3 Wave Physics - exercise sheet 3 he third sheet of exercises addresses sinusoidal and complex exponential solutions to wave equations – and relies, of course, on the mathematical tools practiced in previous weeks. Question 5 is, for this week, optional, but it is a further test of your familiarity with the mathematics of waves, and an intriguing example of more advanced forms of practically-occurring waves. Reading Read about sinusoidal For example Freegarde Pain French Main Coulson & Jeffries waves, standing waves and complex waveforms in your favourite textbook. Introduction to the Physics of Waves chapters 3, 4 The Physics of Vibrations and Waves (6th ed.) pp.112-5, 124-132, 190-1 Vibrations and Waves pp.161-167 Vibrations and Waves in Physics (3rd ed.) pp. 141-148, 156-161 Waves (2nd ed.) pp. 7-10, 16-18, 35-37 1 Waves on a cello string (5 marks) -1 Waves on the lowest (C) string on a correctly tuned ‘cello propagate with a speed of 90 m s . If the string has a mass per unit length of 0.014 kg m-1, find the tension to which it must be subjected. Waves on the highest (A) string of the ‘cello must travel at about 3.4 times the speed of waves on the C string. If the string tensions are approximately equal, find the diameter of the A string if it is made of solid steel with a density of 7850 kg m-3. 2 Waves on a guitar string (5 marks) A taut string, with a mass per unit length M and subject to a tension W, supports transverse waves described by their instantaneous local displacement ψ(x,t). Show that the wave equation  2 W  2  M x 2 t 2 (3.1) can be solved by arbitrary, sinusoidal, and complex-exponential travelling waves,  x, t   f x  vt (3.2)  x, t   f 0 coskx  t    (3.3)  x, t   a0 exp ikx  t  (3.4) In each case, find the constraints upon (i.e., any relationships between) the parameters v, k and . Explain briefly how the different forms of solution are related, and how these travelling waves can be used to construct standing wave solutions. 3 More waves on a guitar string (5 marks) State the boundary condition which must be met at a point where the string of question 2 is fixed. Hence find the real standing wave solutions to the wave equation, and determine the allowed oscillation frequencies, when such a string of length L is fixed at its ends. If the bottom string of a guitar has a mass of 5.4 g m-1 and its length is determined by the distance 0.648 m from the bridge to the nut, find the tension required to tune the string to the note known as E2 (a frequency of 82.4 Hz). How far along the neck must the first fret be, to give a frequency of 87.3 Hz? /continued… PHYS2023 Wave Physics 12/10/2014 2 Exercises sheet 3 4 Dispersion (5 marks) Show that the wave equation im   2  2 t x (3.5) (where i2 = -1 and m is a constant) has complex exponential travelling wave solutions of the form  x, t   f 0 exp ikx  t  (3.6) and determine the dispersion relation between the parameters k and . A travelling wave has two components of the form given in equation (3.6), equal in magnitude, frequencies 0   and wavenumbers k0  k. Show that the wave may be written in the form  x, t   f1 exp ik 0 x  0 t cosk x   t  with (3.7) and thus takes the form of a complex exponential travelling wave that is modulated by a slowly-varying, real periodic function. Show that the velocities of these two functions – i.e. expi(k0x-0t) and cos(kx-t) – differ by a factor of two. 5 Solitons (advanced and optional) The tidal bore that flows most notably along the River Severn is a wave pulse whose shape changes little as it propagates over 30 km upstream1, and is only possible because the river waves show both dispersion and nonlinearity. The phenomenon, famously described by after being observed on the Union Canal near Edinburgh by John Scott Russell2 in 1844, is nowadays categorized a form of soliton; its optical equivalents are of great topical interest to communications engineers trying to achieve the fastest data rates down optical fibres. For systems – such as the River Severn – that show soliton behaviour, the important terms in the wave equation comprise (with appropriate scaling) the Korteweg-de-Vries equation3    3  6  0 t  x  x3 (3.8) 1. Show that, if two waveforms ψ1(x,t) and ψ 2(x,t) are individually solutions to equation (3.8), then an arbitrary superposition ψ(x,t) = a ψ1(x,t) + b ψ2(x,t) will not in general be a solution – i.e., the Korteweg-de-Vries equation is non-linear. 2. Show that travelling wave pulse (with propagation speed c)  x, t   2 2 sech 2  x  ct  (3.9) may, however, be a solution to the equation – but only for a particular value of the parameter .  Find how the parameter  depends upon the propagation speed of the wave pulse c.  Sketch the waveform described by equation (3.9) at time t = 0. 1 See, for example, http://www.severn-bore.co.uk/ See, for example, http://www.macs.hw.ac.uk/~chris/scott_russell.html 3 D. J. Korteweg, and F. de Vries, "On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves", Philosophical Magazine 39 422-443 (1895) 2 12/10/2014 PHYS2023 Wave Physics