Waves and Sound

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Chapter 14

• Waves and Sound

2
Wave Motion

• A wave is a moving self-sustained disturbance of
  a medium either a field or a substance.
• Mechanical waves are waves in a material medium.
• Mechanical waves require
• Some source of disturbance
• A medium that can be disturbed
• Some physical connection between or mechanism
  though which adjacent portions of the medium
  influence each other
• All waves carry energy and momentum

3
Wave Characteristics

• The state of being displaced moves through the
  medium as a wave.
• A progressive or travelling wave is a
  self-sustaining disturbance of a medium that
  propagates from one region to another, carrying
  energy and momentum.
• Examples waves on a string, surface waves on
  liquids, sound waves in air, and compression
  waves in solids or liquids.
• In all cases the disturbance advances and not the
  medium.

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Traveling Waves

• Flip one end of a long rope that is under tension
  and fixed at one end
• The pulse travels to the right with a definite
  speed
• A disturbance of this type is called a traveling
  wave

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Description of a Wave

• A steady stream of pulses on a very long string
  produces a continuous wave
• The blade oscillates in simple harmonic motion
• Each small segment of the string, such as P,
  oscillates with simple harmonic motion

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Amplitude and Wavelength

• Amplitude (A) is the maximum displacement of
  string above the equilibrium position
• Wavelength (?), is the distance between two
  successive points that behave identically

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Longitudinal Waves

• In a longitudinal wave, the elements of the
  medium undergo displacements parallel to the
  motion of the wave
• A longitudinal wave is also called a compression
  wave

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Longitudinal Wave Represented as a Sine Curve

• A longitudinal wave can also be represented as a
  sine curve
• Compressions correspond to crests and stretches
  correspond to troughs
• Also called density waves or pressure waves

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Transverse Waves

• In a transverse wave, each element that is
  disturbed moves in a direction perpendicular to
  the wave motion

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Waveforms

• Wavepulse in taut rope. Shape of pulse is
  determined by motion of driver.
• If driver (hand) oscillates up and down in a
  regular way, it generates a wave train a
  constant frequency carrier whose amplitude is
  modulated (varies with time.)

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Waveform The shape of a Wave

• The high points are crests of the wave
• The low points are troughs of the wave
• As a 2-D or 3-D wave propagates, it creates a
  wavefront

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Velocity of Waves

• Period (T) of a periodic wave - time it takes for
  a single profile to pass a point in space - the
  number of seconds per cycles.
• The inverse of the period (1 /T) is the frequency
  f, the number of profiles passing per second, the
  number of cycles per second.
• The distance in space over which the wave
  executes one cycle of its basic repeated form is
  the wavelength, l the length of the profile.

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Velocity of Waves

• The speed of the wave the rate (in m/s) at
  which the wave advances
• Is derived from the basic speed equation of
  distance/time
• Since a length of wave l passes by in a time T,
  its speed must equal l /T f l
• The speed of any progressive periodic wave
• v fl

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Example 1

• A youngster in a boat watches waves on a lake
  that seem to be an endless succession of
  identical crests passing, with a half-second
  between them. If one wave takes 1.5 s to sweep
  straight down the length of her 4.5 m-long boat,
  what are the frequency, period, and wavelength of
  the waves?
• Given The waves are periodic 0.5 s between
  crests L 4.5 m t 1.5 s
• Find T, f, v, and l

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Transverse Waves Strings

• The speed of a mechanical wave is determined by
  the inertial and elastic properties of the medium
  and not in any way by the motion of the source
• Pulse traveling with a speed v along a
  lightweight, flexible string under constant
  tension FT
• v (11.3)
• When m/L is large, there is a lot of inertia and
  the speed is low. When FT is large, the string
  tends to spring back rapidly, and the speed is
  high

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Example 2

• A 2.0 m-long horizontal string having a mass of
  40 g is slung over a light frictionless pulley,
  and its end is attached to a hanging 2.0 kg mass.
  Compute the speed of the wavepulse on the
  string. Ignore the weight of the overhanging
  length of rope.
• Given A string of length l 2.0 m, m 40 g
  supporting a 2.0 kg load
• Find v

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Reflection, Refraction, Diffraction and Absorption

• End of rope is held stationary energy pumped in
  at the other end, the reflected wave ideally
  carries away all the original energy
• It is inverted 180 out-of-phase with the
  incident wave
• End of the rope is free it will rise up as the
  pulse arrives until all the energy is stored
  elastically.
• The rope then snaps back down, producing a
  reflected wavepulse that is right side up.

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Reflection of Waves Fixed Boundary

• Whenever a traveling wave reaches a boundary,
  some or all of the wave is reflected
• When it is reflected from a fixed end, the wave
  is inverted
• The shape remains the same

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Reflected Wave Open Boundary

• When a traveling wave reaches an open boundary,
  all or part of it is reflected
• When reflected from an open boundary, the pulse
  is not inverted

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Reflection, Refraction, Diffraction and Absorption

• When a wave passes from one medium to another
  having different physical characteristics, there
  will be a redistribution of energy.
• Medium is also displaced, and a portion of the
  incident energy appears as a refracted wave.
• If the incident wave is periodic, the transmitted
  wave has the same frequency but a different speed
  and therefore a different wavelength the larger
  the density of the refracting medium, the smaller
  the length of the wave.

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Reflection, Refraction, Diffraction and Absorption

• When a wave meets a hole or another obstacle, it
  can be bent around it or through itDiffraction
• A wave can lose part or all of its energy when it
  meets a boundary Absorption.

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Reflection, Refraction, Diffraction and Absorption

• A wave passing through a lens will be both
  reflected AND refracted. Examples include light
  (of course) and also sound (through the balloon
  of different gas)
• Absorption can either SUBTRACT (beach sand) or
  ADD (wind) energy to a wave, depending on which
  way the energy is being transferred.

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Superposition of Waves

• Superposition Principle In the region where two
  or more waves overlap, the resultant is the
  algebraic sum of the various contributions at
  each point.
• Superimposing two harmonic waves of the same
  frequency and amplitude at every value of x, add
  the heights of the two sine curves above the
  axis as positive and below it as negative.
• The sum of any number of harmonic waves of the
  same frequency traveling in the same direction is
  also a harmonic wave of that frequency.

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Interference of Waves

• Two traveling waves can meet and pass through
  each other without being destroyed or even
  altered
• Waves obey the Superposition Principle
• If two or more traveling waves are moving through
  a medium, the resulting wave is found by adding
  together the displacements of the individual
  waves point by point
• Actually only true for waves with small amplitudes

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Constructive Interference

• Two waves, a and b, have the same frequency and
  amplitude
• Are in phase
• The combined wave, c, has the same frequency and
  a greater amplitude

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Destructive Interference

• Two waves, a and b, have the same amplitude and
  frequency
• They are 180 out of phase
• When they combine, the waveforms cancel

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Superposition

• When two or more waves interact, their
  amplitudes are added (superimposed) one upon the
  other, creating interference.
• Constructive interference
• occurs when the superposition
• increases amplitude.
• Destructive interference
• occurs when the superposition
• decreases the amplitude.

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Natural Frequency/Harmonics

• If a periodic force occurs at the appropriate
  frequency, a standing wave will be produced in
  the medium.
• The lowest natural frequency in a medium is its
  fundamental harmonic.
• Double this frequency to produce the 2nd
  harmonic.
• Triple this frequency to produce the 3rd harmonic

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Natural Frequency/Harmonics
REQUIRES FIXED BOUNDARIES
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Frequency and Period

• w0 - the natural angular frequency, the specific
  frequency at which a physical system oscillates
  all by itself once set in motion
• natural angular frequency
• and since w0 2pf0
• natural linear frequency
• Since T 1/f0
• Period

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Waves and Energy

• As waves propagate, their energy alternates
  between two froms
• Transverse Waves Potential ltgt Kinetic
• Longitudinal Waves Pressure ltgt Kinetic
• Light Waves Electric ltgt Magnetic

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Waves and Energy

• Generally
• HIGHER FREQUENCY HIGHER ENERGY
• HIGHER AMPLITUDE HIGHER ENERGY

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Nodes and Modes

• Nodes occur/are located at points of
  equilibrium within a wave.
• Anitnodes occur/are located at points of
  greatest displacement (amplitude) within a wave.

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Nodes and Modes

• One-dimensional modes
• Transverse guitar or piano strings
• Rotational jump rope, lasso
• Two- and Three-dimensional modes
• Radial concentric circular nodes and anti-nodes
• Angular linear nodes and anti-nodes radiating
  outward from center.

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Nodes and Modes
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Nodes and Modes