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Chapter 16: Problem 43

(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S-and P-waves, which travel at different speeds. If S- and P-waves travel at 4.00 and \(7.20 \mathrm{km} / \mathrm{s}\), respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined? (b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)

Short Answer

Expert verified
The precision with which the distance to the source of the earthquake can be determined is 0.320 km (320 meters). This might not impose a serious limit to the detection of underground nuclear testing, as it can still provide an approximate location of the test site. However, if the precision is extremely important and there is greater uncertainty about the propagation speeds of the S- and P-waves, it could result in serious limits to the detection of the test site and potential violations of test bans.

Step by step solution

01

Calculate Distance Differences

First, we need to find the distance each wave will travel within the time uncertainty of 0.100 s. Distance = Speed 脳 Time For S-wave: \(D_S = S_{speed} \times \Delta_t\) \(D_S = 4.00 \, km/s \times 0.100 \, s\) \(D_S = 0.4 \, km\) For P-wave: \(D_P = P_{speed} \times \Delta_t\) \(D_P = 7.20 \, km/s \times 0.100 \, s\) \(D_P = 0.720 \, km\)
02

Calculate the difference in distance

Next, we need to find the difference between the distances traveled by the S-wave and P-wave. Difference in Distance, \(D_{diff} = D_P - D_S\) \(D_{diff} = 0.720\, km - 0.4\, km\) \(D_{diff} = 0.320\, km\) So the precision with which the distance to the source of the earthquake can be determined is 0.320 km (320 meters). For part (b), let's discuss if this precision represents a serious limit to the detection of underground nuclear testing: The value calculated in part (a), 320 meters, gives us an idea of how far apart the S-wave and P-wave arrival times can cause the calculated epicenter to be inaccurate. If the underground nuclear bomb testing is detected within this margin of error, it might not impose a serious limit to the detection, as it can still provide an approximate location of the test site. However, in cases where the precision is extremely important, and the uncertainty about propagation speeds of the waves is greater, it could result in serious limits to the detection of the test site and potential violations of test bans.

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Key Concepts

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

S-and P-waves
Seismic waves are critical for understanding earthquakes and are categorized primarily as S-waves (Secondary waves) and P-waves (Primary waves).

P-waves are a type of elastic wave that are the fastest seismic wave and can move through both solid rock and fluids. Due to their speed, they are the first waves to be detected by seismographs after an earthquake occurs. In contrast, S-waves are slower and can only travel through solid materials. This key difference in speed and behavior allows geologists to use the arrival times of both waves to estimate the distance to an earthquake's epicenter.

For example, with S-waves traveling at 4.00 km/s and P-waves at 7.20 km/s, even a small time difference in their detection allows scientists to calculate the distance they have traveled and thus pinpoint the source of the seismic activity.
Seismograph Precision
A seismograph is an instrument used to record vibrations from earthquakes and is designed with high precision to track the exact moments when seismic waves reach a particular location.

The precision of a seismograph is typically measured in tenths of a second, enabling scientists to record the arrival of S-and P-waves very accurately. This timing precision is essential for calculating the distance to an earthquake's epicenter. The precision of 0.100 s used in seismograph readings means that slight time discrepancies between the arrival of S-and P-waves can significantly affect the distance calculation and thus the accuracy in locating an earthquake's origin.
Earthquake Epicenter Determination
Determining the epicenter of an earthquake involves analyzing the time difference between the arrival of S-and P-waves at a seismograph station.

Typically, geologists use three or more seismograph stations to triangulate the location of the epicenter. Each station's data allows for the calculation of the earthquake's distance from that station. With multiple data points, the intersection of the calculated distances narrows down the precise location of the epicenter. The precision of the recorded arrival times directly affects the accuracy of the epicenter determination. Therefore, the 0.320 km precision resulting from the 0.100 second measurement uncertainty directly influences the reliability of the epicenter's location.
Nuclear Test Ban Treaty Monitoring
Monitoring compliance with nuclear test ban treaties relies extensively on seismic wave analysis.

Underground nuclear detonations generate seismic waves similar to those produced by earthquakes, and these can be picked up by a network of international monitoring stations. The speed and accuracy with which these stations can determine the detonation's epicenter are critical for the effectiveness of the treaty enforcement. The precision issue described earlier becomes even more significant in this context, as undetected or inaccurately pinpointed nuclear tests could undermine the treaty's integrity.

Therefore, advances in seismograph precision and an improved understanding of S-and P-wave propagation are invaluable to ensure that movements toward global disarmament are honored and that any clandestine testing is detected and accurately located.

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Most popular questions from this chapter

A string is fixed at both ends to supports 3.50 m apart and has a linear mass density of \(\mu=0.005 \mathrm{kg} / \mathrm{m}\) The string is under a tension of 90.00 N. A standing wave is produced on the string with six nodes and five antinodes. What are the wave speed, wavelength, frequency, and period of the standing wave?

Consider the wave function\(y(x, t)=(3.00 \mathrm{cm}) \sin \left(0.4 \mathrm{m}^{-1} x+2.00 \mathrm{s}^{-1} t+\frac{\pi}{10}\right)\).What are the period, wavelength, speed, and initial phase shift of the wave modeled by the wave function?

In a transverse wave on a string, the motion of the string is perpendicular to the motion of the wave. If this is so, how is possible to move energy along the length of the string?

A quality assurance engineer at a frying pan company is asked to qualify a new line of nonstick-coated frying pans. The coating needs to be \(1.00 \mathrm{mm}\) thick. One method to test the thickness is for the engineer to pick a percentage of the pans manufactured, strip off the coating, and measure the thickness using a micrometer. This method is a destructive testing method. Instead, the engineer decides that every frying pan will be tested using a nondestructive method. An ultrasonic transducer is used that produces sound waves with a frequency of \(f=25 \mathrm{kHz}\). The sound waves are sent through the coating and are reflected by the interface between the coating and the metal pan, and the time is recorded. The wavelength of the ultrasonic waves in the coating is \(0.076 \mathrm{m}\). What should be the time recorded if the coating is the correct thickness \((1.00 \mathrm{mm}) ?\)

If the tension in a string were increased by a factor of four, by what factor would the wave speed of a wave on the string increase?
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