Refraction of sound and waves (Why tsunamis are dangerous at capes)

Otokun

At the beach campsite we went to during summer vacation, there was a sign that said, "The cape is dangerous because tsunami waves gather there and become high, so do not approach it when a tsunami warning is issued." Since it didn't explain why, everyone was wondering why the cape was dangerous.

father

Haven't I taught Oto about wave refraction yet?

Otokun

Refraction? I learned about the refraction of light by lenses in school, but is there any connection between wave refraction and why tsunamis get higher at capes?

father

Yes, it is related. So, let me explain how refraction occurs in the case of the sounds we usually speak.

Otokun

Does sound refract too?

father

That's right. In the case of sound, it's quite difficult to experience the phenomenon of refraction.
Sound travels not only through air, but also through water, as we know from experience when swimming in a pool, where we can hear sounds underwater.

Otokun

I remember Dad telling me before that sound travels several times faster in water than in air.

father

Yes. That's the point.

Otokun

Wait, so the refraction of sound and the difference in the speed of sound in air and water are related?

father

That's right. When sound enters water from the air, or when sound from water enters the air, refraction occurs, and that happens because the speed at which sound propagates through the medium is different.

Otokun

Hmm. I don't really understand.

father

I'll explain by drawing a diagram (Figure 1).

First, let's consider the case where sound enters the water surface from the air, refracts there, and propagates through the water.
Sound is a spherical wave, where the wavefront spreads out on a sphere relative to the sound source (wave source). However, if the sound source is far enough away, or if we consider only a small portion of the wavefront, the wavefront can be thought of as a plane (plane wave).
This diagram shows how a wave propagates from point A on the water surface to point B, which was on the same wavefront as A, and then to point C on the water surface. For example, if the propagation speed in water were the same as in air, the wave would travel from point A to point D without refraction. In reality, the propagation speed of sound in water is faster than in air, so the wave travels a longer distance from point B to point C. It then spreads out as a spherical wave, and because of the relationship between a circle and a tangent line, it propagates to point E as shown in the diagram.

Otokun

Is it the same for light?

father

That's right. When you put a stick in water, it often appears to bend, right? Light and sound are waves, so the same phenomenon of refraction occurs.

Otokun

Hmm. I kind of understand that refraction occurs due to differences in the speed of waves traveling through a medium. But how does wave refraction relate to tsunamis converging and becoming higher at headlands?

father

That's right. Before that, just as refraction occurs when the speed of a wave changes even within the same medium, such as air, if the temperature is very different, the speed at which a wave travels in water changes, causing refraction.

Otokun

What does it mean for the speed at which waves travel in water to change?

father

The propagation speed of tsunami waves is actually influenced by the depth to the seabed. The shallower the water, the slower the tsunami waves propagate.

Otokun

I'm starting to understand. The cape and the depth of the sea are related, aren't they?

father

Just like contour lines on land, contour lines are lines that connect points of the same depth from the water surface to the seabed. If we consider contour lines around a cape, if the contour lines are drawn to follow the tongue-shaped protrusion of the cape, as in this diagram (Figure 2), then the water depth will be shallower closer to the tip of the cape at the position of the horizontal line A. The waves will travel slower there, so refracted waves will converge.