Problem 3 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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:\[1 \text{ hour} = 3600 \text{ seconds}\]

Step 3: 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.

Step 4: Simplify the Calculation

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

Step 5: 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:\[222.22 \text{ m/s} \times 3.6 = 800 \text{ km/h}\]

Step 6: 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.

Key Concepts

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.