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 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. Amplitude A Period (seconds,s) Period, T, is the time for one complete oscillation. (seconds,s) Frequency Hertz (s -1 ) Frequency, f, is the number of complete oscillations per second. Hertz (s -1 )

Review of Simple Harmonic Motion xF 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? xF Period: T = 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. Motion of particles Motion of wave v

Water Waves An ocean wave is a combi- nation of transverse and longitudinal. The individual particles move in ellipses as the wave disturbance moves toward the shore.

Wave speed in a string. v = speed of the transverse wave (m/s) F = tension on the string (N)  or m/L = mass per unit length (kg/m) v = speed of the transverse wave (m/s) F = tension on the string (N)  or m/L = mass per unit length (kg/m) The wave speed v in a vibrating string is determined by the tension F and the linear density , or mass per unit length. L  = m/L

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? 200 g F = (0.20 kg)(9.8 m/s 2 ) = 1.96 N v = 28.0 m/s 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 B A Wavelength  is distance between two particles that are in phase. A vibrating metal plate produces a transverse continuous wave as shown. For one complete vibration, the wave moves a distance of one wavelength as illustrated.

Velocity and Wave Frequency. The period T is the time to move a distance of one wavelength. Therefore, the wave speed is: The frequency f is in s -1 or hertz (Hz). The velocity of any wave is the product of the frequency and the wavelength:

Production of a Longitudinal Wave An oscillating pendulum produces condensations and rarefactions that travel down the spring.An oscillating pendulum produces condensations and rarefactions that travel down the spring. The wave length l is the distance between adjacent condensations or rarefactions.The wave length l is the distance between adjacent condensations or rarefactions.

Velocity, Wavelength, Speed Frequency f = waves per second (Hz) Velocity v (m/s) Wavelength  (m) 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? f = 120 Hz The distance moved during a time of one cycle is the wavelength; therefore: = 0.20 m v = f 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. fA v  = m/L

Example 3. A 2-m string has a mass of 300 g and vibrates with a frequency of 20 Hz and an amplitude of 50 mm. If the tension in the rope is 48 N, how much power must be delivered to the string? P = 2  2 (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.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.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 anti- nodes 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

Possible Frequencies f = v/  : Fundamental, n = 1 1st overtone, n = 2 2nd overtone, n = 3 3rd overtone, n = 4 n = harmonics f = 1/2L f = 2/2L f = 3/2L f = 4/2L f = n/2L

Characteristic Frequencies Now, for a string under tension, we have: Characteristic frequencies:

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? 400 N For three loops: n = 3 f 3 = 224 Hz Third harmonic 2nd overtone

Summary for Wave Motion: