LECTURE 9 STANDING WAVES ON A STRING Instructor: Kazumi Tolich Lecture 9 2 ¨ 16.1 The principle of superposition ¤ Constructive and destructive interference ¨ 16.2 Standing waves ¤ Superposition creates a standing wave ¤ Nodes and antinodes ¨ 16.3 Standing waves on a string ¤ Reflections ¤ Creating a standing wave ¤ The fundamental and the higher harmonics 16.1 The principle of superposition ¨ The principle of superposition states that waves of small amplitude traveling through the same medium combine, or superpose, by simple addition. 16.1 Constructive and destructive interference 4 ¨ If two pulses combine to give a larger pulse, this is constructive interference (a). If they combine to give a smaller pulse, this is destructive interference (b). ¨ During destructive interference, the energy of the wave is in the form of kinetic energy of the medium. Quiz: 16.1-1 ¨ Two identical waves with the same phase are added. Which of the following statements is/are correct about the resultant wave? Choose all that apply. A. Its frequency is the same as that of the original waves. B. Its frequency is doubled. C. Its frequency is zero. D. Its amplitude is the same as that of the original waves. E. Its amplitude is doubled. F. Its amplitude is zero. G. This problem cannot be solved without knowing the wavelengths of the two waves. Quiz: 16.1-1 answer ¨ Its frequency is the same as that of the original waves. ¨ Its amplitude is doubled. Constructive Destructive Quiz: 16.1-2 The diagram shows two waves on a string moving toward each other at t = 0 s. How does the string appear at t = 2 s? m m m m m Quiz: 16.1-2 answer ¨ A ¨ At t = 2 s, the left end of the rectangular wave has moved to 2 m, and the left end of the trapezoidal wave has moved to 3 m. You then need to add the displacements. m m 16.2 Standing waves / Demo ¨ A standing wave is fixed in location, but oscillates with time. ¨ Individual points on a string oscillate up and down, but the wave itself does not travel. ¨ Demo: Standing waves in rubber tubing (vary frequency) ¤ When the right frequencies are reached, the tubing vibrates in various standing wave modes. 16.2 Superposition creates a standing wave ¨ As two sinusoidal waves of equal wavelength and amplitude travel in opposite directions along a string, superposition will occur when the waves interact. 16.2 Nodes and antinodes ¨ Points on the string which never move are called nodes; those which have the maximum displacement are called antinodes. ¨ The intensity is maximum at points of constructive interference and zero at points of destructive interference. Quiz: 16.2-1 ¨ What is the wavelength of this standing wave? A. 0.25 m B. 0.5 m C. 1.0 m D. 2.0 m E. Standing waves don’t have a wavelength. Quiz: 16.2-1 answer ¨ 1.0 m ¨ The wavelength of a standing wave is twice the distance between successive nodes or antinodes. 16.3 Reflections - boundary / Demo 14 ¨ The reflected wave has the same speed, wavelength, and amplitude (and energy) as the incident wave. ¨ For a fixed boundary case, the situation can be simulated as an un-terminated string with positive and negative amplitude waves moving in opposite directions and meeting at the boundary. ¨ For a free-boundary case, the rope is always perpendicular to the rod. The situation can be simulated as an un-terminated string with two waves of the same amplitude moving in the opposite directions and meeting at the rod. ¨ Demo: Spring on table 16.3 Reflections - discontinuity 15 ¨ When a wave encounters a discontinuity, a point where there is a change in the medium, part of the wave is reflected, and part is transmitted. µ1 > µ2 µ1 < µ2 The reflected pulse is upright. The reflected pulse is inverted. The transmitted pulse travels faster than the The transmitted pulse moves slower than the incident pulse. incident pulse. �" > �$ �" < �$ �$ �$ �" < �$ �" > �$ �$ �$ 16.3 Creating a standing wave ¨ Waves reflected off both ends of a string can create standing waves. ¨ The wavelengths and frequencies of standing waves are 2� � = ' � � � �' = = � �' 2� where � = 1, 2, 3, 4, ⋯ are the mode numbers. ¨ The standing-wave modes can be called resonant modes or resonances. Quiz: 16.3-1 ¨ What is the mode number of this standing wave? Quiz: 16.3-1 ¨ 5 ¨ The mode number is equal to the number of antinodes of the standing wave. 16.3 The fundamental and higher harmonics ¨ The fundamental frequency of a string is the first standing-wave mode. � � = $ 2� ¨ Frequencies above the fundamental frequency are referred to as higher harmonics. �' = ��$ Quiz: 16.3-2 ¨ A standing wave on a string vibrates as shown. Suppose the string tension is reduced to 1/4 its original value while the frequency and length are kept unchanged. Which standing wave pattern is produced? Quiz: 16.3-2 answer ¨ When the string tension is reduced to 1/4 its original value, the speed of ; 8 waves on the string halves. � = 9 234567 : ¨ Since the frequency remains constant, the wavelength must halve as well. < �' = => C. Example 16.3-1 A particular species of spider spins a web with silk threads of density 1300 kg/m3 and radius 1.5 μm. A passing insect brushes a 12-cm-long strand of the web, which has a tension of 7.0 mN, and excites the lowest frequency standing wave. With what frequency will the strand vibrate?.