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 & 16.1-2 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 16.3 Reflections - boundary / Demo 10 ¨ 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 11 ¨ 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 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 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?.