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. Period, T, is the time for one complete oscillation. (seconds,s) AmplitudeA 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 Example: The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? x F Period: 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. Motion of particles Motion of wave

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 v in 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)

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/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.

Wavelength l 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. l B A 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: 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. The wave length λ is the distance between adjacent condensations or rarefactions.

Velocity, Wavelength, Speed Frequency f = waves per second (Hz) Wavelength l (m) l Velocity v (m/s) Wave equation

Example 2: An electromagnetic vibrator sends waves down a string 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 = 0.20 m v = 2.40 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 v m = m/L

P = 22(20 Hz)2(0.05 m)2(0.15 kg/m)(17.9 m/s) 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. The resultant displacement of these waves at any point is the algebraic sum (yellow) wave of the two displacements. Constructive Interference Destructive Interference

Add the BLUE wave to the RED wave.

Standing Waves

Reflection of Waves (FREE End Reflection) Free

Reflection of Waves (FIXED End Reflection) Fixed

If you continue to make waves, the returning waves will interfere with the advancing waves.

If you generate waves of the “CORRECT” wavelength, the returning waves will come back “in step” with the waves you are making.

“In Step” for FREE End The returning wave will arrive at a crest the moment you are making the crest of the next wave. Constructive Interference will occur, and the end of the medium will move with twice the amplitude.

Close your eyes and make a standing wave. You can feel the returning wave move your hand.

“In Step” for FIXED End The returning wave will arrive at a trough the moment you are making the crest of the next wave. Destructive Interference will occur, and the end of the medium will not move.

Possible Standing Waves on a Fixed String 12 ft Possible λ’s for a 12 ft medium 24 ft

Possible Standing Waves on a Fixed String 6 ft 6 ft 12 ft Possible λ’s for a 12 ft medium 24 ft , 12 ft

Possible Standing Waves on a Fixed String 4 ft 4 ft 4 ft 12 ft Possible λ’s for a 12 ft medium 24 ft , 12 ft , 8 ft

Possible Standing Waves on a Fixed String 3 ft 3 ft 3 ft 3 ft 12 ft Possible λ’s for a 12 ft medium 24 ft , 12 ft , 8 ft , 6 ft

Possible Standing Waves on a Fixed String 2.4 ft 2.4 ft 2.4 ft 2.4 ft 2.4 ft 12 ft Possible λ’s for a 12 ft medium 24 ft , 12 ft , 8 ft , 6 ft , 4.8 ft

Possible Standing Waves on a Fixed String Write to Learn Question #3 12 ft Find the next TWO possible wavelengths! Possible λ’s for a 12 ft medium 24 ft , 12 ft , 8 ft , 6 ft , 4.8 ft

= Possible Standing Waves on a Fixed String 2 · L n 2.4 ft 2.4 ft 2.4 ft 2.4 ft 2.4 ft 12 ft 2 · L n = Possible λ’s for any length, L 24 ft , 12 ft , 8 ft , 6 ft , 4.8 ft n = 1, 2, 3, 4, . . .

Because each of these waves are POSSIBLE, each one of these waves will be PRESENT.

Because each of these waves are POSSIBLE, each one of these waves will be PRESENT.

Possible Standing Waves on a Fixed String Fundamental Frequency (1st Harmonic)

Possible Standing Waves on a Fixed String 1st Overtone (2nd Harmonic)

Possible Standing Waves on a Fixed String 2nd Overtone (3rd Harmonic)

Possible Standing Waves on a Fixed String 3rd Overtone (4th Harmonic)

Possible Standing Waves on a Fixed String 4th Overtone (5th Harmonic)

Possible Standing Waves (ENDS ARE FREE)

Because each of these waves are POSSIBLE, each one of these waves will be PRESENT.

FREE End FIXED End 12 ft ¼ λ Possible λ’s for a 12 ft medium 48 ft

¾ λ FREE End FIXED End Possible λ’s for a 12 ft medium 48 ft , 16 ft

5/4 λ FREE End FIXED End Possible λ’s for a 12 ft medium 48 ft , 16 ft

7/4 λ FREE End FIXED End Possible λ’s for a 12 ft medium 48 ft , 16 ft

9/4 λ FREE End FIXED End Write to Learn Question #4 1.33 ft 12 ft Find the next TWO possible wavelengths! 9/4 λ Possible λ’s for a 12 ft medium 48 ft , 16 ft , 9.6 ft , 6.86 ft , 5.33 ft , 4.36 ft

= FREE End FIXED End 4 · L 2n-1 Possible λ’s for any length, L

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

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

Third harmonic 2nd overtone 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 = 3 400 N Third harmonic 2nd overtone f3 = 224 Hz