- 1 (210.23 KiB) Viewed 35 times
n-1 n k k 0000000 n+1 к k 00000000 m 00000000 00000000 m 1 1 a a The above diagram shows a section of an infinite chain of identical particles each of mass m. Each particle is connected to its neighbours by springs, of spring constant k. When there is no motion of the particles the distance between adjacent particles is a. Question 1. When the particles are in motion show that there will be a restoring force on the nth particle, given by m dau = k(Un-1 +Um +1 - 20,) dt2 where Un is the displacement of the nth particle and U <<a. Question 2 A solution of the above equation is that of a travelling wave, given by Un = Ae-:{wi-gna) Where w is the angular frequency of the wave and q is its wave vector. Show that the relationship between w and q is 4k 90 w sin (1) m Question 3 Draw a curve of W vs 4 (i.e. the dispersion curve). Question 4. Determine the group velocity of the travelling wave from equation 1 and draw a curve of the velocity of the travelling wave vs 4. Question 5. Compare the results of Questions 3 and 4 with that for a continuous string: