8 General Remarks on Waves 



It is obvious that there exists an antinode at the reflecting wall (n = 0). 

 The next node is at a distance from the wall of one-quarter of the wave length. 



Standing waves not only have their vertical but also horizontal dis- 

 placements. However, in this kind of waves, individual water particles do 

 not move in closed orbits, but the particles return through the same points 

 of their trajectories through which they moved forward. Thus the movement 

 of the individual water particles resembles more an oscillation around a point 

 of equilibrium. The distribution of the horizontal and the vertical components 

 of motion within a standing wave is entirely different from that of progressive 

 waves. The particles in a progressive wave move upward at the front side 

 of a wave crest and downward at the rear side, whereas in a standing wave 

 their movement in the entire antinode is everywhere simultaneously upward 

 or downward. There are no vertical displacements in the nodal points, but 

 here occurs the greatest horizontal water displacements, which disappear 

 again in the antinodes. Standing waves thus have the character of a rocking 

 movement of the entire water-mass around fixed nodal lines. This readily 

 explains that in standing waves the current reverses itself everywhere 

 simultaneously, and this happens when the antinodes have the greatest 

 positive respectively negative displacements, viz. at high and low water. 

 The greatest horizontal velocity is found when the water surface passes 

 through its equilibrium. 



A progressive wave r/ — 2Acos(at — xx) can be imagined to be composed 

 of the interference of two standing waves. r) x = ^cos*.vcoso7 and 

 r] 2 = A sin xx sin at. r\ x and rj 2 are two standing waves of equal wave length 

 having a phase difference of a quarter period, and the amplitude of the two 

 systems be equal; consequently, the relation between them is that the 

 antinodes of the one are superposed upon the nodes of the other. 



It is to be remembered that, if we disregard a harmonic wave profile, 

 each progressive wave can be represented by a function of (at±xx) (equa- 

 tion 1.4); here the negative (positive) sign applies for waves progressing in 

 the positive (negative) .v-direction. For standing waves, the function contain- 

 ing the coordinate for the direction x is separated from the time function. 

 As a rule, both appear in the form of a product (1.5). The mathematical 

 treatment of wave processes is facilitated if we consider standing waves as 

 composed of progressive waves. Therefore, it is important that each train 

 of progressive waves can be represented as a superposition of a system of 

 two standing waves with a phase difference of a quarter of the period. 



(b) Free and forced waves. In every system capable of oscillation we can 

 distinguish free and forced waves. Free waves are generated by a single sudden 

 impulse and for their generation and maintenance they do not require an ex- 

 ternal force. A system with its equilibrium disturbed by a single impulse would 

 continue oscillating ad infinitum in the absence of friction. The amplitude 

 of the oscillations decreases with time according to an ^-function (damped 



