Problem 3 - University Physics with Modern Physics

Question

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 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 m/s and in km/h? Does your answer help you understand why the waves caused such devastation?

Step-by-Step Solution

Answer

The tsunami waves traveled at 222.22 m/s or 800 km/h, explaining their destructive power.

Step 1: Identify Key Variables

We need to identify the key variables that we will use to solve for wave speed. The problem gives us the wavelength \( \lambda \) of 800 km and the period \( T \) of 1.0 hour.

Step 2: Convert Units

To find speed in m/s, first convert the wavelength from kilometers to meters: \( 800 \text{ km} = 800,000 \text{ m} \). Also, convert the period from hours to seconds: \( 1.0 \text{ hour} = 3600 \text{ seconds} \).

Step 3: Use the Wave Speed Formula

The formula for wave speed \( v \) is \( v = \frac{\lambda}{T} \), where \( \lambda \) is the wavelength and \( T \) is the period. Substitute the converted values: \[ v = \frac{800,000 \text{ m}}{3600 \text{ s}} \].

Step 4: Calculate Wave Speed in m/s

Now calculate the speed: \( v = \frac{800,000}{3600} \approx 222.22 \text{ m/s} \).

Step 5: Convert Wave Speed to km/h

Convert the speed from m/s to km/h by multiplying by the conversion factor 3.6 (since 1 m/s is equal to 3.6 km/h). Thus, \( 222.22 \text{ m/s} \times 3.6 = 800 \text{ km/h} \).

Step 6: Interpret the Results

The wave speed is very high at 222.22 m/s (or 800 km/h), indicating why the tsunami caused such massive devastation, as it delivered a large amount of energy over a vast area quickly.

Key Concepts

Tsunami Waves

Tsunamis are huge ocean waves typically generated by underwater earthquakes, volcanic eruptions, or landslides. These waves are known for their incredible speed and energy, traversing entire ocean basins with minimal energy loss. The 2004 Sumatra earthquake-induced tsunami is an example of their destructive power. Despite their immense size, while traveling in the deep ocean, tsunami waves may only be a meter high. However, as they approach shallow coastal areas, their speed decreases, and the waves grow significantly taller, leading to devastating impacts on coastlines.
Understanding the mechanics of tsunami waves is crucial in improving warning systems and minimizing the damage in future events.

Unit Conversion

Unit conversion is an essential aspect of solving physics problems involving real-world scenarios like tsunami wave calculations. In our exercise, we begin by converting the wavelength from kilometers to meters, given that scientific and engineering calculations typically use SI units. Converting kilometers to meters is simple: multiply by 1,000, as 1 kilometer equals 1,000 meters.

• Wavelength: 800 km = 800,000 m

Next, convert the period from hours to seconds. There are 3,600 seconds in an hour, so multiply the number of hours by 3,600 to convert.

• Period: 1 hour = 3,600 seconds

Such conversions are pivotal because they ensure consistency in units, allowing for accurate calculations.

Wavelength and Period

Wavelength and period are fundamental characteristics of waves. Wavelength, denoted as \( \lambda \), refers to the distance between consecutive wave crests. The exercise describes the wavelength as 800 km. The period, \( T \), is the duration it takes for one complete wave to pass a specific point, given as 1 hour here.
The relationship between these properties and wave speed is encapsulated in the formula: \[ v = \frac{\lambda}{T} \]This formula tells us that speed is directly proportional to wavelength and inversely proportional to the period. Understanding these elements not only aids in computing wave speed but also in grasping how waves behave as they travel across different mediums.

Energy of Waves

The energy carried by waves, particularly tsunamis, is vast and impactful. It's this energy that causes significant changes to landscapes and affects human settlements upon hitting coastlines. The energy of a wave is generally a function of its amplitude, speed, and the density of the water it travels through. Tsunamis, even though initially generated with small amplitudes in deep waters, transfer enormous amounts of energy over long distances.
Once a tsunami reaches shallower regions, its speed decreases, and wave height increases, concentrating the energy and causing large-scale destruction. This understanding helps in developing better protective measures against future tsunami events, emphasizing the importance of wave properties and energy in assessments.