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Earthquake waves. Earthquakes produce several types of shock waves. The best known are the P-waves (P for primary or pressure) and the S-waves (S for secondary or shear). In the earth's crust, P-waves travel at around 6.5 \(\mathrm{km} / \mathrm{s}\) while S-waves move at about 3.5 \(\mathrm{km} / \mathrm{s}\) . (The actual speeds vary with the type of material the waves are going through.) The time delay between the arrival of these two types of waves at a seismic recording station tells geologists how far away the earthquake that produced the waves occurred. (a) If the time delay at a seismic station is 33 s, how far from that station did the earthquake that produced the waves occurred. (a) If the time delay at a seismic station is 33 s, how far from that station did the earthquake occur? (b) One form of earthquake warning system detects the faster (but less damaging) P-waves and sounds an alarm when they first arrive, giving people a short time to seek cover before the more dangerous S-waves arrive. If an earthquake occurs 375 \(\mathrm{km}\) away from such a warning device, how much time would people have to take cover between the alarm and the arrival of the S-waves?

Step by step solution

01

Define Variables and Known Quantities

First, we identify the crucial quantities given in the problem: \(v_p = 6.5\, \mathrm{km/s}\) for P-waves, \(v_s = 3.5\, \mathrm{km/s}\) for S-waves, and the time delay \(\Delta t = 33\, \mathrm{s}\) between the arrival of S-waves and P-waves.

02

Calculate the Distance from Station (Part a)

Use the formula for time delay \(\Delta t = \frac{d}{v_s} - \frac{d}{v_p}\) to solve for the distance \(d\). Rearranging this equation gives \(d = \Delta t \times \left(\frac{v_p \times v_s}{v_p - v_s}\right)\). Substituting \(v_p = 6.5\, \mathrm{km/s}\), \(v_s = 3.5\, \mathrm{km/s}\), and \(\Delta t = 33\, \mathrm{s}\) results in the distance \(d\).

03

Solve for Distance

Substitute the values: \(d = 33 \times \left(\frac{6.5 \times 3.5}{6.5 - 3.5}\right)\). Calculate \(d = 33 \times \left(\frac{22.75}{3}\right)\), which simplifies to \(d = 33 \times 7.5833\approx 250.25\, \mathrm{km}\).

04

Calculate Warning Time (Part b)

For part b, calculate the time difference using the known distance \(375\, \mathrm{km}\): The time for P-waves to travel this distance is \(t_p = \frac{375}{6.5}\) and for S-waves is \(t_s = \frac{375}{3.5}\). The warning time is \(t_s - t_p\).

05

Compute Warning Time

Calculate the times: \(t_p = \frac{375}{6.5} \approx 57.7\, \mathrm{s}\) and \(t_s = \frac{375}{3.5} \approx 107.1\, \mathrm{s}\). Thus, the warning time is \(107.1 - 57.7 = 49.4 \, \mathrm{s}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

P-waves

P-waves, or primary waves, are the first type of seismic waves generated by an earthquake. They are called primary because they travel the fastest, reaching seismic recording stations before any other type of wave. P-waves are compressional waves, which means they work by compressing and expanding the material they move through, much like sound waves in air. This allows them to travel through solid rock, liquids, and gases. Their speed typically averages around 6.5 kilometers per second through the Earth's crust, though this can vary depending on the material's properties. The ability of P-waves to travel quickly and through various mediums makes them crucial for early detection systems in earthquake monitoring.

S-waves

S-waves, also known as secondary or shear waves, are the second type of waves to arrive from an earthquake. They are slower than P-waves, with speeds usually around 3.5 kilometers per second in the Earth's crust. Unlike P-waves, S-waves cannot travel through liquids. This limitation occurs because S-waves move through the medium as transverse waves, meaning they transfer energy perpendicular to the direction of wave propagation, similar to shaking a rope up and down. This distinct movement makes them much more damaging than P-waves when they reach the Earth's surface. Being aware of their slower speed, S-waves are crucial in calculating the distance to an earthquake's epicenter.

earthquake warning systems

Earthquake warning systems are vital in reducing the potential damage caused by earthquakes. These systems work by detecting the initial P-waves quickly and using the time before S-waves arrive to alert people. When P-waves are detected, an alert can sound the alarm, giving precious seconds to minutes for people to take cover, halt hazardous activities, or shut down sensitive equipment. The effectiveness of these systems can vary based on factors including the distance from the epicenter and the region’s infrastructure. Generally, closer seismic stations provide more accurate and timely warnings, which can reduce injuries and save lives by providing faster response times. Such systems are constantly being refined to improve their reliability and speed.

seismic recording stations

Seismic recording stations are vital tools in monitoring and studying earthquakes. These stations are equipped with seismometers, which are sensitive instruments that detect the vibrations produced by seismic waves. When an earthquake occurs, the P-waves will be the first to reach these stations, followed by the slower S-waves. The recorded data allows scientists to calculate the distance to the earthquake's epicenter by analyzing the time delay between the arrival of the two types of waves. Widespread networks of seismic stations help triangulate an earthquake's location to pinpoint its origin more accurately. These stations not only aid in immediate earthquake response efforts but also contribute valuable data for research to better understand earthquake dynamics.