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Standing Waves On Strings: Harmonic and Frequency Problem

  1. Jan 25, 2017 #1
    1. The problem statement, all variables and given/known data
    String A is stretched between two clamps separated by distance L. String B, with the same linear density and under the same tension as string A. String B is stretched between two clamps separated by distance 4L. Consider the first eight harmonics of string B. For which of these eight harmonics of B (if any) does the frequency match the frequency of (a) A’s first harmonic, (b) A’s second harmonic, and (c) A’s third harmonic?

    Not sure if I correctly solved the problem (hopefully I did =D). Just need someone to check over my work =D. Thanks!

    2. Relevant equations

    ν = √(T/μ)

    ν = ƒλ

    L = nλ/2

    ν: Velocity
    T: Tension
    μ: Linear Density
    ƒ: Frequency
    λ: Wavelength
    L: Length
    n: nth Harmonic


    3. The attempt at a solution

    Since the tension and linear density on both strings are equal, the velocity is also equal.

    Next I solved for the frequency of system B:

    ν = ƒλ
    ƒ = ν/λ → 1

    L = nλ/2, since L = 4L

    λ = 8L/n → 2

    Subbing 2 → 1

    ƒ = νn/8L

    --------------------------------------------------------------------------------
    First Harmonic of String A:


    L= λ/2 ⇒ λ = 2L
    ƒ=ν/λ ⇒ ƒa1 = ν/2L

    ƒa1 = ƒb

    ν/2L = νn/8L
    n = 4 (between 1-8)

    Second Harmonic of String A:

    L= λ
    ƒ=ν/λ ⇒ ƒa2 = ν/L

    ƒa2 = ƒb

    ν/L = νn/8L
    n = 8 (between 1-8)

    Third Harmonic of String A:

    L= 3λ/2 ⇒ λ = 2L/3
    ƒ=ν/λ ⇒ ƒa3 = 3ν/2L

    ƒa3 = ƒb

    3ν/2L = νn/8L
    n = 12 (not between 1-8)
     
  2. jcsd
  3. Jan 25, 2017 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Hello. Welcome to PF!

    Your work looks correct.

    A similar approach is to note that the harmonic frequencies of A are ##f_A = \frac{n_ {_A} \ v}{2L}## while those of B are ##f_B = \frac{n_{_B} \ v}{8L}##.

    Show that ##f_A =f_B## implies ##n_{_B} = 4n_{_A}##. Then let ##n_{_A}= 1## for the first harmonic of A, etc.
     
  4. Jan 25, 2017 #3
    Thanks for the help! I will try the question both ways =D.
     
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