In this section we shall introduce several of the concepts and quantities needed in the book. Of course, many readers will already be quite familiar with these.
We consider an extended medium or field (in one, two or three dimensions) which is in equilibrium. This could be, for example, a straight stretched wire, the smooth horizontal surface of a pond, air at uniform pressure, or an electric field defined by some boundary configuration. A disturbance is somehow created at a point, for example by throwing a stone into the pond. This disturbance propagates outwards from its point of origin in a way which is determined by the dynamics of how the medium or field responds to changes. In the case of the pond, the disturbance propagates as a growing group of circles around the source (figure 2.4). Although the circles appear to move outwards, the water surface is disturbed locally but there is no outward flow, as you can see by watching a bit of debris floating on the surface, which just bobs up and down as the wave passes it. Just the same, the wave motion transfers energy and momentum from the source.
The waves on the pond can be used to illustrate several other wave concepts. The stone hitting the water surface excites a group of waves, within which the surface profile is approximately sinusoidal. The line following a particular maximum of the wave (a growing circle in this case) is called a wavefront. The wavefront has a velocity called the wave or phase velocity; this can be deduced by comparing figure 2.4(a) and (b) and measuring the distance that the wavefront has traveled in the time between the photographs. It also has a group velocity which is the distance traveled in unit time by the envelope of the group of waves excited by the stone. The group and wave velocities may not be equal; in the case of water surface waves they are not, but for electromagnetic waves in free space the two velocities are equal.
Within the group of waves one can determine the wavelength, X, which is the distance between adjacent wavefrontst. The amplitude, A, is the distance by which the surface at height h deviates from the equilibrium (flat) surface at h 0. For a simple sinusoidal wave the up and down amplitudes have the same value. Notice that the amplitude of the wave gets smaller as the wave propagates outwards; it is not a constant of the motion. If we concentrate on the movement of the bit of debris on the surface at a certain point, we can determine the wave's frequency, f, which is the number of oscillations per second, and its phase, 4>. To measure the phase, we need to define a zero of time (which is arbitrary, but must be the same for all related measurements) and then to express the sinusoidal motion of the surface at that point as
Then a more general definition of a wavefront is a line or surface along which the argument of the cosine is constant (for example zero, where the wave has its maximum value A). Now clearly at a given moment (e.g. one of the snapshots in figure 2.4) the wave repeats itself after distance X in the propagation direction (x). This corresponds to a change of 2n in phase, so that a more general description of the wave is
We can relate the values of x and t on a given wavefront, for which the phase of (2.2) is a constant, by ft — x/X = const., leading to phase or wave velocity v = f X.
t Actually, in figure 2.4 you can see that the wave group is quite complicated, with a longer wavelength at its leading edge and a shorter one at the tail. But this is a property of water waves and is not generally true.
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