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Tsunami! On December \(26,2004,\) a great earthquake occurred off the coast of Sumatra and triggered immense waves (tsunami) that killed some \(200,000\) people. Satellites observing these waves from space measured 800 \(\mathrm{km}\) from one wave crest to the next and a period between waves of 1.0 hour. What was the speed of these waves in \(\mathrm{m} / \mathrm{s}\) and in \(\mathrm{km} / \mathrm{h}\) ? Does your answer help you understand why the waves caused such devastation?

Short Answer

Expert verified
The wave speed was approximately 222.22 m/s or 800 km/h. The high speed explains the devastation.

Step by step solution

01

Identify Given Values

Start by identifying the given values in the problem, which are:- Distance between wave crests (wavelength) = 800 km- Time period between waves = 1.0 hour. We need to express the distance in meters for the \(m/s\) speed calculation. Since 1 km = 1000 meters, this is 800,000 meters.
02

Convert Period to Seconds

To find the speed in meters per second, we need to convert the time period from hours to seconds. Since 1 hour = 3600 seconds:\[1 \text{ hour} = 3600 \text{ seconds}\]
03

Calculate Speed in m/s

Use the formula for speed \( v = \frac{\text{distance}}{\text{time}} \). The distance between wave crests is the wavelength \(\lambda = 800,000 \text{ meters}\), and the time period \( T = 3600 \text{ seconds}\):\[v = \frac{800,000}{3600}\]Carry out the division to find the speed in meters per second.
04

Simplify the Calculation

Perform the calculation:\[v = \frac{800,000}{3600} \approx 222.22 \text{ m/s}\]
05

Calculate Speed in km/h

Convert the speed from meters per second to kilometers per hour. Since 1 m/s is equal to 3.6 km/h:\[222.22 \text{ m/s} \times 3.6 = 800 \text{ km/h}\]
06

Conclusion: Understanding the Devastation

These immense waves traveled at speeds of up to 800 km/h. This high speed, combined with their large size (wavelength), contributed to the immense damage they caused when they struck coastal areas.

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Key Concepts

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

Tsunami
A tsunami is a series of ocean waves with extremely long wavelengths and periods, typically caused by underwater earthquakes, volcanic eruptions, or landslides. These waves travel across the ocean at high speeds and can cause significant destruction upon reaching land. When the 2004 Indian Ocean earthquake struck, it generated a tsunami that devastated coastlines across more than a dozen countries.
The energy from the earthquake transferred to the water, sending waves radiating across the ocean. Tsunamis can travel vast distances with minimal energy loss, making them particularly dangerous for faraway shores. The lethal impact becomes evident when these waves reach shallower coastal waters, where they slow down but increase dramatically in height. This powerful force can lead to serious flooding and destruction.
Wavelength
In the context of waves, wavelength (\( \lambda \)) refers to the distance between consecutive crests or troughs of a wave. It is a crucial parameter for understanding wave behavior.
For the 2004 tsunami, the measured wavelength from satellite data was 800 km. Such immense wavelengths are typical for tsunamis and help differentiate them from ordinary ocean waves. Because they have long wavelengths, tsunamis behave more like rapidly rising tides than traditional breaking waves, making them difficult to detect in deep water without specialized equipment.
  • Long wavelengths can carry massive energy over extensive distances.
  • The longer the wavelength, the more significant the potential impact on coastal areas when the wave makes landfall.
The combination of large wavelength and high speed makes a tsunami a potent natural force.
Seismic Waves
Seismic waves are energy waves generated by the sudden breaking of rock within the earth or an explosion. This energy travels through the earth and is the main cause of earthquakes and tsunamis.
When a tectonic plate shifts beneath the ocean, it displaces a massive volume of water, generating seismic sea waves, or tsunamis. These waves radiate outward from the quake’s epicenter, traveling through the ocean at remarkable speeds.
Understanding seismic wave patterns and speeds helps scientists predict the potential occurrence and impact of tsunamis. This knowledge is crucial for early warning systems, enabling people in vulnerable coastal areas to evacuate in time. By studying past seismic events, like the 2004 earthquake, we can better anticipate future risks.
Speed Calculation
Calculating the speed of waves is essential to understand their potential impact. The formula for speed (\( v \)) is:\[ v = \frac{\text{distance}}{\text{time}} \]Applying this to the tsunami, where the wavelength is given as 800 km and the period between waves as 1 hour, means first converting these units to meters and seconds, respectively:
  • Wavelength: 800 km = 800,000 meters
  • Time Period: 1 hour = 3600 seconds
Plug these values into the formula:\[ v = \frac{800,000}{3600} \approx 222.22 \text{ m/s} \]This speed can also be converted to kilometers per hour (km/h) by multiplying by 3.6, yielding 800 km/h. Such high-speed capabilities help explain the vast distances these waves travel and the destruction they cause upon landfall.
Understanding wave speed helps in modeling potential impact zones, contributing to more effective emergency management and response strategies.

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Most popular questions from this chapter

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is \(y(x, t)=\) 2.30 \(\mathrm{mm} \cos [(6.98 \mathrm{rad} / \mathrm{m}) x+(742 \mathrm{rad} / \mathrm{s}) t] .\) Being more practical, you measure the rope to have a length of 1.35 \(\mathrm{m}\) and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

A piano tuner stretches a steel piano wire with a tension of 800 \(\mathrm{N}\) . The steel wire is 0.400 \(\mathrm{m}\) long and has a mass of 3.00 \(\mathrm{g}\) . (a) What is the frequency of its fundamental mode of vibration? (b) What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to \(10,000 \mathrm{Hz} ?\)

Guitar String. One of the 63.5 -cm-long strings of an ordinary guitar is tuned to produce the note \(B_{3}(\) freguency 245 \(\mathrm{Hz})\) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by \(1.0 \%,\) what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is \(344 \mathrm{m} / \mathrm{s},\) find the frequency and wavelength of the sound wave produced in the air by the vibration of the \(\mathrm{B}_{3}\) string. How do these compare to the frequency and wavelength of the standing wave on the string?

A transverse sine wave with an amplitude of 2.50 \(\mathrm{mm}\) and a wavelength of 1.80 \(\mathrm{m}\) travels from left to right along a long, horizontal, stretched string with a speed of 36.0 \(\mathrm{m} / \mathrm{s} .\) Take the origin at the left end of the undisturbed string. At time \(t=0\) the left end of the string has its maximum upward displacement. (a) What are the frequency, angular frequency, and wave number of the wave? (b) What is the function \(y(x, t)\) that describes the wave? (c) What is \(y(t)\) for a particle at the left end of the string? (d) What is \(y(t)\) for a particle 1.35 \(\mathrm{m}\) to the right of the origin? (e) What is the maximum magnitude of transverse velocity of any particle of the string? (f) Find the transverse displacement and the transverss velocity of a particle 1.35 \(\mathrm{m}\) to the right of the origin at time \(t=0.0625 \mathrm{s}\)

CALC A guitar string is vibrating in its fundamental mode, with nodes at each end. The length of the segment of the string that is free to vibrate is 0.386 \(\mathrm{m} .\) The maximum transverse accelera- tion of a point at the middle of the segment is \(8.40 \times 10^{3} \mathrm{m} / \mathrm{s}^{2}\) and the maximum transverse velocity is 3.80 \(\mathrm{m} / \mathrm{s}\) . (a) What is the amplitude of this standing wave? (b) What is the wave speed for the transverse traveling waves on this string?

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