Question

Earthquakes produce several types of shock waves. The most well 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 about 6.5 km/s and S-waves move at about 3.5 km/s. The time delay between the arrival of these two waves at a seismic recording station tells geologists how far away an earthquake occurred. If the time delay is 33 s, how far from the seismic station did the earthquake occur?

Short answer

The earthquake occurred 250.25 km from the seismic station.

Step-by-step solution

Understand the Problem

The problem asks us to find the distance to an earthquake given the time delay between the arrival of P-waves and S-waves. We know that the P-waves travel faster than S-waves.

Set Up the Formula

Let the distance to the earthquake be denoted by \(d\). If \(t_P\) and \(t_S\) are the travel times for P-waves and S-waves respectively, we know that \(d = v_P \times t_P = v_S \times t_S\), where \(v_P = 6.5\) km/s and \(v_S = 3.5\) km/s. The time delay \(t_S - t_P\) is given as 33 s.

Express the Time Difference

Using the time difference equation, \(t_S = t_P + 33\) seconds. Substitute \(t_S\) in terms of \(t_P\) into the formula for \(d\) for S-waves: \(d = 3.5(t_P + 33)\).

Solve for Common Time \(t_P\)

Equate the expressions for distance \(d\):\[6.5t_P = 3.5(t_P + 33)\]Simplifying, we find:\[6.5t_P = 3.5t_P + 115.5\]\[3.0t_P = 115.5\]\[t_P = \frac{115.5}{3}\]\[t_P = 38.5 \text{ seconds}\]

Calculate the Distance \(d\)

Now use \(t_P\) to calculate the distance \(d\):\[d = 6.5 \times 38.5\]\[d = 250.25 \text{ km}\]

Verify the Solution

With \(t_P = 38.5\) s and \(t_S = t_P + 33 = 71.5\) s, check that both wave speeds yield the same distance:\[d = 3.5 \times 71.5 = 250.25 \text{ km}\]Both calculations give the same distance, confirming that the solution is consistent.

Key concepts

Earthquake Waves

When an earthquake occurs, it releases energy that travels through the Earth's layers in the form of shock waves. These waves are collectively known as earthquake waves. There are several types of earthquake waves, but the most significant ones are:

• P-waves (Primary waves)
• S-waves (Secondary waves)

These waves are crucial for seismologists because they help determine the location and magnitude of an earthquake. As they move through the Earth, they provide valuable information about the seismic event.
Understanding the behavior of these waves allows scientists to better predict the potential impact of earthquakes, thereby improving safety measures and preparedness efforts.

P-waves

P-waves, or primary waves, are the fastest type of seismic waves and are always the first to be detected by a seismic recording station.

• P-waves are compressional waves, which means they compress and expand the material they move through.
• They can travel through solids, liquids, and gases, which makes them incredibly versatile.

The speed of P-waves in the Earth's crust is approximately 6.5 km/s. This high speed is due to the wave's ability to move through different states of matter with ease.
By recording the arrival time of P-waves, seismologists can estimate the location of the earthquake epicenter, which is the point on the Earth's surface directly above the earthquake's origin.

S-waves

S-waves, or secondary waves, follow P-waves and are the second type of waves to be registered by a seismic station. Unlike P-waves, S-waves are shear waves. They move the ground up and down or side-to-side, perpendicular to the direction of wave travel.

• S-waves can only travel through solids because their shear motion requires a rigid medium.
• The speed of S-waves in the Earth's crust is about 3.5 km/s, significantly slower than P-waves.

The arrival time of S-waves is critical for calculating the time delay between P-waves and S-waves, which is used to determine how far away an earthquake has occurred.
This time delay is a key factor in calculating the distance to the earthquake's epicenter, providing essential data for seismological studies.

Seismic Recording

Seismic recording is the practice of capturing and analyzing the waves generated by earthquakes. This process involves seismometers, which are sensitive instruments that detect and record the motion of the ground.
Seismic recording allows scientists to measure various aspects of earthquake waves, including:

• Arrival times
• Amplitude
• Frequency

The difference in arrival times between P-waves and S-waves at a seismic station is used to calculate how far away an earthquake occurred.
By analyzing the wave patterns recorded, seismologists can develop a clearer understanding of the earthquake's characteristics, such as its size and location.
The data gained from seismic recordings are vital for advancing our knowledge of Earth's interior and for improving earthquake preparedness and response strategies.