wave motion a wave is, in general, a disturbance that moves through a medium

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Wave Motion A wave is, in general, a disturbance that moves through a medium.

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Page 1: Wave Motion A wave is, in general, a disturbance that moves through a medium

Wave Motion

A wave is, in general, a disturbance that moves through a medium.

Page 2: Wave Motion A wave is, in general, a disturbance that moves through a medium

A wave carries energy from one location to another without transporting the material of the medium.

Examples of mechanical waves include water waves, waves on a string, and sound waves.

Page 3: Wave Motion A wave is, in general, a disturbance that moves through a medium

What is the medium?

Water

Slinky

Page 4: Wave Motion A wave is, in general, a disturbance that moves through a medium

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).

Page 5: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 6: Wave Motion A wave is, in general, a disturbance that moves through a medium

Longitudinal waves:

Page 7: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 8: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 9: Wave Motion A wave is, in general, a disturbance that moves through a medium

PARTS OF A WAVE

Page 10: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 11: Wave Motion A wave is, in general, a disturbance that moves through a medium

Amplitude: maximum displacement from equilibriumCrest: Top part of the waveTrough: Bottom part of the waveWavelength: Length from crest to crest or trough to trough

Page 12: Wave Motion A wave is, in general, a disturbance that moves through a medium

Wave Motion

Wave velocity v is thevelocity with which the wave crest is propagating.Wave velocity v depends on the medium.

On a string with tensionFT and mass per unit length of the string (linear density) m/Lthe velocity (m/s) of the wave is:

A wave crest travels onewavelength in one period:

Page 13: Wave Motion A wave is, in general, a disturbance that moves through a medium

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 Hz

v = λ f = 0.18 (1900) = 342 m/s

Page 14: Wave Motion A wave is, in general, a disturbance that moves through a medium

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 120 Hz wave is 60 cm?

L = 5 mm = 1.45x10-3 kgf = 120 Hzλ = 0.6 m

vF

m LT/

= 1.5 N

2

T

v mF

L

2 3(72) 1.45 10

5

x

v = λ f = 0.6 (120) = 72 m/s

Page 15: Wave Motion A wave is, in general, a disturbance that moves through a medium

b. How large a mass must be hung from its end to give it this tension?

FT = FG = mg

TFmg

1.5

9.8 = 0.153 kg

Page 16: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 17: Wave Motion A wave is, in general, a disturbance that moves through a medium

BEHAVIOR OF WAVES

When 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.

Page 18: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 19: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 20: Wave Motion A wave is, in general, a disturbance that moves through a medium

PARTIAL REFLECTION

When 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.

Page 21: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 22: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 23: Wave Motion A wave is, in general, a disturbance that moves through a medium

REFRACTIONA water wave traveling from deep water into shallow water changes direction. This phenomenon is known as refraction.

Page 24: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 25: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 26: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 27: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 28: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 29: Wave Motion A wave is, in general, a disturbance that moves through a medium

Second Harmonic

Page 30: Wave Motion A wave is, in general, a disturbance that moves through a medium

Third Harmonic

Page 31: Wave Motion A wave is, in general, a disturbance that moves through a medium

The points of destructive interference (no vibration) are called nodes, and the points of constructive interference (maximum amplitude of vibration) are called antinodes.

Page 32: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 33: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 34: Wave Motion A wave is, in general, a disturbance that moves through a medium
Page 35: Wave Motion A wave is, in general, a disturbance that moves through a medium

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:

Page 36: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 37: Wave Motion A wave is, in general, a disturbance that moves through a medium

9.3 A metal string is under a tension of 88.2 N. Its length is 50 cm and its mass is 0.500 g.a. Find the velocity of the waves on the string.

m = 5x10-4 kgFT = 88.2 N

L = 0.5 m /TFv

m L

4

88.2

5 10 / 0.5x = 297 m/s

b. Determine the frequencies of its fundamental, first overtone and second overtone.

Fundamental:L = 1/2 λ λ = 2L = 2(0.5) = 1 m

vf

297

1 = 297 Hz

Page 38: Wave Motion A wave is, in general, a disturbance that moves through a medium

First overtone fn = n f'

f2 = 2(297)

= 594 Hz Second overtonefn = n f'

f3 = 3(297)

= 891 Hz

Page 39: Wave Motion A wave is, in general, a disturbance that moves through a medium

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 m f = 240 Hz

4 segments: 4th Harmonic

L = 4/2 λ = 2 λλ = ½ L = ½ (2) = 1 m

v = f λ = 240 (1) = 240 m/s

Page 40: Wave Motion A wave is, in general, a disturbance that moves through a medium

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 m f = 256 Hz

Fundamental: L = ½ λλ = 2L = 2(0.3) = 0.6 m

v = f λ = 256 (0.6) = 154 m/s

Page 41: Wave Motion A wave is, in general, a disturbance that moves through a medium

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'

' nffn

460

5 = 92 Hz

b. What frequency will cause it to vibrate in three segments?

fn = n f'

f3 = 3(92)

= 276 Hz

Page 42: Wave Motion A wave is, in general, a disturbance that moves through a medium

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 air

v = 331 + 0.6 T Units: m/sWhere 331 is the speed of sound in m/s at 0°C, and T is the temperature in °C

Page 43: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 44: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 45: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

CIConstructive Interference

DIDestructiveInterference

Page 46: Wave Motion A wave is, in general, a disturbance that moves through a medium

HUMAN EAR:The ear consists of three basic parts:

Outer ear: serves to collect and channel sound to the middle ear.

Middle ear: serves to transform the energy of a sound wave into the internal vibrations of the bone structure of the middle ear and transform these vibrations into a compressional 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.

Page 47: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 48: Wave Motion A wave is, in general, a disturbance that moves through a medium

SOURCES OF SOUND

Sound 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 piano

-Wind Instruments:

Open Pipe: flute and some organ pipes

Closed Pipe: clarinet,oboe and saxophone

Page 49: Wave Motion A wave is, in general, a disturbance that moves through a medium

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 the strings are the same, but their mass per length is different. That changes the velocity and so the frequency changes.

Page 50: Wave Motion A wave is, in general, a disturbance that moves through a medium

WIND INSTRUMENTSWind 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

Page 51: Wave Motion A wave is, in general, a disturbance that moves through a medium

Half Closed Pipe

Page 52: Wave Motion A wave is, in general, a disturbance that moves through a medium

a) For open pipe

The overtones will be multiples of the fundamental

b) For closed pipe

The overtones will be the odd multiples of the fundamental

HARMONICS

Page 53: Wave Motion A wave is, in general, a disturbance that moves through a medium

9.7 A saxophone plays a tune in the key of B-flat. The saxophone has a third harmonic frequency of 466.2 Hz when the speed of sound in air is 331 m/s. What is the length of the pipe that makes up the saxophone?

n = 3f3 = 466.2 Hz

v = 331 m/s

f' = f3/n

= 466.2/3 = 155.4 Hz

Saxophone is closed so: L= 1/4 λ

= 0.53 m 1 1

4 4

vL

f

331

4(155.4)

Page 54: Wave Motion A wave is, in general, a disturbance that moves through a medium

9.8 An organ pipe that is open at both ends has a fundamental frequency of 370.0 Hz when the speed of sound in air is 331 m/s. What is the length of this pipe?

L= 1/2 λf' = 370 Hzv = 331 m/s

331

2(370) = 0.447 m

1 1

2 2

vL

f

Page 55: Wave Motion A wave is, in general, a disturbance that moves through a medium

9.9 What is the fundamental frequency of a viola string that is 35.0 cm long when the speed of waves on this string is 346 m/s?

L = 0.35 mv = 346 m/s L= 1/2 λ and λ = 2L

vf

2

v

L 346

2(0.35) = 494.2 Hz

Page 56: Wave Motion A wave is, in general, a disturbance that moves through a medium

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?

f' = 125 HzL = 1.32 m

L= 1/2 λ and λ = 2L

v = λ f =2 L f = 2(1.32)(125) = 330 m/s

Page 57: Wave Motion A wave is, in general, a disturbance that moves through a medium

9.11 A pipe that is closed on one end has a seventh harmonic frequency of 466.2 Hz. If the pipe is 1.53 m long, what is the speed of the waves in the pipe?

n = 7f7 = 466.2 Hz

L = 1.53 m

L= 7/4 λ and λ = 4/7 L

41(1.53)466.2

7

4

7

Lv f

= 407.6 m/s

Page 58: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 59: Wave Motion A wave is, in general, a disturbance that moves through a medium

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.

Page 60: Wave Motion A wave is, in general, a disturbance that moves through a medium

SHOCK WAVES AND THE SONIC BOOMWhen 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.