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Chapter 17: Problem 5

Earthquakes generate sound waves inside Earth. Unlike a gas, Earth can experience both transverse (S) and longitudinal (P) sound waves. Typically, the speed of S waves is about \(4.5\) \(\mathrm{km} / \mathrm{s}\), and that of \(\mathrm{P}\) waves \(8.0 \mathrm{~km} / \mathrm{s}\). A seismograph records \(\mathrm{P}\) and \(\mathrm{S}\) waves from an earthquake. The first P waves arrive \(3.0 \mathrm{~min}\) before the first S waves. If the waves travel in a straight line, how far away does the earthquake occur?

Short Answer

Expert verified
The earthquake occurs 1851.44 km away.

Step by step solution

01

Convert Time to Seconds

The given time difference between the arrival of P waves and S waves is 3.0 minutes. Convert this into seconds since the velocities are given in km/s. We know 1 minute equals 60 seconds, thus \(3.0 \text{ minutes} = 3.0 \times 60 = 180 \text{ seconds}\).
02

Establish the Relationship Between Distances

Let \(D\) be the distance the waves travel. For P waves, the speed is \(v_P = 8.0 \text{ km/s}\) and for S waves, the speed is \(v_S = 4.5 \text{ km/s}\). Using the formula for distance \(D = vt\), write two equations: \(D = v_P \cdot t_P\) and \(D = v_S \cdot t_S\). Since both waves travel the same distance, \(v_P \cdot t_P = v_S \cdot t_S\).
03

Express Time Difference in Terms of P and S Travel Times

According to the problem, the P waves arrive 180 seconds before the S waves. Therefore, the difference in travel times is \(t_S - t_P = 180\).
04

Substitute and Solve for Distance

From **Step 3**, we have \(t_S = t_P + 180\). Substitute this into the distance equations: \(v_P \cdot t_P = v_S \cdot (t_P + 180)\). This equation becomes:\[8.0 \cdot t_P = 4.5 \cdot (t_P + 180)\]Simplify and solve for \(t_P\):\[8.0t_P = 4.5t_P + 810\]\[3.5t_P = 810\]\[t_P = \frac{810}{3.5} = 231.43 \text{ s}\]The distance is \(D = v_P \cdot t_P = 8.0 \times 231.43 = 1851.44\text{ km}\).
05

Verification

Verify by calculating \(t_S\) using the found \(t_P\): \(t_S = t_P + 180 = 231.43 + 180 = 411.43 \text{ s}\). Calculate distance using S wave speed: \(\text{Distance} = v_S \cdot t_S = 4.5 \times 411.43 = 1851.435 \text{ km}\). Therefore, the calculations are consistent.

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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 fastest type of seismic waves generated by an earthquake. They move through the Earth at a speed of about 8.0 kilometers per second and are typically the first signals to be detected by a seismograph.
Unlike other seismic waves, P waves are longitudinal waves, meaning they compress and expand the ground in the same direction they are traveling, similar to sound waves traveling through air.
  • Fastest seismic waves
  • Travel at about 8.0 km/s
  • Compressional waves
  • First to be detected
This rapid velocity allows them to reach seismographs before S waves, providing crucial data for earthquake detection and analysis.
S waves
S waves, or secondary waves, follow P waves and are slower, moving at about 4.5 kilometers per second. They are transverse waves, which means they move the ground perpendicular to their direction of travel. This shearing motion can cause significant damage to structures during an earthquake.
Unlike P waves, S waves cannot travel through liquids, which is why they are absent when an earthquake occurs across the Earth's liquid outer core.
  • Slower than P waves
  • Travel at about 4.5 km/s
  • Transverse motion
  • Highly destructive
Understanding S waves is essential for assessing the potential damage caused by an earthquake and is a key component in earthquake detection systems.
Seismograph
A seismograph is a sensitive instrument used to detect and record seismic waves from earthquakes. It consists of a mass suspended on a frame that moves with the ground's vibrations. This movement is translated into electrical signals, which can be analyzed to determine various characteristics of the earthquake.
Seismographs are crucial in:
  • Detecting P and S waves
  • Determining the earthquake's epicenter
  • Assessing the earthquake's magnitude
Seismographs around the globe work together to track earthquake activity, providing data that can be used to warn people about potential risks and study the Earth's interior.
Earthquake Detection
Detecting earthquakes involves analyzing the data collected by seismographs around the world. The time difference between the arrival of P waves and S waves helps locate the earthquake's origin or epicenter.
Here's how detection works:
  • P waves arrive first due to their speed
  • S waves follow, with a time gap indicating distance
  • Distance calculations use wave speeds and travel times
This method allows seismologists to pinpoint the earthquake's location and provides vital information for issuing timely alerts. The process is precise, relying on clear data from multiple seismographs to accurately evaluate the earthquake's impact and potential danger.

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Most popular questions from this chapter

A detector initially moves at constant velocity directly toward a stationary sound source and then (after passing it) directly from it. The emitted frequency is \(f .\) During the approach the detected frequency is \(f_{\text {app }}^{\prime}\) and during the recession it is \(f_{\text {rec }}^{\prime}\) If the frequencies are related by \(\left(f_{\text {app }}^{\prime}-f_{\text {rec }}^{\prime}\right) / f=0.500\), what is the ratio \(v_{D} / v\) of the speed of the detector to the speed of sound?

An avalanche of sand along some rare desert sand dunes can produce a booming that is loud enough to be heard 10 \(\mathrm{km}\) away. The booming apparently results from a periodic oscillation of the sliding layer of sand - the layer's thickness expands and contracts. If the emitted frequency is \(90 \mathrm{~Hz}\), what are (a) the period of the thickness oscillation and (b) the wavelength of the sound?

The speed of sound in a certain metal is \(v_{m} .\) One end of a long pipe of that metal of length \(L\) is struck a hard blow. A listener at the other end hears two sounds, one from the wave that travels along the pipe's metal wall and the other from the wave that travels through the air inside the pipe. (a) If \(v\) is the speed of sound in air, what is the time interval \(\Delta t\) between the arrivals of the two sounds at the listener's ear? (b) If \(\Delta t=1.00 \mathrm{~s}\) and the metal is steel, what is the length \(L ?\)

A sound wave travels out uniformly in all directions from a point source. (a) Justify the following expression for the displacement \(s\) of the transmitting medium at any distance \(r\) from the source: $$ s=\frac{b}{r} \sin k(r-v t) $$ where \(b\) is a constant. Consider the speed, direction of propagation, periodicity, and intensity of the wave. (b) What is the dimension of the constant \(b ?\)

Two identical tuning forks can oscillate at \(440 \mathrm{~Hz}\). A person is located somewhere on the line between them. Calculate the beat frequency as measured by this individual if (a) she is standing still and the tuning forks move in the same direction along the line at \(3.00 \mathrm{~m} / \mathrm{s}\), and (b) the tuning forks are stationary and the listener moves along the line at \(3.00 \mathrm{~m} / \mathrm{s}\).
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