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Mechanical Waves

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# Mechanical Waves - PowerPoint PPT Presentation

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

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.

Physics chapters 21-23

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

Physics chapters 21-23

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

Physics chapters 21-23

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
• Determined by the medium and its properties.
• elasticity or restoring force
• inertia

Physics chapters 21-23

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

Waves
• Speed:

Physics chapters 21-23

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
• 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
• 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
• 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

Physics chapters 21-23

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
• 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
• 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
• 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 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
• 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
• 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

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
• 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

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

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

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

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

Physics chapters 21-23

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

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

Transmission

Physics chapters 21-23

Transmission

Physics chapters 21-23

Interference

Physics chapters 21-23

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

Physics chapters 21-23

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

Physics chapters 21-23

Destructive Interference

Physics chapters 21-23

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

Physics chapters 21-23

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
• 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 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 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 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 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 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
• Nodes form at a fixed or closed end.
• Antinodes form at a free or open end.

Physics chapters 21-23

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 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 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 waves
• In general, we can write

Physics chapters 21-23

Standing Waves

Physics chapters 21-23

Standing Waves on a String

Physics chapters 21-23

Overtones

Physics chapters 21-23

String fixed at ONE end

Note: Only the odd harmonics exist!

Physics chapters 21-23

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

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)

Physics chapters 21-23

Example - B)
• v1 = v2
• f1l1 = f2l2

Physics chapters 21-23

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

Physics chapters 21-23

Example 1

= 2.4 x 104 N

Physics chapters 21-23

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

Physics chapters 21-23

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

x

A 0 1 4 9 16

For Spherical Wavefronts: A = 4pr2

Physics chapters 21-23

Intensity
• Power per unit area
• W/m2

Physics chapters 21-23

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)

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

Physics chapters 21-23

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

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

Physics chapters 21-23

EXAMPLE

Physics chapters 21-23

EXAMPLE
• Number of times greater = I1/I2

Physics chapters 21-23

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
• 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 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