### Physics in daily life: brave ducks

L.J.F. (Jo) Hermans
2010 EurophysicsNews
EPN 41/3 29 emember how hard it was to first break the sound barrier? It took several fatal attempts by brave pilots before Charles ('Chuck') Yeager finally managed to fly faster than the speed of sound on the 14 th of October, 1947. e problem was: by the time an aircra approaches the speed of sound, the sound wave crests pile up in front of the plane. It then has to push through this barrier of compressed air in order to go faster than the waves. Once it is faster than the sound waves, an
more » ... sound waves, an interesting situation occurs, quite similar to the case of a bullet moving at supersonic speed. e wave fronts produced have an enveloping circular cone, the 'Mach cone' . It is easy to see that the half apex angle of the cone, θ, is related to the speed of sound c and the speed of the plane ν by sin θ = c/ν. Since there are no sound waves outside the Mach cone, the plane will pass us before we actually hear its sound. Sound waves bear many analogies to water waves. Look at a duck, for example, speeding through a deep pond. See the V-shaped pattern of waves trailing the swimming duck? Doesn't it look like he is fighting the 'wave barrier' of water in front of him and producing a twodimensional version of the Mach cone? Brave duck! is certainly is an appealing thought. But it's wrong. What we may perceive as a 2-D version of a 'Mach cone' actually consists of two envelopes of a feathered pattern of dispersive waves. Despite the analogies between water waves and sound waves, there are a few essential differences. Sound waves in air travel at a fixed speed without dispersion. e phase velocity c is equal for all wavelengths and equal to the group velocity. For supersonic flight this leads to the simple expression for the 'Mach angle' given above. Water waves are much more complicated. ey travel at the interface of two media, and are governed by gravity. Let us look at