Thus from (5) the amplitude of the reflected disturbance is numerically equal to that of the incident one. To interpret (6) we note from (1) that kλ 2π, or the incident wave length is λ = 2πk. Thus, if X' is the wave length of the a reflected train it is 27, k', and (6) then shows that

=

Or in words, by reflection at the moving reflector the wave length is shortened in the ratio (v — u) (v + u).

596. The energy in the wave train is half-potential and half-kinetic, and is given by the integration of p(d/dt) along the train, p being the density. Whence the energy per unit volume is found to be inversely as x2 (see art. 145). Let the energy per unit volume in the incident train be E, and that in the reflected train be E'. Then, since in the two trains the amplitude and density are the same, we have

But in unit time the length v+u of the incident train passes up to the reflector, and a length (v — u) issues from it. Hence, if the energies per second or activities of the incident and reflected trains per unit cross-sectional area are denoted respectively by A and A', we have

A = (v+u)E and A′ = (v — u)E"

(9).

whence the increase of activity in the reflected train is

But this increase in the energy per unit time leaving the

mirror over that reaching it can only arise from work done on it to cause advance against pressure exerted by the radiation. And, just as the product of force and distance. equals energy, so the product of force and speed gives activity. But the force per unit area is the pressure to be evaluated which we will denote by P, and the speed of the mirror is u. Hence we have

or,

597. It is convenient to express this pressure in terms of the total volume density of energy just in front of the reflector. Thus, from (8) we have

Or in words, the pressure on the reflector is (v2 — u2)/(v2 + u2) times the volume density of the total energy of the incident and reflected waves. If u is very small compared with this reduces to

That is, a reflector at rest experiences a radiation pressure equal to the total energy density of radiation in front of it.

1

Larmor illustrates this argument by the transverse vibrations of a tense cord along which, to act as a reflector, a pierced board or ring is made to slide. Lord Rayleigh treats the string problem by a different method. Both show that the constraint, which acts as a reflector, is subject to a force which equals the total longitudinal density of the vibrational energy in front of it.

598. Direct View for Sound Pressure. The above indirect theorem applies, of course, to sound waves, but it seems desirable to see if possible how for any particular kind of wave motion the pressure arises. For the case of sound this was shown very simply by Prof. Poynting in the address already referred to. He said, In sound waves there is at a reflecting surface a node, a point of no motion, but of varying pressure. If the variation of pressure from the undisturbed value were exactly proportional to the displacements of a parallel layer near the surface, and if the displacements were exactly harmonic, then the average pressure would be equal to the normal undisturbed value. But consider a layer of air quite close to the surface. If it moves up a distance y towards the surface, the pressure is increased. If it moves an equal distance y away from the surface, the pressure is decreased, but by a slightly smaller quantity. To illustrate this, take an extreme case, and for simplicity suppose that Boyle's law holds. If the layer advances half-way towards the reflecting surface the pressure is doubled. If it moves an equal distance outwards from its original position the pressure falls, but only by one-third of its original value; and if we could suppose the layer to be moving harmonically, it is obvious that the mean of the increased and diminished pressures would be largely in excess of the normal value. Though we are not entitled to assume the existence of harmonic vibrations when we

1 Phil. Mag., iii. 1902, p. 338.

take into account the second order of small quantities, yet this illustration gives the right idea. The excess of pressure in the compressed half is greater than its defect during the extension half, and the net result is an average excess of pressure a quantity itself of the second order on the reflecting surface. This excess in the compression half of a wave train is connected with the extra speed which exists in that half, and makes the crests of intense sound-waves gain on the troughs."

599. Again, using Boyle's law, Lord Rayleigh has shown1 that for plane sound waves the total force is measured by the longitudinal density of total energy, and that the additional pressure is measured by the volume density of energy. If, however, the aerial vibrations are distributed equally all round, then the pressure due to them on a plane surface would coincide with one-third of the volume density of the total energy.

600. Experimental Confirmation. The pressure of sound waves has been experimentally detected and measured by W. Altberg,2 working in the laboratory of Lebedew. He employed a Kundt tube as the source of sound, and for the receiver a small wooden cylinder 21 mm. diameter. This was fixed at one end of the arm of a delicate torsion balance. One end of the cylinder passed through a hole in a plate with sufficient clearance to allow its free motion. The waves were then received on this plate and cylinder end. In some experiments the pressure rose to 0.24 dyne per sq. cm. The intensity of the sound was independently measured by a telephone plate in a manner devised by Wien. The results of the investigation showed that the pressure upon a reflecting wall is completely analogous to the pressure of light waves.

601. Pressure of Vibrating Membrane.

1 Phil. Mag., iii. 1902, p. 338.

Lord

2 Ann. d. Physik, xi. 2. pp. 405-420, May 14, 1903; Science Abstracts, No. 1471, 1904.

Rayleigh also applied his methods of reasoning to other cases. Thus, if a vibrating membrane has a flexible and extensible boundary capable of shifting along the surface, and the vibrations are equally distributed in the plane, then the force outwards per unit length of contour is onehalf the superficial density of the total energy.

602. Conception of Momentum of Radiation.—In the address already quoted Prof. Poynting also brought forward and emphasised the very valuable conception of momentum in connection with a beam of radiation. We cannot do better than again quote his own words, which run as follows:

Theory and experiment, then, justify the conclusion that when a source is pouring out waves, it is pouring out with them forward momentum as well as energy, the momentum being manifested in the reaction, the back pressure against the source, and in the forward pressure when the waves reach an opposing surface. The wave train may be regarded as a stream of momentum travelling through space. This view is most clearly brought home, perhaps, by considering a parallel train of waves which issues normally from a source for one second, travels for any length of time through space, and then falls normally on an absorbing surface for one second. During this last second momentum is given up to the absorbing surface. During the first second the same amount was given out by the source. If it is conserved in the meanwhile we must regard it as travelling with the train. Since the pressure is the momentum given out or received per second, and the pressure is equal to the energy density in the train, the momentum density is equal to the energy density÷wave velocity.”

603. Thus, if the pressure be P, the momentum per unit volume M, and the energy density E, then the total momentum reaching a stationary surface per unit area per second would be

MX/T=Mv = P.