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  • Standing Waves in Strings

    1

    The two ends must be nodes (because they are fixed).

    Nodes (N) and anti-nodes (A) are separated by /4

    For the nth normal mode: L=n

    2We call this value :n

    n=2L

    n f n=v

    n=n

    2Lv=

    n

    2L T=nf 1 n=1,2,3,...

  • Standing Waves in Strings

    2

    Example 14.3

    A middle C string on a piano has a fundamental frequency of 262 Hz and the A note has a frequency of 440 Hz.

    [A] Calculate the frequencies of the next two harmonics of the C string.

    f n=nf 1

    f 2=2f 1=524Hz

    f 3=3f 1=786Hz

  • Standing Waves in Strings

    3

    Example 14.3

    A middle C string on a piano has a fundamental frequency of 262 Hz and the A note has a frequency of 440 Hz.

    [B] If the strings for the A and the C notes are assumed to have the same mass per unit length and the same length,

    determine the ratio of the tensions in the two strings.

    f 1A=vA

    2L=

    1

    2L T A f 1C= vC2L= 12L T CNote:

    Velocity of the transverse wave for the A note is NOT the same as the velocity for the C note.

    The wavelength of the fundamental is 2L in both cases.

  • Standing Waves in Strings

    4

    Example 14.3

    A middle C string on a piano has a fundamental frequency of 262 Hz and the A note has a frequency of 440 Hz.

    [B] If the strings for the A and the C notes are assumed to have the same mass per unit length and the same length,

    determine the ratio of the tensions in the two strings.

    f 1A

    f 1C= T AT C

    Key: If one takes ratios of frequencies the common constants and L get divided out.

    So: and:T A

    T C=f 1A

    2

    f 1C2=

    4402

    2622=2.82

    To tune it to A you will need to make the string ~ 3 times as taut.

  • Differences with Standing Waves in Strings and Pipes

    5

    Standing waves in strings are transverse waves

    Standing waves in pipes are longitudinal sound waves

    Note: You can only hear longitudinal waves, because transverse waves do not propagate through air or liquid, only through solid.

    To hear the string, the transverse waves on the string have to excite longitudinal sound waves in the air which then reach your

    ear drum

  • Differences with Standing Waves in Strings and Pipes

    6

    Strings: Ends are always nodes

    Pipes: Closed ends are nodes Open ends are anti-nodes

    DidgeridooOne open end & one closed end

    Bamboo fluteTwo open ends

    Image from grapevineroad.orgImage from i235.photobucket.com

  • Standing Waves in Pipes (Air Columns)

    7

    http://www.youtube.com/watch?v=QSXjpWUDvO4

    Wikipedia: For humans, hearing is normally limited to frequencies between about 12 Hz and 20,000 Hz (20 kHz),

    although these limits are not definite.

    The didgeridoo fundamental frequency is closer to the bottom end I estimate the one we heard to be about 80-90 Hz.

    After hearing this, you decide to make yourself a didgeridoo with a PVC pipe from the hardware store.

    How long should this pipe be in order to have a fundamental frequency of 80 Hz?

  • Standing Waves in Open Pipes

    8

    The two ends must be anti-nodes.

    Nodes (N) and anti-nodes (A) are separated by /4

    For the 1st normal mode: L=

    2We call this value :1

    1=2L

    NAA

    L

  • Standing Waves in Open Pipes

    9

    The second normal mode has 1 anti-node in the centre as well.

    Nodes (N) and anti-nodes (A) are separated by /4

    For the 2nd normal mode: L=

    We call this value :22=L

    N NAA A

    L

  • Standing Waves in Open Pipes

    10

    The third normal mode has 2 anti-nodes in the centre as well.

    Nodes (N) and anti-nodes (A) are separated by /4

    For the 3rd normal mode:L=3

    2We call this value :3

    3=2L

    3

    N NA AN AA

    L

  • Standing Waves in Open Pipes

    11

    For the nth normal mode, there will be n-1 antinodes in between.

    Nodes (N) and anti-nodes (A) are separated by /4

    For the nth normal mode: L=n

    2We call this value :n

    n=2L

    nf n=

    v

    n=n

    v

    2L=nf 1 n=1,2,3,...

    NA

    NA

    NA

    NA

    NAA

    L

  • Notation

    12

    Standing Waves in (both-end) Open Pipes

    First Harmonic: n = 1Second Harmonic: n = 2

    Third Harmonic: n = 3

    f n=v

    n=n

    v

    2L=nf 1 n=1,2,3,4,...

  • Standing Waves in Pipes:One end open, one end closed

    13

    Open end: anti-nodeClosed end: node

    Nodes (N) and anti-nodes (A) are separated by /4

    For the 1st normal mode: L=

    4We call this value :1

    1=4L

    AN

  • Standing Waves in Pipes:One end open, one end closed

    14

    Open end: anti-nodeClosed end: node

    Nodes (N) and anti-nodes (A) are separated by /4

    For the 2nd normal mode: L=3

    4We call this value :3

    3=4L

    3

    AN

    AN

  • Standing Waves in Pipes:One end open, one end closed

    15

    Open end: anti-nodeClosed end: node

    Nodes (N) and anti-nodes (A) are separated by /4

    For the 3rd normal mode: L=5

    4We call this value :5

    5=4L /5

    AN

    A ANN

  • Standing Waves in Pipes:One end open, one end closed

    16

    Open end: anti-nodeClosed end: node

    Nodes (N) and anti-nodes (A) are separated by /4

    In general for n = 1, 3, 5, 7 ....L=n

    4We call this value :n

    n=4L /n

    AN

    A ANN

    f n=v

    n=n

    v

    4L=nf 1 n=1,3,5,...

    AN

  • Notation

    17

    Standing Waves in (both-end) Open Pipes

    First Harmonic: n = 1Second Harmonic: n = 2

    Third Harmonic: n = 3

    f n=v

    n=n

    v

    2L=nf 1 n=1,2,3,4,...

    Standing Waves in one-end-Open one-end-Closed Pipes

    f n=v

    n=n

    v

    4L=nf 1 n=1,3,5,7,...

    First Harmonic: n = 1Third Harmonic: n = 3Fifth Harmonic: n = 5

    Only odd harmonics are present !

  • The PVC Didgeridoo

    18

    We want the fundamental frequency to be 80 Hz.

    One end of the pipe is effectively closed (at your mouth)

    f n=v

    n=n

    v

    4L=nf 1 n=1,3,5...

    L=v

    4f 1=

    343m.s1

    320 s1

    =1.1m

  • A PVC middle-C Flute

    19

    We want the fundamental frequency to be middle-C : 262 Hz.

    You blow across the hole: so the hole is actually open to the atmosphere. So this would be a two-end-open pipe.

    f 1=v

    2L

    L=v

    2f 1=

    343m.s1

    2262 s1=0.65m

  • Beats: The Sum of Two Sine Waves of Slightly Different Frequency

    y1t=Asin kx1 t

    1=2 f 1 2=2 f 2 =12=2 f 1 f 2

    y2 t =Asin kx2 t

    =A sinkxt

  • Beats: The Sum of Two Sine Waves of Slightly Different Frequency

    21

  • Beats: The Sum of Two Sine Waves of Slightly Different Frequency

    22

    y1t=Asin kx1 ty2 t =A sin kx2 t= A sin kx

    1t

    0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0

    1 0

    5

    5

    1 0

    0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0

    1 0

    5

    5

    1 0

    +

  • Beats: The Sum of Two Sine Waves of Slightly Different Frequency

    23

    y1t=Asin kx1 ty2 t =A sin kx2 t= A sin kx

    1t

    0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0

    2 0

    1 0

    1 0

    2 0

    =

  • How about Square Waves (instead of sines and cosines)?

    24

  • 25

    0.09 A1 sin25f 1t0.0027 A1 sin27f 1 t

  • Constructing a Square Wave from a sum of sines...

    26

    Harmonics :f 1 ,3 f 1 y t =A1 sin2 f 1t0.3A1 sin23f 1t

  • Constructing a Square Wave from a sum of sines...

    27

    Harmonics :f 1 ,3 f 1, 5f 1y t =A1 sin2 f 1t0.3A1 sin23f 1t

    0.09 A1 sin25f 1t

  • Constructing a Square Wave from a sum of sines...

    28

    Harmonics :f 1 ,3 f 1, 5f 1y t =A1 sin2 f 1t0.3A1 sin23f 1t

    0.09 A1 sin25f 1t0.0027 A1 sin27f 1 t

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