Scientists observing the wave on the open ocean measured the time between crests

Question

Scientists observing the wave on the open ocean measured the time between crests to be 1.00h and the speed of the wave to be 800km/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.

The distance between the southern tip of Africa and northern Antarctica is about 4500km , while the distance between the southern end of Australia and Antarctica is about 3700km . 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.

Explanation / Answer

The fraunhofer equation for first minimum angle is

d*sin()=, where d is the width of the slit, the angle of the first minimum intensity, ie angle where the waves cancel themselves, and the wavelength of the wave.

Using this, we know that the period of the wave is 1.00hr (time between crest to crest or trough to trough). So we know that the frequency is 1 cycle per hour. This means that for a wave traveling at 800km/h, the wavelength would be 800km/h divided by the frequency with wave speed formula of

v=f*.

So we know the wavelength is 800km, which is indeed huge.

Now we just plug into the fraunhofer equation above, for 4500km and 3700km. We put these values in d,

For tip of Africa and northern Antartica, which has the slit distance of 4500km, the angle will be

=arcsin(/d)=10.24034 degrees

For between Antartica and Australia, the angle will be

=arcsin(800km/3700km)=12.48689 degrees.

Hope it helps!

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