Chapter 21 – Mechanical Waves

Objectives: After completion of this module, you should be able to: • Demonstrate your understanding of transverse and longitudinal waves. • Define, relate and apply the concepts of frequency, wavelength, and wave speed. • Solve problems involving mass, length, tension, and wave velocity for transverse waves. • Write and apply an expression for determining the characteristic frequencies for a vibrating string with fixed endpoints.

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.

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

It might be helpful for you to review Chapter 14 on Simple Harmonic Motion. Many of the same terms are used in this chapter. x F Review of Simple Harmonic Motion

x F Example:The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? Period: T = 0.500 s Frequency: f = 2.00 Hz

Motion of wave Motion of particles A Transverse Wave In a transverse wave, the vibration of the individual particles of the medium is perpendicular to the direction of wave propagation.

v 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.

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. The wave speed vin a vibrating string is determined by the tension F and the linear density m, or mass per unit length. 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)

200 g 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 = 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.

l A B Wavelengthl is distance between two particles that are in phase. 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.

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:

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

Frequency f = waves per second (Hz) l Wavelength l (m) Velocity v (m/s) Velocity, Wavelength, Speed 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: v = fl v = (120 Hz)(0.02 m) l = 0.020 m v = 2.40 m/s

m = m/L f A v 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 amplitudeA of the wave.

Example 3.A 2-m string has a mass of 300 g and vibrates with a frequency of 20 Hz and an amplitude of 50mm. 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. • 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

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

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

400 N 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 Third harmonic 2nd overtone f3 = 224 Hz

Summary for Wave Motion:

CONCLUSION: Chapter 21Mechanical Waves

Chapter 21 – Mechanical Waves