Transcript Longitudinal Waves

Waves and Sound
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
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.
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
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
Amplitude and Wavelength
 Amplitude (A) is the
maximum
displacement of string
above the equilibrium
position
 Wavelength (λ), is the
distance between two
successive points that
behave identically
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
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
Transverse Waves
 In a transverse wave, each element that is disturbed
moves in a direction perpendicular to the wave
motion
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.)
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
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.
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
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
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

FT
m/ L
(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
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
Example 2
 Solution: This is a problem about waves on a
string, use equation 11.3.
 The tension is the load in Newtons, so FT = mg =
(2.0 kg)(9.81 m/s2) = 19.62 N
v
FT
19.62 N

 31 m / s
mL
0.040kg / 2.0 m
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.
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
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
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.
Reflection, Refraction, Diffraction
and Absorption
 When a wave meets a hole or another obstacle, it can
be bent around it or through it—Diffraction
 A wave can lose part or all of its energy when it meets a
boundary – Absorption.
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.
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.
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
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
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
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.
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
Natural Frequency/Harmonics
REQUIRES FIXED BOUNDARIES
Frequency and Period
 w0 - the natural angular frequency, the specific
frequency at which a physical system oscillates all
by itself once set in motion
k
natural angular frequency
w0 
m
 and since w0 = 2pf0
natural linear frequency
1
2p
k
m
T  2p
m
k
f0 
 Since T= 1/f0
Period
Waves and Energy
As waves propagate, their energy alternates between
two froms:
 Transverse Waves – Potential <> Kinetic
 Longitudinal Waves – Pressure <> Kinetic
 Light Waves – Electric <> Magnetic
Waves and Energy
Generally –
 HIGHER FREQUENCY = HIGHER ENERGY
 HIGHER AMPLITUDE = HIGHER ENERGY
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.
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.
Nodes and Modes
Nodes and Modes