Harmonic Superposition and Standing Waves · 2016. 1. 4. · A: The wave speed on a string is given...

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Harmonic Superposition and Standing Waves 11 January 2016 PHYC 1290 Department of Physics and Atmospheric Science

Transcript of Harmonic Superposition and Standing Waves · 2016. 1. 4. · A: The wave speed on a string is given...

  • Harmonic Superposition and Standing Waves

    11 January 2016

    PHYC 1290 Department of Physics and Atmospheric Science

  • Superposition of Harmonic Waves

    y1 y2

    y1 = A sin(kx - ωt) y2 = A sin(kx + ωt)⊖ for wave moving right ⊕ for wave moving left

    Let them superpose:

    ➞ y = y1 + y2 = A sin(kx - ωt) + A sin(kx + ωt)

    Consider 2 waves with same amplitude, wavelength and period:

    A

    -A0

  • t = π/ωt = 2π/3ω

    t = π/3ωt = 0

    Use the trig identity

    sin B + sin C = 2 sin�

    12(B + C)

    ⇥cos

    �12(B � C)

    with B = kx - ωt and C = kx + ωt .

    ➞ y = 2A sin(kx)cos(-ωt)

    But cos(-θ) = cos(θ).

    ➞ y = 2A sin(kx)cos(ωt)

    It doesn’t move left or right!

    It is a “standing wave” of amplitude 2A.

    y

    xπ/k 2π/k2A

    -2A0

  • Nodes Anti-nodes

    Simulation

  • How to make a standing wave

    fixed end

    drivesinusoidally

    When the wave is reflected from the fixed end, it will be inverted but will have the same k, ω and A.

    ➞ y = ... = 2A cos(kx) sin(ωt) (a standing wave)

    ➞ y = A sin(kx - ωt) - A sin(kx + ωt)

    inverted moving leftmoving right

  • DemoStanding wave on a spring

    By shaking the spring at different frequencies, we can see different “harmonics” m:

    m = 1

    m = 2

    m = 3

    m = 4

  • DemoStanding longitudinal wave on an aluminum rod

    Running fingers along an aluminum rod (using lots of resin) excites longitudinal modes that we can hear.

  • LEnds can move(free, not fixed)

    Hold herefor m=1

    Hold here for m=2

    How Does it Work?

    yx

    Node

    m=1:

    x

    Node Node

    y

    m=2:

  • Standing wave modes:

    L

    m = 1

    m = 2

    m = 3

    m = 4

    Fixed-fixed ends

    �2 =2L2

    �3 =2L3

    �4 =2L4

    Harmonic numbers:

    f2 = 2v

    2L

    f3 = 3v

    2L

    f4 = 4v

    2L

    �1 =2L1

    Wavelengths:Frequencies

    (using v = fλ):

    f1 = 1v

    2L

  • GeneralizationFor standing waves with fixed ends, the wavelength for

    harmonic (or mode) number m is

    �m =2Lm

    The lowest frequency (f1) is called the “fundamental harmonic”:

    f1 =v

    2Lwhich allows us to write

    fm = mf1

    and the frequency is

    fm = mv

    2L

    with m = 1, 2, 3, ...

  • Standing wave modes

    L

    m = 1

    Free-free ends

    Harmonic numbers:

    m = 2

    m = 3

    The generalizations are the same as before:

    �m =2Lm

    f1 =v

    2L andfm = mf1, with m = 1, 2, 3, ...

  • Standing wave modes

    L

    m = 1

    m = 3

    m = 5

    Fixed-free ends.

    Harmonic numbers:

    The generalizations for fixed-free ends are different:

    �m =4Lm

    f1 =v

    4L andfm = mf1, with m = 1, 3, 5, ...

  • Extra Material

  • Standing wave modes

    L

    m = 1 �1 =2L1

    Free-free ends

    Harmonic numbers:

    m = 2 �2 =2L2

    f2 = 2v

    2L

    m = 3 �3 =2L3

    f3 = 3v

    2L

    Wavelengths:Frequencies

    (using v = fλ):

    f1 = 1v

    2L

    Conclusion: We can use the same generalizations as before.

    �m =2Lm

    f1 =v

    2L andfm = mf1, with m = 1, 2, 3, ...

  • Standing wave modes

    L

    m = 1

    m = 3

    m = 5

    Fixed-free ends

    Harmonic numbers: Wavelengths:

    Frequencies (using v = fλ):

    �1 =4L1

    �3 =4L3

    �5 =4L5

    f1 = 1v

    4L

    f3 = 3v

    4L

    f5 = 5v

    4L

    Conclusion: The generalizations for fixed-free ends are different.

    �m =4Lm

    f1 =v

    4L andfm = mf1, with m = 1, 3, 5, ...

  • Standing wave on a stringHere is a 60 Hz standing wave (m=2) on a string. Part way through a strobe light is used to isolate the string’s motion.

    http://youtu.be/rkeIubVQ0N8

    http://youtu.be/rkeIubVQ0N8

  • ExampleMiddle-C (C4) on a piano has fundamental frequency of 262 Hz.

    Q: What are the frequencies of the Harmonics?

    A: The string is fixed at both ends, so fm = mf1.

    m=1: f1 = 262 Hz (given)

    ... and so on.

    m=2: f2 = 2×f1 = 524 Hzm=3: f3 = 3×f1 = 786 Hz

  • Q: If the string is 62.5 cm long, what are the wavelengths?

    A: For fixed ends, λm = 2L/m , som=1: λ1 = 2L/1 = 125 cmm=2: λ2 = 2L/2 = 62.5 cmm=3: λ3 = 2L/3 = 41.67 cm

    Q: What are the wave speeds on the string?

    A: Using, v = fλ,m=1: v1 = f1×λ1 = 32750 cm/sm=2: v2 = f2×λ2 = 32750 cm/s m=3: v3 = f3×λ3 = 32750 cm/s

    The speeds are all the same! This is expected because the wave speed v depends only on the medium’s properties.

  • Q: What tension does the middle-C string have?

    A: The wave speed on a string is given by

    v =

    �T

    µ� T = µv2

    We need to first determine the linear density μ=m/L.

    ➞ μ = m/L = density × πr 2 = 0.00614 kg/m

    ➞ T = 659 N

    v =

    �T

    µ� T = µv2

    Piano wire is made of steel which has density 7820 kg/m3. The middle-C string is 1 mm in diameter.

    ➞ m = density × volume = density × πr 2L

  • 659 N is a lot of tension for one string! This is equivalent to having the tension on one string

    drawn by a 67 kg (148 lb) mass.

    There are normally about 230 strings in a piano (although there are 88 keys, some keys have

    multiple strings).

    This means there is a combined tension of over 150,000 N (15,000 kg or 30,000 lbs) in a piano!!

    For this reason, at the heart of a piano is a cast-iron frame (or harp, pictured left) needed to

    sustain the massive tension.

    Image souce: http://commons.wikimedia.org/wiki/File:Fluegel-Rahmen.jpg

    http://commons.wikimedia.org/wiki/File:Fluegel-Rahmen.jpg

  • Here is a view inside a grand piano. You can see the strings drawn across the cast-iron frame.

    Image source: http://commons.wikimedia.org/wiki/File:Bosendorfer_185.JPG

    http://commons.wikimedia.org/wiki/File:Bosendorfer_185.JPG