Mechanical waves
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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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Mechanical Waves

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Mechanical waves

Mechanical Waves

Ch 21-23


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


Mechanical waves

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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