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 Chapter 15 Wave Motion Dr . Armen Kocharian Lecture 3 and 4

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  • Chapter 15

    Wave Motion

    Dr. Armen Kocharian

    Lecture 3 and 4

  • Goals for Chapter 15 To study the properties and varieties of mechanical waves

    To relate the speed, frequency, and wavelength of

    periodic waves

    To interpret periodic waves mathematically

    To calculate the speed of a wave on a string

    To calculate the energy of mechanical waves

    To understand the interference of mechanical waves

    To analyze standing waves on a string

    To investigate the sound produced by stringed

    instruments

  • Types of mechanical waves

    A mechanical wave is a disturbance traveling through a medium.

    Figure below illustrates transverse waves and longitudinal waves.

  • Longitudinal waves

    a. Longitudinal waves (sound in air)

    b. Longitudinal waves (spring vibrations)

  • Transverse waves

    a. Transverse waves (sound in air)

    b. Longitudinal waves (spring vibrations)

  • Periodic waves

    For a periodic wave, each particle of the medium undergoes periodic motion.

    The wavelength of a periodic wave is the length of one complete wave pattern.

    The speed of any periodic wave of frequency f is v = f.

  • Periodic transverse waves For the transverse waves

    shown here in Figures 15.3 and 15.4, the particles move up and down, but the wave moves to the right.

  • Periodic longitudinal waves For the longitudinal waves

    shown here in Figures 15.6 and 15.7, the particles oscillate back and forth along the same direction that the wave moves.

    Follow Example 15.1.

  • Mathematical description of a wave

    The wave function, y(x,t), gives a

    mathematical description of a

    wave. In this function, y is the

    displacement of a particle at time

    t and position x.

    The wave function for a

    sinusoidal wave moving in the +x-

    direction is y(x,t) = Acos(kx t), where k = 2/ is called the wave number.

    Figure 15.8 at the right illustrates

    a sinusoidal wave.

  • Types of Waves

    There are two main types of waves

    Mechanical waves

    Some physical medium is being disturbed

    The wave is the propagation of a disturbance through a

    medium

    Electromagnetic waves

    No medium required

    Examples are light, radio waves, x-rays

    Matter waves

    Quantum mechanics

    Particle propagation

  • General Features of Waves

    In wave motion, energy is transferred over a

    distance

    Matter is not transferred over a distance

    All waves carry energy

    The amount of energy and the mechanism

    responsible for the transport of the energy differ

  • Mechanical Wave

    Requirements

    Some source of disturbance

    A medium that can be disturbed

    Some physical mechanism through which

    elements of the medium can influence each

    other

  • Pulse on a String

    The wave is generated by a flick on one end of the string

    The string is under tension

    A single bump is formed and travels along the string The bump is called a

    pulse

  • Pulse on a String

    The string is the medium through which the pulse travels

    The pulse has a definite height

    The pulse has a definite speed of propagation along the medium

    The shape of the pulse changes very little as it travels along the string

    A continuous flicking of the string would produce a periodic disturbance which would form a sinusoidal type wave

  • Transverse Wave

    A traveling wave or pulse

    that causes the elements of

    the disturbed medium to

    move perpendicular to the

    direction of propagation is

    called a transverse wave

    The particle motion is

    shown by the blue arrow

    The direction of propagation

    is shown by the red arrow

  • Longitudinal Wave

    A traveling wave or pulse that causes the elements of the disturbed medium to move parallel to the direction of propagation is called a longitudinal wave

    The displacement of the coils is parallel to the propagation

  • Complex Waves

    Some waves exhibit a combination of transverse

    and longitudinal waves

    Surface water waves are an example

    Use the active figure to observe the displacements

  • Complex Waves

  • Example: Earthquake Waves

    P waves P stands for primary

    Fastest, at 7 8 km / s

    Longitudinal

    S waves S stands for secondary

    Slower, at 4 5 km/s

    Transverse

    A seismograph records the waves and allows determination of information about the earthquakes place of origin

  • Traveling Pulse

    The shape of the pulse at t = 0 is shown

    The shape can be represented by

    y (x,0) = f (x) This describes the

    transverse position y of the element of the string located at each value of x at t = 0

  • Traveling Pulse, 2

    The speed of the pulse is v

    At some time, t, the pulse

    has traveled a distance vt

    The shape of the pulse

    does not change

    Its position is now

    y = f (x vt)

  • Traveling Pulse, 3

    For a pulse traveling to the right y (x, t) = f (x vt)

    For a pulse traveling to the left y (x, t) = f (x + vt)

    The function y is also called the wave function:

    y (x, t)

    The wave function represents the y coordinate of any element located at position x at any time t The y coordinate is the transverse position

  • Traveling Pulse, final

    If t is fixed (snapshot) then the wave function

    as a function of position is called the

    waveform

    It defines a curve representing the actual

    geometric shape of the pulse at that time

  • Sinusoidal Waves

    The wave represented by

    the curve shown is a

    sinusoidal wave

    It is the same curve as sin q plotted against q

    This is the simplest

    example of a periodic

    continuous wave

    It can be used to build more

    complex waves

  • Sinusoidal Waves, cont

    The wave moves toward the right In the previous example, the brown wave represents the

    initial position

    As the wave moves toward the right, it will eventually be at the position of the blue curve

    Each element moves up and down in simple harmonic motion

    It is important to distinguish between the motion of the wave and the motion of the particles of the medium

  • Wave Model

    The wave model is a new simplification

    model

    Allows to explore more analysis models for

    solving problems

    An ideal wave has a single frequency

    An ideal wave is infinitely long

    Ideal waves can be combined

  • Terminology: Amplitude and

    Wavelength

    The crest of the wave is the location of the maximum displacement of the element from its normal position This distance is called

    the amplitude, A

    The wavelength, , is the distance from one crest to the next

  • Terminology: Wavelength and

    Period

    More generally, the wavelength is the

    minimum distance between any two identical

    points on adjacent waves

    The period, T , is the time interval required for

    two identical points of adjacent waves to pass

    by a point

    The period of the wave is the same as the period

    of the simple harmonic oscillation of one element

    of the medium

  • Terminology: Frequency

    The frequency, , is the number of crests (or any point on the wave) that pass a given

    point in a unit time interval

    The time interval is most commonly the second

    The frequency of the wave is the same as the

    frequency of the simple harmonic motion of one

    element of the medium

  • Terminology: Frequency, cont

    The frequency and the period are related

    When the time interval is the second, the

    units of frequency are s-1 = Hz

    Hz is a hertz

    1

    T

  • Terminology, Example

    The wavelength, , is 40.0 cm

    The amplitude, A, is

    15.0 cm

    The wave function can

    be written in the form y

    = A cos(kx t)

  • Speed of Waves

    Waves travel with a specific speed

    The speed depends on the properties of the medium being disturbed

    The wave function is given by

    This is for a wave moving to the right

    For a wave moving to the left, replace x vt

    with x + vt

    2( , ) cosy x t A x vt

  • Wave Function, Another Form

    Since speed is distance divided by time,

    v = / T

    The wave function can then be expressed as

    This form shows the periodic nature of y

    y can be used as shorthand notation for y(x, t)

    ( , ) cos 2

    x ty x t A

    T

  • Wave Equations

    We can also define the angular wave number

    (or just wave number), k

    The angular frequency can also be defined

    2k

    22

    T

  • Sinusoidal Wave on a String

    To create a series of pulses, the string can be attached to an oscillating blade

    The wave consists of a series of identical waveforms

    The relationships between speed, velocity, and period hold

  • Sinusoidal Wave on a String, 2

    Each element of the string

    oscillates vertically with

    simple harmonic motion

    For example, point P

    Every element of the string

    can be treated as a simple

    harmonic oscillator vibrating

    with a frequency equal to

    the frequency of the

    oscillation of the blade

  • Sinusoidal Wave on a String, 3

    The transverse speed

    of the element is

    or vy = A SIN(kx t)

    This is different than

    the speed of the wave

    itself

    constant

    y

    x

    dyv

    dt

  • Sinusoidal Wave on a String, 4

    The transverse

    acceleration of the

    element is

    or ay = 2A COS(kx t)

    constant

    y

    y

    x

    dva

    dt

  • Sinusoidal Wave on a String, 5

    The maximum values of the transverse speed

    and transverse acceleration are

    vy, max = A

    ay, max = 2A

    The transverse speed and acceleration do

    not reach their maximum values

    simultaneously

    v is a maximum at y = 0

    a is a maximum at y = A

  • The Linear Wave Equation

    The wave functions y (x, t) represent solutions of an

    equation called the linear wave equation

    This equation gives a complete description of the

    wave motion

    From it you can determine the wave speed

    The linear wave equation is basic to many forms of

    wave motion

  • Linear Wave Equation, General

    The equation can be written as

    2 2

    2 2 2

    1y y

    x v t

    , cosy x t A kx t

    2

    2 2

    2

    ,, cos ,y

    y x ta x t A kx t y x t

    t

    Wave function

    Velocity,

    ,, siny

    y x tv x t A kx t

    t

    Acceleration,

    The second derivative is,

    22 2

    2cos ,

    yk A kx t k y x t

    x

    2 2 2

    2 2 2

    y k y

    x t

    v

    k

  • Linear Wave Equation, General

    The equation can be written as

    This applies in general to various types of traveling waves y represents various positions

    For a string, it is the vertical displacement of the elements of the string

    For a sound wave, it is the longitudinal position of the elements from the equilibrium position

    For em waves, it is the electric or magnetic field components

    2 2

    2 2 2

    1y y

    x v t

  • Linear Wave Equation, General

    cont

    The linear wave equation is satisfied by any

    wave function having the form

    y = f (x vt)

    The linear wave equation is also a direct

    consequence of Newtons Second Law applied to any element of a string carrying a

    traveling wave

  • Example 15.1 and 15.2

    15.1

    15.2

  • Graphing the wave function

    The graphs in Figure 15.9 a

    and b to the right look similar,

    but they are not identical.

    Graph (a) shows the shape of

    the string at t = 0, but graph

    (b) shows the displacement y

    as a function of time at t = 0.

  • Particle velocity and acceleration in a sinusoidal

    wave The graphs below show the velocity and acceleration

    of particles of a string carrying a transverse wave.

  • The speed of a wave on a string

    q

    1 tany Ax

    F y

    F x

    q

    2 tany Bx x

    F y

    F x

    1 2y y y

    x x x

    y yF F F F

    x x

    The net y component of force q q (tan tan )y B AF F

  • The speed of a wave on a string

    .Fv

    2

    2y

    x x x

    y y yF F x

    x x t

    The Newton Law:

    2

    2y y

    yF ma x

    t

    Dividing both sides by Fx

    2

    2x x x

    y y

    yx x

    x F t

    2 2

    2 2

    y y

    x F t

    0x

    2 22

    2 2

    y yv

    x twave equation

  • Calculating wave speed

    Follow Example 15.3 and refer to Figure 15.14 below.

    220.0 9.8 / 196boxF m g kg m s N

    2.000.0250 /

    80.0

    ropem kgkg m

    L m

    The ropes linear density is

    The waves speed is

    19688.5 /

    0.0250 /

    F Nv m s

    kg m

    The wavelength is

    1

    88.5 /44.3

    2,00

    v m sm

    f s

  • Energy in a wave motion

    A wave transfers power along a string because it transfers

    energy.

    The average power is equal:

    ,,y

    y x tF x t F

    x

    , ( , ) ,y yP x t F x t v x t

    where

    ,,y

    y x tv x t

    t

    , cosy x t A kx t For sinusoidal wave,

    ,sin

    y x tkA kx t

    x

    ,sin

    y x tA kx t

    t

    2 2, sinP x t Fk A kx t

  • Energy in a wave motion, cont

    By using relationships vk

    Average Power

    2 2 2 2 2 2, sin sinF

    P x t A kx t F A kx tv

    2 /v F

    2 2, sinP x t Fk A kx t

    2max, sinP x t P kx t

    2 2maxP F A

    Power is reduced to

    where

    2 2 maxmax max, sin sin2

    PP x t P kx t P kx t

  • Useful relationships

    2 2

    2 2

    2 2

    1

    2

    1

    2

    1

    2

    P A v

    FP A

    P A F

  • Power in a wave

  • Wave intensity

    The intensity of a wave is the

    average power it carries per unit

    area.

    If the waves spread out uniformly in

    all directions and no energy is

    absorbed, the intensity I at any

    distance r from a wave source is

    inversely proportional to r2: I 1/r2.

    1 2

    14

    PI

    r 2 2

    24

    PI

    r 2 21 1 2 24 4r I r I

    2

    1 22

    2 1

    I r

    I r

  • Wave intensity: Example 15.5

  • Boundary conditions

    When a wave

    reflects from a fixed

    end, the pulse

    inverts as it reflects.

    See Figure

    15.19(a) at the

    right.

    When a wave

    reflects from a free

    end, the pulse

    reflects without

    inverting. See

    Figure 15.19(b) at

    the right.

  • Reflection of Pulse on

    a String

    Reflection from a HARD boundary

    Propagation of pulse

    Reflection from a

    SOFT boundary

  • Wave interference and

    superposition

    Interference is the

    result of overlapping

    waves.

    Principle of super-

    position: When two

    or more pulses

    (waves) overlap, the

    total displacement is

    the sum of the

    displacements of the

    individual pulses

    (waves).

    1 2, , ,y x t y x t y x t

  • Wave interference and

    superposition

    Interference is the

    result of overlapping

    pulses (waves).

    Principle of super-

    position: When two

    or more waves

    overlap, the total

    displacement is the

    sum of the displace-

    ments of the

    individual pulses

    (waves).

    1 2, , ,y x t y x t y x t

    Two pulses with different

    shapes

  • Wave interference and

    superposition

    Interference is the

    result of overlapping

    waves.

    Principle of super-

    position: When two

    or more waves

    overlap, the total

    displacement is the

    sum of the displace-

    ments of the

    individual waves.

    1 2, , ,y x t y x t y x t

  • Standing waves on a String

    Interference is the result of overlapping waves.

    Principle of super-position: When two or more waves

    overlap, the total displacement is the sum of the

    displacements of the individual waves

    1 , cosy x t A kx t

    Incident wave traveling to the left

    Reflected wave traveling to the right

    2 , cosy x t A kx t

    Standing wave

    1 2, , , cos cosy x t y x t y x t A kx t kx t

  • Standing waves on a String

    Use relationship

    Standing wave is equal to

    Standing wave amplitude 2SWA A

    Nodes of standing wave is defined

    1 2, , , 2 sin siny x t y x t y x t A kx t

    cos cos cos sin sina b a b a b

    , 0 2 siny x t A kx

    This implies, that positions of zero amplitudes or nodes at

    sin 0 ,2

    n nkx x

    k 0,1,2,3,...n

    The positions of antinodes at

    3... , 1,3,5,...

    4 4 4

    nx n

    1 2, , , cos cosy x t y x t y x t A kx t kx t

  • Standing waves on a String

    The diagrams above show standing-wave patterns produced at various

    times by two waves of equal amplitude traveling in opposite directions.

    In a standing wave, the elements of the medium alternate between the

    extremes shown in (a) and (c).

  • Standing waves on a String

  • Standing waves on a string

    The distance between adjacent antinodes is /2. The distance between adjacent nodes is /2. The distance between a node and an adjacent antinode is /4

  • Standing waves on a string Waves traveling in opposite directions on a taut

    string interfere with each other.

    The result is a standing wave pattern that does

    not move on the string.

    Destructive interference occurs where the wave

    displacements cancel, and constructive

    interference occurs where the displacements add.

    At the nodes no motion occurs, and at the

    antinodes the amplitude of the motion is greatest.

    Figure below shows several standing wave

    patterns.

  • The formation of a standing wave

    A wave to the left

    combines with a

    wave to the right

    to form a standing

    wave.

  • Example 15.6 Standing wave on a guitar string

  • Normal modes of a string

    For a taut string fixed at

    both ends, the possible

    wavelengths are n = 2L/n and the possible frequencies

    are fn = n v/2L = nf1, where

    n = 1, 2, 3,

    f1 - is the minimal

    fundamental frequency,

    f2 - is the second harmonic

    (first overtone),

    f3 is the third harmonic

    (second overtone), etc.

    Right figure illustrates the

    first four harmonics.

  • Standing waves and musical

    instruments

    A stringed instrument is tuned to the correct frequency

    (pitch) by varying the tension. Longer strings produce bass

    notes and shorter strings produce treble notes. (See Figure

    below.)

    .

  • Example 15.7

    2n

    n Ff

    L 1

    1

    2

    Ff

    L 2 214F L f

    1 1,2,32

    n

    vf n nf n

    L

  • Example 15.8

    n nv f

  • Speed of a Wave on a String

    The speed of the wave depends on the physical characteristics of the string and the tension to which the string is subjected

    Restoring force (tension, T) and inertia (mass per unit length, )

    This assumes that the tension is not affected by the pulse

    This does not assume any particular shape for the pulse

    tension

    mass/length

    Tv

  • Resonance on a String

    Because an oscillating system exhibits a large amplitude when driven at any of its

    natural frequencies, these frequencies are referred to as resonance frequencies.

    If the system is driven at a frequency that is not one of the natural frequencies, the

    oscillations are of low amplitude and exhibit no stable pattern

  • Speed of a Wave on a String

    Tv

    v f

    One end of the string is attached to a vibrating blade f. The other end passes over a pulley with a hanging mass attached to the end. This produces the tension in the string and increase of speed v. The string is vibrating in its second harmonic, with larger .

  • Reflection of a Wave, Fixed

    End

    When the pulse

    reaches the support,

    the pulse moves back

    along the string in the

    opposite direction

    This is the reflection of

    the pulse

    The pulse is inverted

  • Reflection of a Wave, Free End

    With a free end, the

    string is free to move

    vertically

    The pulse is reflected

    The pulse is not

    inverted

    The reflected pulse has

    the same amplitude as

    the initial pulse

  • Transmission of a Wave

    When the boundary is

    intermediate between

    the last two extremes

    Part of the energy in

    the incident pulse is

    reflected and part

    undergoes transmission

    Some energy

    passes through the

    boundary

    Similar to fixed end.

  • Transmission of a Wave, 2

    Assume a light string is attached to a heavier string

    The pulse travels through the light string and reaches the boundary

    The part of the pulse that is reflected is inverted

    The reflected pulse has a smaller amplitude

  • Transmission of a Wave, 3

    Assume a heavier

    string is attached to a

    light string

    Part of the pulse is

    reflected and part is

    transmitted

    The reflected part is not

    inverted

    Similar to free end.

  • Transmission of Pulse on

    a String

    Propagation of pulse from light to heavy string

    Propagation of wave from

    heavy to light string

    a) How do the amplitudes of the reflected and transmitted waves compare to the

    amplitude of the incident wave?

    b) How do the widths of the reflected and transmitted waves compare to the width of

    the incident wave?

  • Transmission of a Wave, 4

    Conservation of energy governs the pulse When a pulse is broken up into reflected and transmitted

    parts at a boundary, the sum of the energies of the two pulses must equal the energy of the original pulse

    When a wave or pulse travels from medium A to medium B and vA > vB, it is inverted upon reflection B is denser than A

    When a wave or pulse travels from medium A to medium B and vA < vB, it is not inverted upon reflection A is denser than B

  • Introduction Earthquake waves

    carry enormous power as they travel through the earth.

    Other types of mechanical waves, such as sound waves or the vibration of the strings of a piano, carry far less energy.

    Overlapping waves interfere, which helps us understand musical instruments.

  • Conceptual

    Checkpoints

    Chapter 15

    Waves

  • If you double the wavelength of a wave on a string, what happens to

    the wave speed v and the wave frequency f?

    A. v is doubled and f is doubled.

    B. v is doubled and f is unchanged.

    C. v is unchanged and f is halved.

    D. v is unchanged and f is doubled.

    E. v is halved and f is unchanged.

    Q15.1

  • If you double the wavelength of a wave on a string, what happens to

    the wave speed v and the wave frequency f?

    A. v is doubled and f is doubled.

    B. v is doubled and f is unchanged.

    C. v is unchanged and f is halved.

    D. v is unchanged and f is doubled.

    E. v is halved and f is unchanged.

    A15.1

  • Which of the following wave functions describe a wave that moves in

    the x-direction?

    A. y(x,t) = A sin (kx t)

    B. y(x,t) = A sin (kx + t)

    C. y(x,t) = A cos (kx + t)

    D. both B. and C.

    E. all of A., B., and C.

    Q15.2

  • Which of the following wave functions describe a wave that moves in

    the x-direction?

    A. y(x,t) = A sin (kx t)

    B. y(x,t) = A sin (kx + t)

    C. y(x,t) = A cos (kx + t)

    D. both B. and C.

    E. all of A., B., and C.

    A15.2

  • A wave on a string is moving to the right. This

    graph of y(x, t) versus coordinate x for a specific

    time t shows the shape of part of the string at that

    time.

    At this time, what is the velocity of a particle of the

    string at x = a?

    A. The velocity is upward.

    B. The velocity is downward.

    C. The velocity is zero.

    D. not enough information given to decide

    Q15.3

    x

    y

    0 a

  • A wave on a string is moving to the right. This

    graph of y(x, t) versus coordinate x for a specific

    time t shows the shape of part of the string at that

    time.

    At this time, what is the velocity of a particle of the

    string at x = a?

    A. The velocity is upward.

    B. The velocity is downward.

    C. The velocity is zero.

    D. not enough information given to decide

    A15.3

    x

    y

    0 a

  • A wave on a string is moving to the right. This

    graph of y(x, t) versus coordinate x for a specific

    time t shows the shape of part of the string at that

    time.

    At this time, what is the acceleration of a particle of

    the string at x = a?

    A. The acceleration is upward.

    B. The acceleration is downward.

    C. The acceleration is zero.

    D. not enough information given to decide

    Q15.4

    x

    y

    0 a

  • A wave on a string is moving to the right. This

    graph of y(x, t) versus coordinate x for a specific

    time t shows the shape of part of the string at that

    time.

    At this time, what is the acceleration of a particle of

    the string at x = a?

    A. The acceleration is upward.

    B. The acceleration is downward.

    C. The acceleration is zero.

    D. not enough information given to decide

    A15.4

    x

    y

    0 a

  • A wave on a string is moving to the right. This

    graph of y(x, t) versus coordinate x for a specific

    time t shows the shape of part of the string at that

    time.

    At this time, what is the velocity of a particle of the

    string at x = b?

    A. The velocity is upward.

    B. The velocity is downward.

    C. The velocity is zero.

    D. not enough information given to decide

    Q15.5

    x

    y

    0 b

  • A wave on a string is moving to the right. This

    graph of y(x, t) versus coordinate x for a specific

    time t shows the shape of part of the string at that

    time.

    At this time, what is the velocity of a particle of the

    string at x = b?

    A. The velocity is upward.

    B. The velocity is downward.

    C. The velocity is zero.

    D. not enough information given to decide

    A15.5

    x

    y

    0 b

  • A wave on a string is moving to the right. This

    graph of y(x, t) versus coordinate x for a specific

    time t shows the shape of part of the string at that

    time.

    A. only one of points 1, 2, 3, 4, 5, and 6.

    B. point 1 and point 4 only.

    C. point 2 and point 6 only.

    D. point 3 and point 5 only.

    E. three or more of points 1, 2, 3, 4, 5, and 6.

    Q15.6

    At this time, the velocity of a particle on the string is upward at

  • A wave on a string is moving to the right. This

    graph of y(x, t) versus coordinate x for a specific

    time t shows the shape of part of the string at that

    time.

    A. only one of points 1, 2, 3, 4, 5, and 6.

    B. point 1 and point 4 only.

    C. point 2 and point 6 only.

    D. point 3 and point 5 only.

    E. three or more of points 1, 2, 3, 4, 5, and 6.

    A15.6

    At this time, the velocity of a particle on the string is upward at

  • A. 4 times the wavelength of the wave on string #1.

    B. twice the wavelength of the wave on string #1.

    C. the same as the wavelength of the wave on string #1.

    D. 1/2 of the wavelength of the wave on string #1.

    E. 1/4 of the wavelength of the wave on string #1.

    Q15.7

    Two identical strings are each under the same tension. Each string has a

    sinusoidal wave with the same average power Pav.

    If the wave on string #2 has twice the amplitude of the wave on string #1, the

    wavelength of the wave on string #2 must be

  • Two identical strings are each under the same tension. Each string has a

    sinusoidal wave with the same average power Pav.

    If the wave on string #2 has twice the amplitude of the wave on string #1, the

    wavelength of the wave on string #2 must be

    A. 4 times the wavelength of the wave on string #1.

    B. twice the wavelength of the wave on string #1.

    C. the same as the wavelength of the wave on string #1.

    D. 1/2 of the wavelength of the wave on string #1.

    E. 1/4 of the wavelength of the wave on string #1.

    A15.7

  • The four strings of a musical instrument are all made of the same material and

    are under the same tension, but have different thicknesses. Waves travel

    A. fastest on the thickest string.

    B. fastest on the thinnest string.

    C. at the same speed on all strings.

    D. not enough information given to decide

    Q15.8

  • The four strings of a musical instrument are all made of the same material and

    are under the same tension, but have different thicknesses. Waves travel

    A. fastest on the thickest string.

    B. fastest on the thinnest string.

    C. at the same speed on all strings.

    D. not enough information given to decide

    A15.8

  • While a guitar string is vibrating, you gently touch the midpoint of the

    string to ensure that the string does not vibrate at that point.

    The lowest-frequency standing wave that could be present on the

    string

    A. vibrates at the fundamental frequency.

    B. vibrates at twice the fundamental frequency.

    C. vibrates at 3 times the fundamental frequency.

    D. vibrates at 4 times the fundamental frequency.

    E. not enough information given to decide

    Q15.9

  • While a guitar string is vibrating, you gently touch the midpoint of the

    string to ensure that the string does not vibrate at that point.

    The lowest-frequency standing wave that could be present on the

    string

    A. vibrates at the fundamental frequency.

    B. vibrates at twice the fundamental frequency.

    C. vibrates at 3 times the fundamental frequency.

    D. vibrates at 4 times the fundamental frequency.

    E. not enough information given to decide

    A15.9

  • The graph at the top is the history graph at x = 4 m of a

    wave traveling to the right at a speed of 2 m/s. Which is

    the history graph of this wave at x = 0 m?

    (1) (2) (3) (4)

  • The graph at the top is the history graph at x = 4 m of a

    wave traveling to the right at a speed of 2 m/s. Which is

    the history graph of this wave at x = 0 m?

    (3) (4) (1) (2)

  • Which of the following actions would make a pulse travel

    faster down a stretched string?

    1. Move your hand up and down more quickly as you

    generate the pulse.

    2. Move your hand up and down a larger distance as

    you generate the pulse.

    3. Use a heavier string of the same length, under the

    same tension.

    4. Use a lighter string of the same length, under the

    same tension.

    5. Use a longer string of the same thickness, density

    and tension.

  • Which of the following actions would make a pulse travel

    faster down a stretched string?

    1. Move your hand up and down more quickly as you

    generate the pulse.

    2. Move your hand up and down a larger distance as

    you generate the pulse.

    3. Use a heavier string of the same length, under the

    same tension.

    4. Use a lighter string of the same length, under the

    same tension.

    5. Use a longer string of the same thickness, density

    and tension.

  • What is the frequency of this

    traveling wave?

    1. 10 Hz

    2. 5 Hz

    3. 2 Hz

    4. 0.2 Hz

    5. 0.1 Hz

  • What is the frequency of this

    traveling wave?

    1. 10 Hz

    2. 5 Hz

    3. 2 Hz

    4. 0.2 Hz

    5. 0.1 Hz

  • What is the phase difference between the crest of a

    wave and the adjacent trough?

    1. 0

    2. /4

    3. /2

    4. 3 /2

    5.

  • What is the phase difference between the crest of a

    wave and the adjacent trough?

    1. 0

    2. /4

    3. /2

    4. 3 /2

    5.

  • A graph showing wave displacement versus position at a

    specific instant of time is called a

    1. snapshot graph.

    2. history graph.

    3. bar graph.

    4. line graph.

    5. composite graph.

  • A graph showing wave displacement versus position at a

    specific instant of time is called a

    1. snapshot graph.

    2. history graph.

    3. bar graph.

    4. line graph.

    5. composite graph.

  • A graph showing wave displacement versus time at a

    specific point in space is called a

    1. snapshot graph.

    2. history graph.

    3. bar graph.

    4. line graph.

    5. composite graph.

  • A graph showing wave displacement versus time at a

    specific point in space is called a

    1. snapshot graph.

    2. history graph.

    3. bar graph.

    4. line graph.

    5. composite graph.

  • A wave front diagram shows

    1. the wavelengths of a wave.

    2. the crests of a wave.

    3. how the wave looks as it moves toward

    you.

    4. the forces acting on a string thats under

    tension.

    5. Wave front diagrams were not discussed in

    this chapter.

  • A wave front diagram shows

    1. the wavelengths of a wave.

    2. the crests of a wave.

    3. how the wave looks as it moves toward

    you.

    4. the forces acting on a string thats under

    tension.

    5. Wave front diagrams were not discussed in

    this chapter.

  • The waves analyzed in this chapter are

    1. sound waves.

    2. light waves.

    3. string waves.

    4. water waves.

    5. all of the above.

  • The waves analyzed in this chapter are

    1. sound waves.

    2. light waves.

    3. string waves.

    4. water waves.

    5. all of the above.

  • Introduction Earthquake waves

    carry enormous power as they travel through the earth.

    Other types of mechanical waves, such as sound waves or the vibration of the strings of a piano, carry far less energy.

    Overlapping waves interfere, which helps us understand musical instruments.