4 Transverse and Longitudinal Waves There are two types of mechanical waves:Transverse waves: The particles of the medium vibrate up and down (perpendicular to the wave).
5 Longitudinal waves: The particles in the medium vibrate along the same direction as the wave (parallel). The medium undergoes a series of expansions and compressions. The expansions are when the coils are far apart and compressions are when they are when the coil is close together.Expansions and compressions are the analogs of the crests and troughs of a transverse wave.
11 Amplitude: maximum displacement from equilibrium Crest: Top part of the waveTrough: Bottom part of the waveWavelength: Length from crest to crestor trough to trough
12 Wave Motion Wave velocity v is the velocity with which the wave crest is propagating.Wave velocity v depends on the medium.A wave crest travels onewavelength in one period:On a string with tensionFT and mass per unit length ofthe string (linear density) m/Lthe velocity (m/s) of the wave is:
13 9.1 Measurements show that the wavelength of a sound wave in a certain material is 18.0 cm. The frequency of the wave is 1900 Hz. What is the speed of the sound wave?λ = 0.18 mf = 1900 Hzv = λ f= 0.18 (1900)= 342 m/s
14 9. 2 A horizontal cord 5. 00 m long has a mass of 1. 45 g. a 9.2 A horizontal cord 5.00 m long has a mass of 1.45 g a. What must be the tension in the cord if the wavelength of a Hz wave is 60 cm?L = 5 mm = 1.45x10-3 kgf = 120 Hzλ = 0.6 mv = λ f= 0.6 (120)= 72 m/s= 1.5 N
15 b. How large a mass must be hung from its end to give it this tension?FT = FG = mg= kg
16 REFLECTIONA wave front of a wave striking a straight barrier follows the law of reflection. This law states that: "The angle of incidence equals the angle of reflection".The angle of incidence is the angle between the direction of the incoming wave and a normal drawn to the surface. The angle of reflection is the angle between the direction of the reflected wave and the normal to the surface.
17 BEHAVIOR OF WAVESWhen a wave travels from one medium to another, the wave is both reflected and transmitted.When a wave passes into a new medium, its speed changes. The wave must have the same frequency in the new medium as in the old medium, thus, the wavelength adjusts.
20 PARTIAL REFLECTIONWhen a wave travels from one medium to another, the wave is both reflected and transmitted.When a wave passes into a new medium, its speed changes. The wave must have the same frequency in the new medium as in the old medium. Thus, the wavelength adjusts.
23 REFRACTIONA water wave traveling from deep water into shallow water changes direction. This phenomenon is known as refraction.
24 Interference: What happens when two waves pass through the same region? When two crests overlap it is called constructive interference. The resultant displacement is larger then the individual ones.When a crest and a trough interfere, it is called destructive interference. The resultant displacement is smaller.
26 DIFFRACTIONWaves tend to bend as they pass an obstacle. For example, water waves passing a log in a river will tend to bend around the log after they pass. This bending is known as diffraction. The amount of diffraction depends on the wavelength and the size of the obstacle.
27 STANDING WAVES IN STRINGS At certain frequencies standing waves can be produced in which the waves seem to be standing still rather than traveling. This means that the string is vibrating as a whole. This is called resonance and the frequencies at which standing waves occur are called resonant frequencies or harmonics.
28 A standing wave is produced by the superposition of two periodic waves having identical frequencies and amplitudes which are traveling in opposite directions. In stringed musical instruments, the standing wave is produced by waves reflecting off a fixed end and interfering with oncoming waves as they travel back through the medium.
31 The points of destructive interference (no vibration) are called nodes, and the points of constructive interference (maximum amplitude of vibration) are called antinodes.
32 When a wave reflects off of something, it can interfere with its own reflection. The interference is alternately constructive or destructive as the two waves move past each other. This creates a standing wave.
33 There can be more than one frequency for standing waves. Frequencies at which standing waves can be produced are called the natural (or resonant) frequencies.When a string on an instrument is plucked, vibrations, that is, waves, travel back and forth through the medium being reflected at each fixed end. Certain sized waves can survive on the medium. These certain sized waves will not cancel each other out as they reflect back upon themselves. These certain sized waves are called the harmonics of the vibration. They are standing waves. That is, they produce patterns which do not move.
35 Standing Waves in Strings A string can be fixed in both sides, like a guitar or piano string.When the string is plucked, many frequency waves will travel in both directions. Most will interfere randomly and die away. Only those with resonant frequencies will persist.Since the ends are fixed, they will be the nodes.The wavelengths of the standing waves have a simple relation to the length of the string.The lowest frequency called the fundamental frequency has only one antinode. That corresponds to half a wavelength:
36 The other natural frequencies are called overtones The other natural frequencies are called overtones. They are also called harmonics and they are integer multiples of the fundamental.The fundamental is called the first harmonic.The next frequency has two antinodes and is called the second harmonic.
37 = 297 m/s L = 1/2 λ λ = 2L = 2(0.5) = 1 m = 297 Hz Fundamental: 9.3 A metal string is under a tension of 88.2 N. Its length is 50 cm and its mass is g.a. Find the velocity of the waves on the string.m = 5x10-4 kgFT = 88.2 NL = 0.5 m= 297 m/sb. Determine the frequencies of its fundamental, first overtone and second overtone.Fundamental:L = 1/2 λλ = 2L= 2(0.5)= 1 m= 297 Hz
38 First overtonefn = n f'f2 = 2(297)= 594 HzSecond overtonef3 = 3(297)= 891 Hz
39 4 segments: 4th Harmonic L = 4/2 λ = 2 λ λ = ½ L = ½ (2) = 1 m v = f λ 9.4 A string 2.0 m long is driven by a 240 Hz vibrator at its end. The string resonates in four segments. What is the speed of the waves on the string?L = 2 mf = 240 Hz4 segments: 4th HarmonicL = 4/2 λ = 2 λλ = ½ L = ½ (2) = 1 mv = f λ= 240 (1)= 240 m/s
40 Fundamental: L = ½ λ λ = 2L = 2(0.3) = 0.6 m v = f λ = 256 (0.6) 9.5 A banjo string 30 cm long resonates in its fundamental to a frequency of 256 Hz. What is the speed of the waves on the string?L = 0.3 mf = 256 HzFundamental: L = ½ λλ = 2L = 2(0.3) = 0.6 mv = f λ= 256 (0.6)= 154 m/s
41 fn = n f' = 92 Hz fn = n f' f3 = 3(92) = 276 Hz 9.6 A string vibrates in five segments to a frequency of 460 Hz.a. What is its fundamental frequency?f5 = 460 Hzfn = n f'= 92 Hzb. What frequency will cause it to vibrate in three segments?fn = n f'f3 = 3(92)= 276 Hz
42 SOUND WAVESSound is a longitudinal wave produced by a vibration that travels away from the source through solids, liquids, or gases, but not through a vacuum. The speed of sound is independent of the pressure, frequency, and wavelength of the sound. However, the speed of sound in a gas is proportional to the temperature T. The following equation is useful in determining the speed of sound in airv = T Units: m/sWhere 331 is the speed of sound in m/s at 0°C, and T is the temperature in °C
43 There are two main characteristic of sound: pitch and loudness. Pitch refers to how high (or low) the sound is. It is measured by the frequency.We hear frequencies in the range of 20 Hz to 20,000 Hz.This is called the audible range.Frequencies above this range are called ultrasonic.Sound waves whose frequency is lower than the audible range are called infrasonic.
44 INTERFERENCE OF SOUND WAVES: BEATS If two sources are close in frequency, the sound from them interferes and what we hear is an alternating sound level. The level rise and falls. If the alternating sound is regular, it is called beats.
45 The beat frequency equals the difference in frequencies between the sources. This is a way to tune musical instruments. Compare a tuning fork to a note and tune until the beats disappear.CIConstructiveInterferenceDIDestructive
46 HUMAN EAR:The ear consists of three basic parts:Outer ear: serves to collect and channelsound to the middle ear.Middle ear: serves to transform theenergy of a sound wave into theinternal vibrations of the bonestructure of the middle ear andtransform these vibrations into acompressional wave in the inner ear.Inner ear: serves to transform the energy of a compressional wave within the inner ear fluid into nerve impulses which can be transmitted to the brain.
47 Hearing:A compression forces the eardrum inward and an expansion forces the eardrum outward, thus vibrating the eardrum at the same frequency of the sound wave.
48 SOURCES OF SOUNDSound comes from a vibrating object. If an object vibrates with frequency and intensity within the audible range, it produces sound we can hear.MUSICAL INSTRUMENTS- String Instruments: guitar, violin and pianoWind Instruments:Open Pipe: flute and some organ pipesClosed Pipe: clarinet,oboe and saxophone
49 STRING INSTRUMENTSThe sounds produced by vibrating strings are not very loud. Many stringed instruments make use of a sounding board or box, sometimes called a resonator, to amplify the sounds produced. The strings on a piano are attached to a sounding board while for guitar strings a sound box is used. When the string is plucked and begins to vibrate, the sounding board or box begins to vibrate as well. Since the board or box has a greater area in contact with the air, it tends to amplify the sounds.On a guitar or a violin, the length of thestrings are the same, but their mass perlength is different. That changes thevelocity and so the frequency changes.
50 Open pipe WIND INSTRUMENTS Wind instruments produce sound from the vibrations ofstanding waves in columns of air inside a pipe or a tube.In a string, the ends are nodes. In air columns the ends can be either nodes or antinodes.Open pipe
52 The overtones will be multiples of the fundamental HARMONICSa) For open pipeThe overtones will be multiples of the fundamentalb) For closed pipeThe overtones will be the odd multiples of the fundamental
53 Saxophone is closed so: L= 1/4 λ 9.7 A saxophone plays a tune in the key of B-flat. The saxophone has a third harmonic frequency of Hz when the speed of sound in air is 331 m/s. What is the length of the pipe that makes up the saxophone?f' = f3/n= 466.2/3= HzSaxophone is closed so: L= 1/4 λn = 3f3 = Hzv = 331 m/s= 0.53 m
54 9.8 An organ pipe that is open at both ends has a fundamental frequency of Hz when the speed of sound in air is 331 m/s. What is the length of this pipe?f' = 370 Hzv = 331 m/sL= 1/2 λ= m
55 9. 9 What is the fundamental frequency of a viola string that is 35 9.9 What is the fundamental frequency of a viola string that is cm long when the speed of waves on this string is 346 m/s?L = mv = 346 m/sL= 1/2 λ and λ = 2L= Hz
56 L= 1/2 λ and λ = 2L v = λ f =2 L f = 2(1.32)(125) = 330 m/s 9.10 A pipe that is open at both ends has a fundamental frequency of 125 Hz. If the pipe is 1.32 m long, what is the speed of the waves in the pipe?L= 1/2 λ and λ = 2Lv = λ f=2 L f= 2(1.32)(125)= 330 m/sf' = 125 HzL = 1.32 m
57 9.11 A pipe that is closed on one end has a seventh harmonic frequency of Hz. If the pipe is 1.53 m long, what is the speed of the waves in the pipe?n = 7f7 = HzL = 1.53 mL= 7/4 λ and λ = 4/7 L= m/s
58 DOPPLER EFFECTWhen a source of sound waves and a listener approach one another, the pitch of the sound is increased as compared to the frequency heard if they remain at rest. If the source and the listener recede from one another, the frequency is decreased. This phenomenon is known as the Doppler effect.
59 Light waves also exhibit the Doppler effect Light waves also exhibit the Doppler effect. The spectra of stars that are receding from us is shifted toward the longer wavelengths of light. This is known as the red shift.Measurement of the red shift allows astrophysicists to calculate the speed at which stars are moving away. Since almost all stars and galaxies exhibit a red shift, it is believed that the universe is expanding.
60 SHOCK WAVES AND THE SONIC BOOM When the speed of a source of sound exceeds the speed of sound, the sound waves in front of the source tend to overlap and constructively interfere. The superposition of the waves produce an extremely large amplitude wave called a shock wave.The shock wave contains a great deal of energy. When the shock wave passes a listener, this energy is heard as a sonic boom.The sonic boom is heard only for a fraction of a second; however, it sounds as if an explosion has occurred and can cause damage.