Problem 57 - College Physics

Question

\(\bullet\bullet\) Assume that \(P\) and \(S\) (primary and secondary) waves from an earthquake with a focus near the Earth's surface travel through the Earth at nearly constant but different average speeds. A monitoring station that is \(1000 \mathrm{~km}\) from the epicenter detected the S wave to arrive at \(42 \mathrm{~s}\) after the arrival of the P wave. If the P wave has an average speed of \(8.0 \mathrm{~km} / \mathrm{s},\) what is the average speed of the \(\mathrm{S}\) wave?

Step-by-Step Solution

Answer

The average speed of the S wave is approximately 5.99 km/s.

Step 1: Define Given Information

We are given that the distance from the epicenter to the monitoring station is \(1000\,\text{km}\). The S wave arrives \(42\,\text{s}\) later than the P wave, and the P wave travels at \(8.0\,\text{km/s}\).

Step 2: Determine Time Taken by P Wave

Calculate the time \(t_p\) it takes for the P wave to travel \(1000\,\text{km}\) using the formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \). Thus, \(t_p = \frac{1000}{8} = 125\,\text{s}\).

Step 3: Calculate Time Taken by S Wave

Since the S wave arrives \(42\,\text{s}\) after the P wave, the time \(t_s\) it takes for the S wave is \(t_p + 42\). Thus, \(t_s = 125 + 42 = 167\,\text{s}\).

Step 4: Determine Average Speed of S Wave

Use the formula \( \text{speed} = \frac{\text{distance}}{\text{time}} \) to find the average speed of the S wave. Substitute \(1000\,\text{km}\) for distance and \(167\,\text{s}\) for time. The average speed of the S wave is \(\frac{1000}{167} \approx 5.99\,\text{km/s}\).

Key Concepts

P Waves

In seismology, P waves, or Primary waves, are the fastest type of seismic wave. These waves are first to be detected by monitoring stations following an earthquake. P waves travel through the Earth by compressing and expanding the material they pass through, moving in a push-pull motion. This ability allows them to travel through both solid and liquid layers of the Earth.
Their speed depends on various factors, including the type of material they travel through. For example, in the exercise example, the P wave travels at a swift speed of 8.0 km/s across a distance of 1000 km. Seismologists utilize the arrival time of P waves to quickly assess the location of an earthquake epicenter and its magnitude. By understanding the characteristics of P waves, scientists can gain valuable information about the Earth's inner layers.

S Waves

S waves, also known as Secondary waves, follow after the P waves and are slower. These waves cause the ground to move perpendicular to the direction of wave propagation, creating a side-to-side motion. Unlike P waves, S waves can only travel through solid materials and are unable to move through liquids.
In seismic recordings, the S wave's delayed arrival helps calculate critical information about an earthquake. For instance, in the exercise, the S wave arrives 42 seconds after the P wave. This delay indicates their slower speed, allowing seismologists to determine both the wave's speed and characteristics of the materials the wave has traversed. S waves play a vital role in understanding the structure and composition of the Earth's interior.

Wave Speed

Wave speed is a fundamental aspect of seismology that determines how quickly seismic waves travel through different layers of the Earth. The speed of a wave is calculated using the formula \( \text{speed} = \frac{\text{distance}}{\text{time}} \). Each type of wave (P or S) has a distinct speed, influenced by the medium through which it travels.
In the example provided, the P wave has a recorded speed of 8.0 km/s, making it much faster than the S wave, which has a calculated speed of approximately 5.99 km/s. Wave speed is essential when analyzing the data from seismographs since it aids in determining the time it takes for both P and S waves to reach a specific location. Understanding wave speeds can also assist seismologists in identifying the layers within Earth's interior by observing changes in the speed and direction of the waves.

Distance Calculation

Calculating the distance between an earthquake's epicenter and a monitoring station involves understanding the relationship between wave speed and time. The formula \( \text{distance} = \text{speed} \times \text{time} \) is used to assess the distance traveled by seismic waves.
In the exercise scenario, knowing the P wave speed (8.0 km/s) and arrival delay of S waves (42 seconds) helps determine distances efficiently.

• P waves travel 1000 km in 125 seconds, which is calculated by dividing the distance by the wave speed.
• The S wave's travel time, calculated as 167 seconds (125 seconds plus 42 seconds), gives the comparative travel speed through the same distance.

Understanding these calculations is imperative when locating an earthquake’s epicenter and evaluating its impact on specific areas. It allows for more informed decision-making around safety measures and disaster preparedness.