pushing from the left. On the other hand, there is pressure in the water on the right also, exerting an opposite force on the region in question, which is, by the same kind of analysis, ipghi. Now we must balance the forces against the rate of change of the momentum. Thus we have to figure out how much more momentu there is in situation(b)in Fig. 51-5 than there was in(a). We see that the additional ed v is just ph2u△t-ph2△t( multiplying this by v gives the additional momentum to be equated to the impulse ph2b△)=(lgh2-bpgh3) If we eliminate v from this equation by substituting uh2= u(h2 -h,), already found, and simplify, we get finally that u2=gh2h1 + h2)/2h1 If the height difference is very small, so that hi and h2 are nearly equal, this says that the velocity =vgh. As we will see later, that is only true provided the wavelength of the wave is longer than the depth of the channel We could also do the analogous thing for sound waves-including the con servation of internal energy, not the conservation of entropy, because the shock is irreversible. In fact, if one checks the conservation of energy in the bore problem, one finds that energy is not conserved. If the height difference is small, it is almost perfectly conserved, but as soon as the height difference becomes very appreciable there is a net loss of energy. This is manifested as the falling water and the churning wn in Fig. 51 In shock waves there is a corresponding apparent loss of energy, from the point of view of adiabatic reactions. The energy in the sound wave, behind the shock, goes into heating of the gas after shock passes, corresponding to churning of ater in the bore In working for the out to be necessary for solution, and the temperature behind the shock is not the same as the temperature in front, as we have try to make a bore that is upside down(h2 hi), then we find that the energy loss per second is negative. Since energy is not available from anywhere, that bore cannot then maintain itself it is unstable. If e to start a wave of that sort, it would flatten out, because the speed dependence on height that resulted ng in the case we di 51-3 Waves in solids The next kind of waves to be discussed are the more complicated waves in direct analog to a sound wave in a solid. If a sudden push is applied to a solid, it is compressed. It resists the compression, and a wave analogous to sound is started However there is another kind of wave that is possible in a solid, and which is not possible in a fluid. If a solid is distorted by pushing it sideways(called shearing), then it tries to pull itself back. That is by definition what distinguishes a solid from a liquid: if we distort a liquid (internally), hold it a minute so that it calms down and then let go, it will stay that way, but if we take a solid and push it, like shearing a piece of"Jello, and let it go, it flies back and starts a shear wave, travelling in the same way the compressions travel. In all cases, the shear wave speed is less than the speed of longitudinal waves. The shear waves are somewhat more anal- ogous, so far as their polarizations are concerned, to light waves. Sound has no polarization, it is just a pressure wave. Light has a characteristic orientation per- In a solid, the waves are of both kinds. First, there is a compl n wave analogous to sound, that runs at one speed. If the solid is not crystalline, then a shear wave polarized in any direction will propagate at a characteristic speed (Of course all solids are crystalline, but if we use a block made up of microcrystals of all orientations, the crystal anisotropies average out. Another interesting question concerning sound waves is the following: What happens if the wavelength in a solid gets shorter, and shorter, and shorter? How short can it get? It is interesting that it cannot get any shorter than the space