By Coulson, C.A. & Jeffrey, A.
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Extra resources for Waves. A mathematical approach to the common types of wave motion
In each case we shall find that an incident wave gives rise to a and a wave. 2 Consider first, then, the case of two semi-infinite strings and joined at the origin as in Fig. 6. Let the densities peru nit length of the two strings be and Denote the displacements in the two strings by discontinuity reflected 1 transmitted P1 P2 · 0 X Fig. 6 Y1 Y2 · and Let u s su ppose that a train of harmonic waves is incident from the negative direction of When these waves meet the change of string material, they will su ffer partial reflection and partial transmis sion.
3. Find the reflection coefficient for two strings which are joined together and whose line densities are 2·5 kg m- 1 and 0·9 kg m- 1 • 4. An infinite string lies along the x axis. At t = 0 that part of it between x = ±a is given a transverse velocity a 2 - x 2 • Describe, with the help of equation (9), the subsequent motion of the string, the speed of wave motion being c. 5. Investigate the same problem as in question (4) except that the string is finite and of length 2a, fastened at the points x = ±a.
The same resu lt applies to the localised progressive wave y = g(x + ct), moving to the left with speed c, bu t it does not, in general, apply to the stationary type waves y = f(x -ct) + g (x +ct). V We can now decide whether ou r initial assu mption is correct, that the tension remains effectively constant. If the string is elastic, the change in tension will be proportional to the change in length. :) 2 8x. ay is of the first order of small qu antities, the ax change of tension is of the second order, and may safely be neglected.