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Chapter 16: Problem 47

When an earthquake occurs, two types of sound waves are gen- erated and travel through the earth. The primary, or P, wave has a speed of about 8.0 km/s and the secondary, or S, wave has a speed of about 4.5 km/s. A seismograph, located some distance away, records the arrival of the P wave and then, 78 s later, records the arrival of the S wave. Assuming that the waves travel in a straight line, how far is the seismograph from the earthquake?

Short Answer

Expert verified
The seismograph is approximately 802 km away from the earthquake.

Step by step solution

01

Define the relationship for distance

Recall that distance is given by the formula: \( \text{Distance} = \text{Speed} \times \text{Time} \). We will use this formula for both P and S waves.
02

Establish the equations for P and S waves

Let \( d \) be the distance to the seismograph. For the P wave, \( d = v_p \times t_p \) where \( v_p = 8.0 \text{ km/s} \) (speed of P wave) and \( t_p \) is the travel time for the P wave. Similarly, for the S wave, \( d = v_s \times t_s \) where \( v_s = 4.5 \text{ km/s} \) (speed of S wave) and \( t_s = t_p + 78 \text{ s} \).
03

Set up an equation with the given time difference

Since \( t_s = t_p + 78 \), substitute in the S wave equation to get: \( d = 4.5(t_p + 78) \).
04

Equate the distance formulas

Set the two expressions for \( d \) equal: \[ 8.0t_p = 4.5(t_p + 78) \]. Simplify this equation to find \( t_p \).
05

Solve for t_p

Expand and simplify the equation: \[ 8.0t_p = 4.5t_p + 351 \]. Subtract \( 4.5t_p \) from both sides: \[ 3.5t_p = 351 \]. Solve for \( t_p \) by dividing both sides by 3.5: \[ t_p = \frac{351}{3.5} \approx 100.29 \text{ s} \].
06

Calculate the distance

Now that you have \( t_p \), substitute back into the P wave distance formula: \( d = 8.0 \times 100.29 \approx 802.32 \text{ km} \). Therefore, the seismograph is approximately 802 km from the earthquake.

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Key Concepts

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

Earthquake Waves
Earthquake waves are fascinating and critical components in understanding seismic events. When an earthquake occurs, it releases energy in the form of seismic waves, which travel through the Earth's layers. The two primary types of seismic waves include:
  • Primary waves (P waves) - These are longitudinal waves, meaning the particle motion is parallel to the wave direction. They are compressional and can travel through solids, liquids, and gases, making them the fastest seismic waves.
  • Secondary waves (S waves) - These are transverse waves, which move the ground perpendicular to the wave direction. S waves only travel through solids and are slower than P waves.
Both waves are crucial for geologists to understand and locate the epicenter of an earthquake, helping in predicting possible damage zones.
Seismograph
A seismograph is an essential tool for detecting and recording the motion of the Earth caused by seismic waves. These instruments are crucial in seismology and are generally composed of:
  • A mass suspended on a spring inside a rigid frame securely fixed to the ground.
  • When the ground shakes, the frame moves while the mass remains stationary due to inertia.
This relative motion between the frame and the mass is then recorded as a seismogram. Seismographs help to identify not only the time but also the intensity and duration of seismic waves, assisting scientists in analyzing earthquake characteristics and distances.
Physics Problem Solving
Physics problem solving often involves breaking down complex phenomena into simpler parts, using mathematical tools and models. To solve the problem of determining the distance from an earthquake to a seismograph:
  • Use carefully defined formulas, like distance calculation where distance equals speed multiplied by time: \( d = v \times t \).
  • Establish equations for each type of wave based on their speeds and recorded time differences.
By isolating and manipulating variables, we arrive at solutions that predict or explain physical scenarios with precision.
Wave Speed Calculation
The calculation of wave speed is a fundamental concept in physics and is particularly relevant in seismology to determine how far a seismic event is from detection points. When calculating wave speeds for P and S waves:
  • Recognize given speeds: 8.0 km/s for P waves and 4.5 km/s for S waves.
  • Understand that the different speeds result in varied arrival times, crucial for measuring distance.
  • Apply the distance formula for each wave type and use the time difference to set equations that can be solved for time and, eventually, distance.
These calculations are crucial for interpreting seismic data and enhancing our response to and understanding of earthquakes.

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Most popular questions from this chapter

A woman is standing in the ocean, and she notices that after a wave crest passes, five more crests pass in a time of 50.0 s. The distance between two successive crests is 32 m. Determine, if possible, the wave鈥檚 (a) period, (b) frequency, (c) wavelength, (d) speed, and (e) amplitude. If it is not possible to determine any of these quantities, Then so state.

You are riding your bicycle directly away from a stationary source of sound and hear a frequency that is 1.0% lower than the emitted frequency. The speed of sound is 343 m/s. What is your speed?

To navigate, a porpoise emits a sound wave that has a wavelength of 1.5 cm. The speed at which the wave travels in seawater is 1522 m/s. Find the period of the wave.

A typical adult ear has a surface area of \(2.1 \times 10^{-3} \mathrm{m}^{2}\) The sound intensity during a normal conversation is about 3.2 \(\times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\) at the listener's ear. Assume that the sound strikes the surface of the ear perpendicularly. How much power is intercepted by the ear?

A source of sound is located at the center of two concentric spheres, parts of which are shown in the drawing. The source emits sound uniformly in all directions. On the spheres are drawn three small patches that may or may not have equal areas. However, the same sound power passes through each patch. The source produces 2.3 W of sound power, and the radii of the concentric spheres are \(r_{\mathrm{A}}=0.60\) m and \(r_{\mathrm{B}}=0.80 \mathrm{m} .\) (a) Determine the sound intensity at each of the three patches. \((\mathrm{b})\) The sound power that passes through each of the patches is \(1.8 \times 10^{-3} \mathrm{W} .\) Find the area of each patch.
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