between the atoms, because if there is supposed to be a wave in which one point goes up and the next down, etc, the shortest possible wavelength is clearly the atom spacing. In terms of the modes of oscillation, we say that there are longi tudinal modes, and transverse modes, long wave modes, short wave modes. As we consider wavelengths comparable to the spacing between the atoms, then the speeds are no longer constant; there is a dispersion effect where the velocity is not independent of the wave number. But, ultimately, the highest mode of transverse vaves would be that in which every atom is doing the opposite of neighboring Now from the point of view of atoms, the situation is like the two pendulums that we were talking about, for which there are two modes, one in which they both go together, and the other in which they go apart. It is possible to analyze the olid waves another way, in terms of a system of coupled harmonic oscillators, like an enormous number of pendulums, with the highest mode such that they oscillate oppositely, and lower modes with different relationships of the timing The shortest wavelengths are so short that they are not usually available technically. However they are of great interest because, in the theory of thermo- dynamics of a solid, the heat properties of a solid, for example specific heats, can be analyzed in terms of the properties of the short sound waves. Going to extreme of sound waves of ever shorter wavelength, one necessarily comes to the individual motions of the atoms; the two things are the same ultimately a very interesting example of sound waves in a solid, both longitudinal and transverse, are the waves that are in the solid earth, who makes the noises we do not know, but inside the earth, from time to time, there are earthquakes-some ck slides past some other rock. That is like a little noise So waves like sound waves start out from such a source very much longer in wavelength than one usu- ally considers in sound waves, but still they are sound waves, and they travel around in the earth. The earth is not homogeneous, however, and the properties of pressure, density, compressibility, and so on, change with depth, and therefore the speed varies with depth. Then the waves do not travel in straight lines---there is a kind of index of refraction and they go in curves. The longitudinal waves and the mm数 way the thing jiggles after there has been an earthquake somewhere else, then we do not just get an irregular jiggling. We might get a jiggling, and a quieting down, and then another jiggling-what happens depends upon the location. If it were close enough, we would first receive longitudinal waves from the disturbance, and then, a few moments later, transverse waves, because they travel more slowly. By measuring the time difference between the two, we can tell how far away the earth- quake is, if we know enough about the speeds and composition of the interior egions involved ANSVLA An example of the behavior pattern of waves in the earth is shown in Fig. 51-6 The two kinds of waves are represented by different symbols. If there were an earth Fig. 51-6. Schematic of the earth quake at the place marked"source, "the transverse waves and longitudinal waves longi would arrive at difTerent times at the station by the most direct routes, and there verse sound waves would also be reflections at discontinuities, resulting in other paths and times. It turns out that there is a core in the earth which does not carry transverse waves. If the station is opposite the source, transverse waves still arrive, but the timing not right. What happens is that the transverse wave comes to the core, and when er the transverse waves come to a surface which is oblique between two materials, two new waves are generated, one transverse and one longitudinal. But inside the core of the earth, a transverse wave is not propagated (or at least, there is no evi- dence for it, only for a longitudinal wave); it comes out again in both forms and comes to the station It is from the behavior of these earthquake waves that it has been de that transverse waves cannot be propagated within the inner circle. This means that the center of the earth is liquid in the sense that it cannot propagate transverse waves.The only way we know what is inside the earth is by studying earthquakes So, by using a large number of observations of many earthquakes at different