Question

On December 26, 2004, a violent earthquake of magnitude 9.1 occurred off the coast of Sumatra. This quake triggered a huge tsunami (similar to a tidal wave) that killed more than 150,000 people. Scientists observing the wave on the open ocean measured the time between crests to be 1.0 h and the speed of the wave to be 800 km/h. Computer models of the evolution of this enormous wave showed that it bent around the continents and spread to all the oceans of the earth. When the wave reached the gaps between continents, it diffracted between them as through a slit. (a) What was the wavelength of this tsunami? (b) The distance between the southern tip of Africa and northern Antarctica is about 4500 km, while the distance between the southern end of Australia and Antarctica is about 3700 km. As an approximation, we can model this wave's behavior by using Fraunhofer diffraction. Find the smallest angle away from the central maximum for which the waves would cancel after going through each of these continental gaps.

Short answer

(a) 800 km; (b) 10.23° for Africa-Antarctica, 12.52° for Australia-Antarctica.

Step-by-step solution

Understand the Given Problem

We need to calculate the wavelength of the tsunami wave using its speed and the time interval between wave crests. Then we will use Fraunhofer diffraction to find the smallest angle for wave cancellation between two continental gaps.

Calculate the Wavelength of the Tsunami

To find the wavelength, use the formula: $\lambda =\frac{v}{f}$. Given the speed $v=800\text{km/h}$ and the period of the wave $T=1.0\text{h}$, and knowing that frequency $f=\frac{1}{T}$, we have:$$\lambda =\frac{800\text{km/h}}{1.0/h}=800\text{km}$$.Thus, the wavelength is $800\text{km}$.

Use Fraunhofer Diffraction for Continental Gaps

For Fraunhofer diffraction, the formula for the angle $\theta$ where destructive interference occurs is given by $a\sin \theta =m\lambda$, where $a$ is the width of the gap, $\lambda$ is the wavelength, and $m$ is the order of the minimum (for the first minimum, $m=1$).

Calculate Minimum Angle for the First Gap

For the gap between Africa and Antarctica, $a=4500\text{km}$.Using $\lambda =800\text{km}$ in the diffraction formula:$$4500\sin \theta =1\times 800$$$$\sin \theta =\frac{800}{4500}\approx 0.1778$$Thus, $\theta =\arcsin (0.1778)\approx {10.23}^{\circ }$.

Calculate Minimum Angle for the Second Gap

For the gap between Australia and Antarctica, $a=3700\text{km}$.Using $\lambda =800\text{km}$ in the diffraction formula:$$3700\sin \theta =1\times 800$$$$\sin \theta =\frac{800}{3700}\approx 0.2162$$Thus, $\theta =\arcsin (0.2162)\approx {12.52}^{\circ }$.

Key concepts

Wavelength Calculation

To determine the wavelength of a tsunami, we apply basic wave physics. The wavelength ($\lambda$) of a wave is calculated using the formula:

• $\lambda =\frac{v}{f}$

where $v$ is the speed of the wave, and $f$ is its frequency. For our tsunami, the speed $v$ is given as 800 km/h. The time between wave crests is 1.0 hour, representing the period $T$. Frequency is the reciprocal of the period:

• $f=\frac{1}{T}$

For this tsunami, $f$ equals 1/h, substituting into the formula, the wavelength is:

• $\lambda =\frac{800\text{km/h}}{1/h}=800\text{km}$

So, this wave travels with crests 800 km apart in open ocean, highlighting its enormous nature.

Fraunhofer Diffraction

Fraunhofer diffraction describes how waves behave when they encounter an obstacle, such as the continental gaps here. It's crucial for understanding how the tsunami spread through narrow passages without losing much energy. This model is typically applied when wave sources and observation points are far from the diffracting object, which is the case with vast oceanic distances. In such diffraction scenarios, destructive interference – when waves cancel each other out – occurs at specific angles. These angles ($\theta$) can be calculated using the formula:

• $a\sin \theta =m\lambda$

Here, $a$ is the width of the gap, $\lambda$ is the wavelength (800 km here), and $m$ is the order of the minimum (starting at $m=1$ for the first minimum). By solving this, you can find angles where the wave impact is least felt.

Wave Speed Calculation

The speed of a wave tells us how fast it moves across a medium, and it's an integral part of understanding tsunamis. For this problem, we calculated the speed of the wave ($v$) and were given as 800 km/h. This value helps determine other crucial wave properties, such as wavelength and frequency.Wave speed ($v$) is related to wavelength ($\lambda$) and frequency ($f$) by the fundamental wave equation:

• $v=\lambda \cdot f$

Given the massive scale of a tsunami, such a high speed plays a critical role in how far and fast a wave can travel, impacting vast regions and posing serious challenges to coastal safety.

Destructive Interference

Destructive interference is a fascinating phenomenon where two waves of similar amplitude meet, and effectively "cancel" each other out. In our context, as the tsunami wave passes through continental gaps, it diffuses and can interfere with itself destructively. For destructive interference to occur, differences in path length between two waves must be an odd multiple of half-wavelengths. In simpler terms, this means waves need to be out of phase, causing the heights of crests and troughs to match oppositely. Using the Fraunhofer diffraction formula, angles where this cancellation occurs can be found, explaining why some areas might experience reduced wave impact. Thus, even a massive, energetic tsunami can occasionally be less destructive, thanks to the principles of interference.