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Mechanical Waves. Ch 21-23. Waves. A wave is a disturbance in a medium which carries energy from one point to another without the transport of matter. The medium allows the disturbance to propagate . Transverse Wave.

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waves
Waves
  • A wave is a disturbance in a medium
  • which carries energy from one point to another
  • without the transport of matter.
  • The medium allows the disturbance to propagate.

Physics chapters 21-23

transverse wave
Transverse Wave
  • Particles oscillate at right angles to the direction of motion.

Physics chapters 21-23

longitudinal waves
Longitudinal Waves
  • Particles oscillate parallel to the direction of motion.

Physics chapters 21-23

periodic waves pulses
Periodic Waves & Pulses
  • A wave pulse is a single disturbance.
  • A periodic wave is a series of disturbances or wave train.

Physics chapters 21-23

transverse wave speed
Transverse Wave Speed
  • Determined by the medium and its properties.
  • elasticity or restoring force
  • inertia

Physics chapters 21-23

wave on a medium with tension
Wave on a mediumwith tension.
  • String, rope, wire, etc…
  • T is the tension, & m is the linear density, m = m/L = mass per unit length.

Physics chapters 21-23

waves1
Waves
  • Speed:

Physics chapters 21-23

wave terminology
Wave Terminology
  • Frequency (f) - cycles per second. (Hz)
  • Period (T) - Seconds per cycle.
  • Amplitude (A) - Maximum displacement from equilibrium.
  • The distance that a wave travels in one period is the wavelength (l).

Physics chapters 21-23

example 1
Example 1
  • A wave travels along a string. The time for a particular point to move from a maximum displacement to zero is 0.170 s. The wavelength is 1.40 m. What are the period, frequency, and wave speed?

Physics chapters 21-23

example 1 continued
Example 1 continued
  • It takes 0.680 s for one cycle, so T = 0.680 s
  • f = 1/T, so f = 1.47 Hz

Physics chapters 21-23

example 2
Example 2
  • What is the speed of a transverse wave in a rope of length 2.00 m and mass 60.0 g under a tension of 500 N?

Physics chapters 21-23

example 2 continued
Example 2 continued

Physics chapters 21-23

polarization
Polarization
  • Most transverse waves are linearly polarized
    • They either move just up and down
      • Vertically polarized
    • Or just side to side
      • Horizontally polarized

Physics chapters 21-23

circular polarization
Circular polarization
  • If we combine two perpendicular waves that have equal amplitude but are out of step by a quarter-cycle, the resulting wave is circularly polarized.

Physics chapters 21-23

polarizing filters
Polarizing filters
  • Only let through waves that are polarized one way.
  • Like passing a rope through a slot in a board – only waves in the direction of the slot will get through.

Physics chapters 21-23

longitudinal wave speed
Longitudinal Wave Speed
  • Depends on the pressure change and the fractional volume change
  • Where r is the density. B is the bulk modulus of a fluid. Y is young’s modulus for a solid. See tables 12-1 and 12-2. B = 1/k

Physics chapters 21-23

longitudinal waves1
Longitudinal waves
  • Don’t have polarization
  • When the frequency is within the range of human hearing, it is called sound.

Physics chapters 21-23

sound waves in gases
Sound waves in gases
  • Temperature doesn’t remain constant as sound waves move through air.
  • So, we use the equation
  • Where g is the ratio of heat capacities (ch 18), R is the ideal gas constant (8.314 J/mol∙K), T is temperature in K, and M is the molecular mass (ch 17).

Physics chapters 21-23

sound waves
Sound waves
  • Humans can hear from about 20 Hz to about 20 000 Hz.
  • Air is not continuous – it consists of molecules.
  • Like a swarm of bees.
  • Also sort of like wave/particle duality.

Physics chapters 21-23

mathematical wave description
Mathematical wave description

y(x, t) = A sin(wt – kx)

(Motion to right)

or y(x, t) = A sin(wt + kx)

(Motion to left)

Physics chapters 21-23

reflection
Reflection
  • When a wave comes to a boundary, it is reflected.
  • Imagine a string that is tied to a wall at one end.
    • If we send a single wave pulse down the string,
    • when it reaches the wall, it exerts an upward force on the wall.

Physics chapters 21-23

reflection1
Reflection
  • By Newton’s third law,
    • the wall exerts a downward force that is equal in magnitude.
  • This force generates a pulse at the wall, which travels back along the string in the opposite direction.

Physics chapters 21-23

reflection2
Reflection
  • In a ‘hard’ reflection like this,
  • there must be a node at the wall
  • because the string is tied there.
  • The reflected pulse is inverted from the incident wave.

Physics chapters 21-23

reflection3
Reflection
  • Now imagine that instead of being tied to a wall
  • the string is fastened to a ring which is free to move along a rod.
  • When the wave pulse arrives at the rod, the ring moves up the rod
  • and pulls on the string.

Physics chapters 21-23

reflection4
Reflection
  • This sort of ‘soft’ reflection
  • creates a reflected pulse
  • that is not inverted.

Physics chapters 21-23

transmission
Transmission
  • When a wave is incident on a boundary that separates two regions of different wave speeds
    • part of the wave is reflected
    • and part is transmitted.

Physics chapters 21-23

transmission1
Transmission
  • If the second medium is denser than the first
    • the reflected wave is inverted.
  • If the second medium is less dense
    • the reflected wave is not inverted.
  • In either case, the transmitted wave is not inverted.

Physics chapters 21-23

transmission2
Transmission

Physics chapters 21-23

transmission3
Transmission

Physics chapters 21-23

interference
Interference

Physics chapters 21-23

interference1
Interference
  • The effect that waves have when they occupy the same part of the medium.
  • They can add together or cancel each other out.
  • After the waves pass each other, they continue on with no residual effects.

Physics chapters 21-23

constructive interference
Constructive Interference

Physics chapters 21-23

constructive interference1
Constructive Interference
  • l out of phase = 360° = 1 cycle = 2p rad

Physics chapters 21-23

destructive interference
Destructive Interference

Physics chapters 21-23

destructive interference1
Destructive Interference
  • l/2 out of phase = 180° = 1/2 cycle = p rad

Physics chapters 21-23

superposition of waves
Superposition of waves
  • If two waves travel simultaneously along the same string
  • the displacement of the string when the waves overlap
  • is the algebraic sum of the displacements from each individual wave.

Physics chapters 21-23

standing waves
Standing Waves
  • Consider a string that is stretched between two clamps, like a guitar string.
  • If we send a continuous sinusoidal wave of a certain frequency along the string to the right
  • When the wave reaches the right end, it will reflect and travel back to the left.

Physics chapters 21-23

standing waves1
Standing waves
  • The left-going wave the overlaps with the wave that is still traveling to the right.
  • When the left-going wave reaches the left end
  • it reflects again and overlaps both the original right-going wave and the reflected left-going wave.
  • Very soon, we have many overlapping waves which interfere with each other.

Physics chapters 21-23

standing waves2
Standing waves
  • For certain frequencies
  • the interference produces a standing wave pattern
  • with nodes and large antinodes.
  • This is called resonance
  • and those certain frequencies are called resonant frequencies.

Physics chapters 21-23

standing waves3
Standing waves
  • A standing wave looks like a stationary vibration pattern,
  • but is the result of waves moving back and forth on a medium.

Physics chapters 21-23

standing waves4
Standing waves
  • Superposition of reflected waves which have a maximum amplitude and appear to be a stationary vibration pattern.

y1 + y2 = -2Acos(wt)sin(kx)

Physics chapters 21-23

standing waves5
Standing Waves
  • If the string is fixed at both ends
  • there must be nodes there.
  • The simplest pattern of resonance that can occur is one antinode at the center of the string.

Physics chapters 21-23

standing waves on strings
Standing Waves on Strings
  • Nodes form at a fixed or closed end.
  • Antinodes form at a free or open end.

Physics chapters 21-23

standing waves6
Standing waves
  • For this pattern, half a wavelength spans the distance L.
  • This is called the 1st harmonic.
  • It is also called the fundamental mode of vibration.

Physics chapters 21-23

standing waves7
Standing waves
  • For the next possible pattern, a whole wavelength spans the distance L.
  • This is called the 2nd harmonic, or the 1st overtone.

Physics chapters 21-23

standing waves8
Standing Waves
  • For the next possible pattern, one and a half wavelengths span the distance L.
  • This is called the 3rd harmonic, or the 2nd overtone.

Physics chapters 21-23

standing waves9
Standing waves
  • In general, we can write

Physics chapters 21-23

standing waves10
Standing Waves

Physics chapters 21-23

standing waves on a string
Standing Waves on a String

Physics chapters 21-23

overtones
Overtones

Physics chapters 21-23

string fixed at one end
String fixed at ONE end

Note: Only the odd harmonics exist!

Physics chapters 21-23

example
Example
  • The A-string of a violin has a linear density of 0.6 g/m and an effective length of 330 mm.
  • (a) Find the tension required for its fundamental frequency to be 440 Hz.
  • (b) If the string is under this tension, how far from one end should it be pressed against the fingerboard in order to have it vibrate at a fundamental frequency of 495 Hz, which corresponds to the note B?

Physics chapters 21-23

example1
Example
  • m = 0.6 g/m = 6 x 10–4 kg/m
  • L = 330 mm = 0.33 m
  • a) Ft = ?
  • b) 0.33 m – L2 = ?

Physics chapters 21-23

example a
Example - A)

Physics chapters 21-23

example b
Example - B)
  • v1 = v2
  • f1l1 = f2l2

Physics chapters 21-23

wave example 1
Wave Example 1
  • The stainless steel forestay of a racing sailboat is 20 m long, and its mass is 12 kg. To find its tension, it is struck by a hammer at the lower end and the return of the pulse is timed. If the time interval is 0.20 s, what is the tension in the stay?

Physics chapters 21-23

example 11
Example 1
  • L = 20 m, t = 0.20 s, m = 12 kg
  • Find: F

Physics chapters 21-23

example 12
Example 1

= 2.4 x 104 N

Physics chapters 21-23

slide60
Note:
  • Wave speed is determined by the medium.
  • Wave frequency is determined by the source.

Physics chapters 21-23

sound waves1
Sound Waves
  • p = BkAcos(wt - kx)
  • If y is written as a sine function, P is written as a cosine function because the displacement and the pressure arep/2 rad out of phase.
  • pmax = BkA

Physics chapters 21-23

waves in 3 dimensions
Waves in 3 Dimensions

x

A 0 1 4 9 16

For Spherical Wavefronts: A = 4pr2

Physics chapters 21-23

intensity
Intensity
  • Power per unit area
  • W/m2

Physics chapters 21-23

loudness of sound
Loudness of Sound
  • Also called intensity level
  • Determined by the intensity
  • which is a function of the sound\'s amplitude.
  • The human ear does not have a linear response to the intensity of sound.
  • The response is nearly logarithmic.

Physics chapters 21-23

decibel scale db
Decibel Scale (dB)

Where: Io = 1 x 10-12 W/m2

Physics chapters 21-23

common decibel levels
Common decibel levels
  • Threshold of hearing
    • 0 dB = 1 x 10-12 W/m2
  • Whisper
    • 20 dB = 1 x 10-10 W/m2
  • Conversation
    • 65 dB = 3.2 x 10-6 W/m2
  • Threshold of pain
    • 120 dB = 1 W/m2

Physics chapters 21-23

example2
EXAMPLE
  • How many times more intense is an 80-dB sound than a 40-dB sound?

Physics chapters 21-23

example3
EXAMPLE

Physics chapters 21-23

example4
EXAMPLE
  • Number of times greater = I1/I2

Physics chapters 21-23

beats
Beats
  • When two sound waves that are at nearly the same frequency interfere with each other, they form a beat pattern.
  • It is an amplitude variation.
  • The beat frequency

Physics chapters 21-23

the doppler effect
The Doppler effect
  • When a source of sound is moving towards you, it sounds higher pitched (higher frequency).
  • When it moves away, it sounds lower pitched.

Physics chapters 21-23

the doppler effect1
The Doppler Effect
  • The S’s stand for the source of the sound.
  • The L’s stand for the listener.
  • v by itself stands for the speed of sound.
  • Be careful with the signs on your velocities!!
    • The direction from listener toward source is positive
    • The direction from source toward listener is negative

Physics chapters 21-23

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