The duck's wake

This is the first translation I'm doing of one of my French science popularization blog posts.

When I was preparing my PhD thesis a few years ago, I was also doing weekly hour-long preparation sessions for groups of three students to train them for the engineering schools contests. In these sessions, each student is given a problem that he must solve on a chalkboard.

The students were often brilliant and showed a great physical sense.

One of my favorite problems was the following: "the duck's wake". The student was supposed to figure out both the questions and the answers. Great exercise if you ask me and very revealing of the student's qualities or lack thereof. Here are some of the things you could say on this subject...

Let's simplify and suppose that the duck is a dimensionless point. When it moves on the water surface, it casts waves. Let's imagine these circular waves travel at an approximately constant speed (in reality they don't, more on that later). There are two possibilities: the duck can move slower than the waves, in which case the first wave will always be ahead of all the others, or the duck is "supersonic" and the waves will form a triangular wake. The first question you can ask yourself is to determine the relation between the angle of the wake and the speed of the duck. As could be expected, the faster the duck, the sharper the angle. The angle would be 180 degrees if the duck had the exact speed of the waves: the wavefront would be a line perpendicular to the direction of the duck and it would move with it.

The triangle that the wave's enveloppe creates is really a shockwave similar to the one that a supersonic plane emits. The supersonic bang is just the shockwave. Contrary to popular belief, the bang is not emitted by the plane as it passes the speed of sound but is continually emitted as long as it travels faster than sound. The thing is that the places where the bang can be heard move at the speed of sound. What you hear when a supersonic plane flies by is thus, to summarize: silence as long as you're outside the sound cone, a bang as you enter it, and the noise of the plane after that.

The second question you could ask is the repartition of energy in the shockwave. The result of the calculation is really surprising: the energy diverges and becomes infinite at the tip of the cone. This means that you would need infinite energy to go above the speed of sound. I suppose that's one reason why people used to think it was impossible. Of course, it's not really infinite in practice, just very expensive because no plane or duck is a point.

So now we know that any object that travels faster than the waves it emits emits these waves as a shockwave that packs most of its energy in its enveloppe. This phenomenon is commonly observed for surface waves (a duck's or a boat's wake) as well as for sound (supersonic planes). Now can it be observed for lightwaves?

A priori, no physical object can travel faster than the speed of light in a vacuum, so this phenomenon looks like something that would be out of the question. Nevertheless, light doesn't always travel in a vacuum. In any medium, light travels slower than the speed of light in a vacuum. It is thus possible to travel faster than light in a medium. The shockwave that the theory predicts does exist and is a commonly observed phenomenon called Cerenkov radiation. It is this radiation that is used in some neutrino detection devices.

Neutrinos are subtle particles. They have a very low mass (it's only recently that we've discovered thay have one), they don't have an electrical charge, don't participate in the strong force and the only way they interact with anything is through improbable weak interactions (gravitation can be neglected for detection purposes, the neutrino masses being so small). Despite their being very common particles in the universe (billions of neutrinos go through us every second), as they rarely interact with ordinary matter that is made from electrons, protons and neutrons, their detection is very difficult. Some detectors use the Cerenkov effect. When a neutrino interacts inside the detector, a particle such as an electron can be emitted with a faster than light speed. This particle emits light while decelerating (any charged particle accelerating emits electromagnetic waves). The resulting shockwave is then measured by detectors, from which we can deduce the trajectory of the charged particle, which gives us the direction of the neutrino that created it.

This is how you can start from the duck's wake and end up discussing the detection of neutrinos. That's just one illustration of the extraordinary explicative power of physics...

UPDATE: comments pointed out that the angle of the wake of the duck (or of a boat, or of whatever moves fast enough) does not depend on the speed of the object, as the good Lord Kelvin showed. It is constant at approximately 39 degrees. This is because the speed of the waves depends on the wavelength in such a subtle way as to cancel the dispersion that a single wavelength wave would show. Our simplistic calculation is still perfectly valid for cases where such dispersion doesn't exist, such as Çerenkov radiation.


  • Oh my brain hurts. I never did well in physics in college :)

  • Ca y'es, c'est reparti ;-)))))

  • Mais j'ai oublié: ca fait d bien dans ce monde de brute!

  • Does anyone know the angle of the wake, what controls the angle, and who first discovered it? It's for a physics assignment, so any help is appreciated.

  • Lindsey: I don't want to patronize you or anything but if this is an assignment, that's all the more reason not to give you a full answer, you need to work it out yourself. I'll give you some clues though.

    Draw the following: one dot for the duck at instant 0, one for the duck at instant t. The duck emitted a wave front at instant 0 so that makes a circle centered on the first point. The speed of the wave front is the speed of waves at the surface, let's call this c. So the radius of the circle at instant t is c times t. At this moment, the radius of the circle the ducks emits at instant t is still zero. The distance between both points (duck at instant 0 and duck at instant t) is v times t where v is the speed of the duck.

    The wake is formed by the enveloppe of all circles so its summit is the second point (duck at t) and it must be tangent to the circle emitted at zero. So if you draw a line that's tangent to the circle and goes through the second point, you've drawn the wake.

    If you draw a line between the two points and a line from the first point to the point where the wake tangents the circle, you have drawn a triangle. This triangle has a 90 degree angle at the tangent point. You know one angle of the triangle, as well as the length of two sides. It's elementary trigonometry to find the angle of the wake.

  • The angle of the wake of a body moving steadily in deep water is always 2arcsin(1/3)

  • Right. I should have done my homework more seriously. Updating the post.

  • few days ago we had some sort of sonic bang in our area,could feel it even inside a house,it happend twice in interval of about 30-40mins,but there was no other noise before or after,which made me think it can not be a supersonic airoplane?anyone have an idea what else this could be???

  • Yup, with a supersonic bang, there should be noise after, as you are inside the cone. On the other hand, that noise is always considerably weaker than the bang itself, as the boundary of the cone packs most of the energy so it may be that there was noise afterwards but you didn't notice. Those could have been plain explosions too. Did you check the local news?

  • From the hint to Lindsey on calculating the angle of the wake: Why is the wave_speed/duck_speed = 1/3? Thanks.

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