Bauhaus-Universität Weimar

which it impinged on the surface, and thence will result a system of reflected portions 
of waves which will together form a reflected wave. The direction of the reflected 
wave may be either the same or different from that of the original wave. The re¬ 
flected and incident waves have the same direction, when straight waves are produced 
in a channel, as above described, and when these waves impinge in a perpendicular 
direction on the reflecting surface, and also when circular concentric waves issuing 
from one point strike upon a surface which itself forms a concentric circle around that 
point; in the latter case the reflected waves move back agaia towards the central point 
whence they arose. 
A circular wave or undulation is reflected from a plane surface in the direction which 
a wave would have, that issued from a point at the same distance behind the surface 
as the central point of the original wrave lies in front. 
Waves passing through an opening in any solid body do not retain the form which 
they had in the opening; on the contrary, their extremities which have passed the 
margins of the opening receive a circular inflexion around these margins, in conse¬ 
quence of which the waves, after their exit, not merely extend forwards, but also in the 
lateral direction. This is the “inflexion” of waves. 
B. • Undulations of inflexion or flexion-waves in solid bodies.—The cause of the undu¬ 
lations of this kind in fluids is the force of gravity; in solid bodies it is the force of 
cohesion and elasticity. These undulations are much quicker in solid bodies than in 
water, and in elastic bodies are the cause of sound. 
If a stretched cord or string of a musical instrument is struck, not at its middle, but 
near one extremity, an extension of it is produced at this part, and is communicated as 
a wave or oscillation to the whole cord, travelling from one end to the other where it 
is reflected back again like the undulations of fluids. 
If the cord or string is struck several times in succession, a regular series of undu¬ 
lations is produced, as upon the surface of water. The reflection of these undulations 
at the extremity of the cord, and the meeting of the incident and reflected undulations, 
give rise to the stationary vibrations; and the parts of the cord wThich remain at rest 
between these stationary waves or vibrations are the “nodal points.” 
The simplest stationary vibration of a cord or string is, however, that which re¬ 
sults, not from the meeting of progressive undulations, but from the transverse vibra¬ 
tion or movement from side to side of the whole cord between its fixed extremities, 
which here constitute the nodal points. This kind of vibration is induced most readily 
by striking a string with the finger, or with a violin-bow. The transverse oscillation 
of solid bodies which do not owe their elasticity to tension, such as metal rods fixed 
at one extremity only, is also an instance of stationary vibration. 
C. Undulations of condensation in liquids, gases, and solid bodies.—The undulations 
of inflexion in water are not attended with any condensation and rarefaction, nor are 
they necessarily so in a cord or string. If the cord is not extensile, or not elastic, the 
undulations of inflexion may be produced by the mere displacement, and the striving 
of the parts of the cord to regain the straight direction. Generally, however, the 
waves of inflexion in strings are attended with alternations of condensation and rare¬ 
faction. The peculiarity of the undulations of inflexion consists in many parts of the 
body having imparted to them so considerable a movement in the direction perpen¬ 
dicular to its surface, that the form of the surface undergoes a visible change. 
Undulations excited in the air consist of progressive condensations and rarefactions. 
The part where the condensation exists is analogous to the elevation; that where the 
rarefaction is, to the depression of an undulation of inflexion. A progressive undula¬ 
tion travelling through a column of air in a tube is reflected at the extremity of the 
tube if this is closed, and its retrograde course retains its original properties. The 
wave is also imperfectly reflected even when the extremity of the tube is open; but, 
experience has shown that in that case the properties of the undulations are reversed, 
its rarefaction taking place where its condensation should be, and vice versa. Undu¬ 
lations in the open air have a spherical form. {Weber, op. cit. § 276.) 
II. Of the stationary and progressive undulations of sonorous bodies. 
The vibrations of sonorous bodies are either undulations of inflexion, or undulations 
of condensation. Either of these kinds of undulations, or both simultaneously, may 
take place in strings and solid bodies which give sound. Columns of air, in yielding 
sound, are the seat of undulations of condensation only. The undulations of sonorous 
bodies may be either stationary or progressive.


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