when they pass through different media. It states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant and is equal to the ratio of the velocities of the wave in the two media. This principle is crucial for determining the angle of refraction when an earthquake wave transitions between different types of rock.

Specific Gravity

Specific gravity is a dimensionless quantity that compares the density of a substance to the density of water. In the context of seismic waves, the specific gravity of the rocks affects their density and, consequently, the speed of the waves traveling through them. Understanding the specific gravities of the two types of rock is essential for applying Snell's Law to find the angle of refraction.

Elastic Modulus

The elastic modulus is a measure of a material's stiffness or resistance to deformation under stress. It is a critical property in determining how seismic waves propagate through different materials. In this scenario, the assumption that the elastic modulus is the same for both types of rock simplifies the analysis, allowing us to focus on the effects of specific gravity on wave behavior at the boundary.