HCI 09 Waves Solutions (Lecture Examples)

Wave Lecture Examples Suggested Solution Example 9.3.1: Consider the snapshot of the wave shown below. Other than the distance between the 2 successive crests or troughs, (i) identify another point (B') that pairs with B that will correspond to a single wavelength. (ii) Choose another pair of points on the wave that correspond to one wavelength. (iii) What do you notice about the pairs of points that represent one wavelength? Each pair of points not only have the same distance but also the same displacement from the equilibrium position. They are said to be in the same state of oscillation or in phase. Direction of wave travel  B B’  A A’

Example 9.3.2 : Transverse wave The diagram below shows an instantaneous position of a string as a transverse progressive wave travels along it from left to right. Fig. 9.3.8. (a) Taking upwards to be positive, for each of the points 1, 2, and 3 on the string, (i) state whether the displacement of each point is positive or negative. 1: positive 2: negative 3: neither (zero displacement) (ii) What are the directions of the velocities of the points? By drawing the wave when it is at the next instance (refer to diagram above), you can deduce that: 1: downwards 3: downwards For 2, its velocity is zero since it is at maximum displacement. (iii) Of the three points, which one has the largest speed? 3 has the highest speed since it is at the equilibrium position. (b) Mark on the string a point A with zero velocity. Refer to diagram above. (c) What is the direction of acceleration of point A? The direction of the acceleration is downwards, towards the equilibrium position. (Recall SHM equation a = -2x)    1 2 3 A 

Example 9.3.3: Longitudinal Wave Fig. 9.3.11 shows a 'snapshot' of a horizontal slinky spring when it is (a) at rest and (b) carrying a longitudinal wave moving from left to right. There is no energy dissipation along the slinky spring. Fig. 9.3.11 State and explain whether each of the statement is true or false. Statement A: The distance between X and Y is one wavelength. False. XY is distance between centre of compression C and rarefaction R. Hence XY corresponds to 1/2 . ( XZ corresponds to 1) Statement B: The amplitude of oscillation of Y is greater than the amplitude of oscillation of X. False. Since no energy is dissipated, amplitude of oscillation for all points is the same for a 1-D progressive wave. Hence amplitude of X = amplitude of Y. Statement C: The displacement of X at this instant is zero. True. X is at its equilibrium position. Hence its displacement at this instant is zero. (same at Y, Z) Example 9.3.4: Speed of a Wave A fisherman notices that wave crests passes the bow of his anchored boat every 6.0 s. He measures the distance between the two crests to be 15.0 m. How fast are the waves travelling? Time taken for a crest