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Chapter 21 – Mechanical Waves A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University © 2007 Mechanical Waves A mechanical wave is a physical disturbance in an elastic medium. Consider a stone dropped into a lake. Energy is transferred from stone to floating log, but only the disturbance travels. Actual motion of any individual water particle is small. Energy propagation via such a disturbance is known as mechanical wave motion. Periodic Motion Simple periodic motion is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time. 1 f T Amplitude A Period, T, is the time for one complete oscillation. (seconds,s) Frequency, f, is the number of complete oscillations per second. Hertz (s-1) Review of Simple Harmonic Motion x F It might be helpful for you to review Chapter 14 on Simple Harmonic Motion. Many of the same terms are used in this chapter. Example: The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? 15 s T 0.50 s 30 cylces x F Period: T = 0.500 s 1 1 f T 0.500 s Frequency: f = 2.00 Hz A Transverse Wave In a transverse wave, the vibration of the individual particles of the medium is perpendicular to the direction of wave propagation. Motion of particles Motion of wave Longitudinal Waves In a longitudinal wave, the vibration of the individual particles is parallel to the direction of wave propagation. v Motion of particles Motion of wave Water Waves An ocean wave is a combination of transverse and longitudinal. The individual particles move in ellipses as the wave disturbance moves toward the shore. Wave speed in a string. The wave speed v in a vibrating string is determined by the tension F and the linear density m, or mass per unit length. v F m FL m L m = m/L v = speed of the transverse wave (m/s) F = tension on the string (N) m or m/L = mass per unit length (kg/m) Example 1: A 5-g section of string has a length of 2 M from the wall to the top of a pulley. A 200-g mass hangs at the end. What is the speed of a wave in this string? F = (0.20 kg)(9.8 m/s2) = 1.96 N v FL (1.96 N)(2 m) m 0.005 kg v = 28.0 m/s 200 g Note: Be careful to use consistent units. The tension F must be in newtons, the mass m in kilograms, and the length L in meters. Periodic Wave Motion A vibrating metal plate produces a transverse continuous wave as shown. For one complete vibration, the wave moves a distance of one wavelength l as illustrated. l A B Wavelength l is distance between two particles that are in phase. Velocity and Wave Frequency. The period T is the time to move a distance of one wavelength. Therefore, the wave speed is: v l T 1 but T f so v fl The frequency f is in s-1 or hertz (Hz). The velocity of any wave is the product of the frequency and the wavelength: v fl Production of a Longitudinal Wave l l • An oscillating pendulum produces condensations and rarefactions that travel down the spring. • The wave length l is the distance between adjacent condensations or rarefactions. Velocity, Wavelength, Speed l Frequency f = waves per second (Hz) v s t Velocity v (m/s) Wavelength l (m) v fl Wave equation Example 2: An electromagnetic vibrator sends waves down a string. The vibrator makes 600 complete cycles in 5 s. For one complete vibration, the wave moves a distance of 20 cm. What are the frequency, wavelength, and velocity of the wave? 600 cycles f ; 5s f = 120 Hz The distance moved during a time of one cycle is the wavelength; therefore: l = 0.20 m v = fl v = (120 Hz)(0.2 m) v = 24.0 m/s Energy of a Periodic Wave The energy of a periodic wave in a string is a function of the linear density m , the frequency f, the velocity v, and the amplitude A of the wave. f A m = m/L v E 2 2 f 2 A2 m L P 2 2 f 2 A2 m v Example 3. A 2-m string has a mass of 300 g and vibrates with a frequency of 20 Hz and an amplitude o 50 mm. If the tension in the rope is 48 N, how much power must be delivered to the string? m 0.30 kg m 0.150 kg/m L 2m v F (48 N) 17.9 m/s m 0.15 kg/m P 2 2 f 2 A2 m v P = 22(20 Hz)2(0.05 m)2(0.15 kg/m)(17.9 m/s) P = 53.0 W The Superposition Principle • When two or more waves (blue and green) exist in the same medium, each wave moves as though the other were absent. • The resultant displacement of these waves at any point is the algebraic sum (yellow) wave of the two displacements. Constructive Interference Destructive Interference Formation of a Standing Wave: Incident and reflected waves traveling in opposite directions produce nodes N and antinodes A. The distance between alternate nodes or antinodes is one wavelength. Possible Wavelengths for Standing Waves Fundamental, n = 1 1st overtone, n = 2 2nd overtone, n = 3 3rd overtone, n = 4 n = harmonics 2L ln n n 1, 2, 3, . . . Possible Frequencies f = v/l : Fundamental, n = 1 f = 1/2L 1st overtone, n = 2 f = 2/2L 2nd overtone, n = 3 f = 3/2L 3rd overtone, n = 4 f = 4/2L n = harmonics f = n/2L nv fn 2L n 1, 2, 3, . . . Characteristic Frequencies Now, for a string under tension, we have: v F m Characteristic frequencies: FL m and n fn 2L nv f 2L F m ; n 1, 2, 3, . . . Example 4. A 9-g steel wire is 2 m long and is under a tension of 400 N. If the string vibrates in three loops, what is the frequency of the wave? For three loops: n = 3 n fn 2L 3 f3 2L F m ; n3 FL 3 m 2(2 m) Third harmonic 2nd overtone 400 N (400 N)(2 m) 0.009 kg f3 = 224 Hz Summary for Wave Motion: v F m v fl FL m n fn 2L E 2 2 f 2 A2 m L F m 1 f T ; n 1, 2, 3, . . . P 2 2 f 2 A2 m v