Waves may be classified according to the direction of the vibration of the medium as compared with the direction the wave is moving. According to this mode of classification there are two types: (1) transverse waves, those in which the direction of vibration is at right angles to the direction of propagation of the wave; (2) longitudinal waves, those in which the direction of vibration is parallel to the direction of propagation. (In a few cases (waterwaves, for example) the motion is a combination of these two types.) 311. Transverse waves. Imagine a stretched cord with waves traveling along it. Fig. 181 shows what might be termed an instantaneous picture of it. Each particle of the rope is vibrating up and down with the same frequency. At a, e, and i the particles have reached the highest points of their vibration, while those at c and g have reached their lowest points. At points lying between and so on. Looking at this figure and thinking of these motions, one can see that in a short time the particles at a will have moved downward and those at b will have moved up and formed a crest, with the result that the crest will have moved to the right. In the same time c has moved up and d down, so that the trough has also moved to the right. Thus by a transverse vibratory motion of the cord the crests and troughs travel to the right. There is no motion of the cord to the right. It is an important fact in wave-motion that the medium is not carried along with the wave. The particles at a, e, and i are in the same phase (sect. 117). Those at b and f are also in one phase, but opposite in phase to those at d and h. A wave-length is the least distance between particles which are in the same phase of vibration. The distances ae, bf, cg, are wave-lengths. The form of apparatus known as Kelvin's wave-apparatus is useful for showing a transverse wave. Rods with heavy balls on their ends are clamped to a steel wire or ribbon (Fig. 182), which is hung from the ceiling. When seen at rest and "edge on," one row of balls lies in a vertical straight line. But when the balls in this row are vibrating with the same amplitude and frequency but with a progressive change of phase, a wave is traveling up or down (Fig. 183). The student should try to visualize this to-and-fro vibration and to see how the crests travel up and down. In what direction must the balls of Fig. 183 be moving in order that the wave may be traveling upward? 312. Longitudinal waves. A simple model for longitudinal waves is a long spiral spring hung from the ceiling. Small pieces of white cloth tied at frequent intervals increase the visibility of the motions. If a short portion of this spring is compressed by the hands and then suddenly released, a wave of compression will travel along the spring; or if a part is stretched and suddenly released, a wave of stretch, or extension, will travel up and down the spring. In each of these cases the motion is parallel to the length of the spring. It is therefore a longitudinal motion. → FIG. 182 FIG. 183 It is usually difficult to follow with 'the eye the motions of an actual wave. Hence it is best to observe the motions in a slowmoving mechanical model or to study a figure like Fig. 184. The top line of this figure represents an instantaneous view of the particles in a longitudinal wave traveling to the right. For convenience only a few particles are shown. Each of these particles is vibrating to and fro in a horizontal direction, the zero position for each being shown by a short vertical line. The arrows show the direction of motion of the particles. At this particular time the first and ninth particles are at a maximum displacement and are momentarily at rest. The third and eleventh particles are in "compressions," while the seventh is in a "rarefaction." From the first to the ninth or from the third to the eleventh is one wave-length. The second horizontal line shows the position of the same particles one eighth of the time of a complete vibration (or period) later. As shown in this line, the condensations and rarefactions have moved to the right during this interval of time. The third line shows the particles at a time two eighths of the period later than the top line, the fourth at a time three eighths of the period later, and so on. The last line gives the position of the par 1 2 3 4 5 6 7 8 9 10 11 12 plete wave-length to 4 + ۰۰۰۰ the right. Careful study of 5 | + |||||| + |→ 2. The particles in the wave do not vibrate together; there is a progressive change in phase from one particle to the next. 3. Most important is it to see that the to-and-fro motion of the particles produces a movement of the condensation and the rarefaction to the right. 4. In the condensation the particles are swinging to the right, in the direction the wave is traveling. 5. In the rarefaction the particles are moving to the left, opposite to the direction the wave is traveling. 6. If it be imagined that these particles are connected by spiral springs (in a real wave there is something equivalent to an elastic bond between them), each may be regarded as exerting a force on the one in front of it. In the condensations each spring will be pushing the particles in front forward; but in the rarefactions, where the springs will be stretched, they will be pulling the particles in front back. Since the particles move in the direction of these forces (see 4 and 5 above), each spring is doing work on the particle ahead of it. It is an important fact, true in all wavemotions, that there is always a transfer of energy in the direction the wave travels. 313. Transfer of energy in a wave. While there is no transfer of the medium in the direction a wave is traveling, there is always a transfer of energy in the direction of the propagation of the wave. In the preceding paragraph, attention was called to the fact that the forces and motions of the vibrating particles are always in such a direction that each particle is doing work on the one in front of it. The same thing is true of the forces and motions in transverse waves. Referring to Fig. 181, it will be seen that the segment of the cord at e is pulling upon f in the direction f is moving; hence it is doing work on the part at f. The segment at e also exerts a force on d; but d is not moving in the direction of the force, and hence does not receive energy from e. Energy is transferred only forward. Wave-motion is one of the important methods of the transfer of energy. For example, it is the only method by which the earth receives energy from the sun. 314. Water-waves. The waves along the surface of the water are a combination of the two types of waves just discussed. Each FIG. 185 particle of the water moves up and down (that is, transversely), and at the same time moves to and fro in the direction the wave is traveling. Water is flowing in the direction of the wave-motion on the crest and in the opposite direction in the trough. The resultant motion of any particle (for the case of deep water) is circular. In Fig. 185 are shown the paths of a number of particles which lie on the surface of water, and the wave-form resulting from their motions. Each of the particles is rotating in a clockwise circular motion. One can see from this figure, by imagining the position of the particles a short time later, that both the crest and the trough are moving to the right. Here, as in other cases, while the wave moves to the right, there is no general movement of the water in that direction. The speed of a surface wave in a deep liquid can be computed from the following formula: gλ 2 πς where g is the acceleration of a falling body, A the wave-length, S the surface tension, and d the density of the liquid. For water-waves longer than about 10 centimeters the first term on the right is so much larger than the second that the last may be neglected entirely; but for ripples which have a wave-length less than about 2 millimeters the first term may be neglected and only the second used. In long waves gravitational force predominates, but in short waves surface tension is the more important factor. When the depth of water is small compared with the wave-length, the motion of the particles is not in simple circular orbits, as it is in deep water. In the case of shallow water it has been found that for waves longer than about 10 centimeters the speed is independent of the wave-length but varies inversely as the square root of the depth. The formula is Speed in shallow water = √gh, (2) where g is the acceleration of gravity and h the depth. Equation (2) enables us to point out one thing which tends to make sea waves curl over and break as they travel toward a beach. Since each crest is in deeper water than the trough in front of it, it travels faster than the trough and ultimately falls over into the trough. The formula for short waves in shallow water is 2 πη ελ 2 πς Ad), (3) Speed of short waves in shallow water = where the symbols have the same meaning as in equations (1) and (2). The derivation of these equations will be found only in advanced treatises. 315. Relation between velocity, period, frequency, and wavelength. The period of vibration has been defined as the time required for a complete vibration. It is also the time required for a |