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ACOUSTICS WEEK 4 PHYSICS

Lecture on Acoustics and Air Columns

General Information

  • Presenter: University of Guelph - Physics

  • The lecture includes various topics such as harmonic vibrations, resonant lengths, sound intensity, and the Doppler Effect.

Harmonics and Vibrations

Wine Glass Vibration

  • Question: What harmonic did the wine glass vibrate in before it broke?

    • Options: A) 1st, B) 2nd, C) 3rd, D) 4th

Harmonics Defined

  • When visualizing a vibrating string or air column, nodes (points of no displacement) and antinodes (points of maximum displacement) are identified.

  • 4th Harmonic: Observations show that a setup with 4 antinodes between nodes indicates the 4th harmonic is present.

Resonant Lengths for Air Columns
Open at One End

  • The first resonant length (
    𝐿_1 ext{1})

    is given by:

    rac{1}{4} ext{ of a wavelength}(λ)

  • The second resonant length adds a half wavelength, summing to:


    • rac{3}{4} ext{ of a wavelength}(λ)

Open at Both Ends

  • For an air column open at both ends, the first harmonic involves:


    • rac{1}{2} ext{ wavelength}(λ)

    • The second harmonic results in:


    • 1 ext{ wavelength}(λ)

Fixed Wavelengths and Frequencies

  • Resonant Lengths: Change in length of an air column leads to adjustments in frequency and the harmonic produced. Resonant lengths have fixed wavelengths and frequencies.

  • Harmonics: They are multiples of the fundamental (first) harmonic. For example, the third harmonic frequency is 3 times that of the first harmonic,


    • 3 f_1

Sample Problems

Speed of Sound Calculation

  • Consider a 426.7 Hz tuning fork above a hollow PVC pipe submerged partially in water. Lengths of air column:

    • Resonates at lengths of:

      • 18.0 cm ± 0.5 cm and 58.0 cm ± 0.5 cm.

    • Formula:


      • v = f λ

  • For adjacent resonant lengths

dealt with as:


  • L{n+1} - Ln = rac{λ}{2}

Calculation Steps

  • Calculating the wavelength from given lengths:


    • L{n+1} = 58.0 ext{ cm} ± 0.5, Ln = 18.0 ext{ cm} ± 0.5 cm


    • L{n+1} - Ln = 58.0 - 18.0 = 40.0 cm

  • From error propagation:

    • -

ightarrow λ = 80.0 ± 1.4 cm

  • Assuming tuning fork error:


    • f = 426.7 Hz ± 0.05.

  • Calculated speed of sound:
    v = 341.3 m/s ± 1.75%

Finding Air Column Length Based on Harmonics

  • If an air column closed at one end resonates at the 3rd harmonic (336 Hz), calculate length: (speed of sound = 345 m/s)

    • 3rd harmonic implies:


    • L = rac{3λ}{4}

    • Solve for:


      • rac{λ = rac{v}{f}}{λ = rac{345}{336} = 1.025 ext{ m}}

    • Therefore,
      L = 0.77 m or 77.0 cm.

Additional Sample Problems on Harmonics

5th Harmonic Frequency Calculation

  • Calculation methods involve:

    1. Wavelength based -

    • use
      f5 = rac{v}{λ5}

    1. Multiples -

    • 5f_1 = 5(112 Hz) = 560 Hz

Second Harmonic Frequency of Air Column

  • For air columns open at one end, only the odd harmonics exist, complicating woodwind performance.

Frequency with Alterated Air Column Length

  • Example: A trumpet acts as an open-ended column:

    1. Length = 148.0 cm, speed of sound = 345 m/s leads to calculated frequencies.

    2. If length increases by 17.9 cm, the new frequency was found to be approx 104 Hz for the new length.

Sound Intensity and Level

Hearing Sensitivity Calculation

  • Audible range: 20 Hz to 20,000 Hz; intensity threshold = 1.0 x 10⁻¹² W/m².

- Calculated amplitude at hearing threshold for low frequency.

A = 2√(I/(ρvω²)) which leads to an astonishingly small amplitude, demonstrating human acoustic sensitivity.

Speaker Power Calculation

  • If an underwater speaker at 8.0 m producing intensity = 0.0187 W/m², the power produced is derived using
    I = P/A.

Beat Frequency Problem Scenario

  • Given: Tuning fork f = 512 Hz with beats produced between it and a guitar string.

    • If guitar string's pitch varies, beat frequencies changed from 5 to 4 Hz during adjustments. The calculation involved helps to reverse-engineer original string frequencies.

Understanding Doppler Effect

Change in Frequency Due to Motion

  • Use formula involving source and observer velocities:
    fR = f × rac{v + vR}{v - v_S}

Examples Implementing Doppler Effect

  • Ranks frequency detected changes based on the movement of source/receiver.

  • Sample problems involving bats include echolocation leveraging Doppler principles to discern movement of prey.

Sample Problems Involving Doppler Effect and Frequency Changes

  • Problems require recursive application of frequency shift equations under the rules of classic and modern Doppler principles.

Final Remarks: The detailed calculations and sample problems emphasize a deep understanding in physics regarding acoustics, resonances, and Doppler effects, relevant across numerous applications in both theoretical and practical domains of physics.