TRANSMISSION OF SOUND It is important now to consider what takes place in an ex tended elastic medium (solid, liquid or gas) containing a source of sound. Sound waves are the inevitable result when vibratory stresses are set up by any means at any point of an elastic me dium. Such sound waves consist of alternations of condensation and rarefaction, corresponding to the successive forward and backward movements of the source. The state of compression is passed on from layer to layer of the medium, with the velocity of sound; this Loing followed in turn by a rarefaction, another compression, and so on, as long as the source continues to vibrate. The phenomena of propagation of such waves of condensation and rarefaction may be demonstrated very simply by means of a long helical spring supported at suitable intervals by thin threads. Simple harmonic longitudinal displacement of one end of such a spring results in the generation of waves which travel along the spring at a definite speed. In such waves, as in sound-waves, the displacements of individual particles are in the direction of propaga tion of the wave—the motion is consequently termed longitudinal, as distinct from transverse wave motion in which the displace ments are at right angles to the direction in which the wave travels (e.g., ripples on water, or waves travelling along a stretched string). If the condensational wave travels with a uniform velocity c cms./ sec and the source of sound has a frequency of N periods per second, it will be clear, without formal proof, that there are N condensation and N rarefactions in the distance c covered by the wave in one second. Now the distance, by which one condensation is ahead of the next, is called the wave-length X of the sound in the medium; consequently we have NX = c. When a simple harmonic vibration is transmitted through the medium the linear density of the particles, or their state of compression or expansion, is at any instant represented by a simple harmonic curve which repeats itself at regular intervals of a wave-length.
sin[27r/X.(ct— x)] (3) and = a sinn (t — x/c) (4) where n has the usual significance and is equal to 27rN. These four alternative expressions for the particle displacement in a progressive plane wave are convenient for most purposes. Comparison of these relations shows that the phases, in the case of progressive waves, may be expressed in terms of fractions of a wave length X which corresponds to a phase-angle of 27. Thus the difference of phase between the vibrations of two particles at and respectively from the origin will be 27 (x2—xi)/X. It will be evident that a system of waves travelling in the negative direction of x will be represented by the introduction of a positive instead of a negative sign inside the brackets in the above expres sions for A system of simple harmonic progressive waves may, of course, be represented graphically with the displacement plotted as ordinates and the time t as abscissae.
Velocity and Acceleration of Particles in the IV ave.—Differen tiatmg E with respect to t we obtain the particle velocity, thus Relations (5) and (6) give the particle velocity and acceleration in terms of wave velocity c and the slope and curvature of the displacement curve. Equation (6) is the differential equation which characterises wave motion. Its complete solution is =f x) + F (ct + x) (6a) which represents two independent systems of waves travelling in opposite directions with the same velocity c. This velocity is, within certain limits entirely independent of the form of the wave, being independent of wave-length X and amplitude a. Within these limits, the velocity is determined solely by the physical properties of the medium. As we shall see, these properties are density and elasticity, corresponding to the factors mass and stiffness in the case of the vibrations of a particle. As the wave travels through the medium the volume and density fluctuate locally, these fluctuations being controlled by the properties of the medium and the applied forces. The following definitions are important : Dilatation A is the ratio of the increment of volume by to the original volume thus = av/vo and v= v„(i +A). (7)Condensation s is the ratio of increment of the density ap to the original density thus s= and p= (8)since pv= and (1+s) (I +A) = I (8a) and s= —A neglecting sA as a small second order quantity. Volume or Cubic Elasticity K sometimes known as "bulk modu lus" of elasticity.