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Standing waves?

  1. Oct 21, 2003 #1
    hello i have got to these questions in my homework and have no idea how to start them, if anyone can point me in the correct direction i will be greatfull. these questions are to do with a guitar.

    the fundemental standing wave pattern shown produces a note of frequency 280hz

    is a rough picture down the page i think

    1)by placing a finger lightly at certain places on the string it is possible to produce standing waves with other specific frequences
    a)sketch one of these standing wave patterns
    b)state the frequency of the pattern you have drawn

    for this i thought i could just sketch a graph thing with like 2 anti nodes and one nodes to represent if i places my finger in the middle, and then that'd make the frequency double to 560hz, i think. but i'm not too sure at all

    2) the speed, v, of a transverse wave along a strectched string is given by

    v= √(T/μ)

    where T is the tension and μ is the mass per unit length of the string. show that the fundamental frequency is given by


    where L is the vibrating length of the string between the nut and bridge

    i'm not sure what it means to do, am i supposed to sub in the value i got in question one or something like that? grrr i've read the book but it doesn't give worked examples [which is how i usually learne things]

    3)assuming both L and μ remmain constant, calculate the new fundamental mode of vibration if the tension were halfed

    4)in practice μ , the mass per unit length, changes because the string contracts when the tension is reduced

    if the tension is halved, explain whether the mass per unit length will increase or decrease.

    i think if the tension halfs, the MPUL will increase

    this is really really hard alot of people in the class probably did not understand because the teacher hasn't really covered this yet, but i guess if i know what some of these questions mean i can have a go in the morning, thanks for any starters guys
    Last edited: Oct 21, 2003
  2. jcsd
  3. Oct 22, 2003 #2


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    Science Advisor

    Your picture should show a full "sine wave" (both above and below the axes) as opposed to the fundamental frequency which shows only half.
    I'll bet your book has a formula that relates wave length and frequency- something like "frequency* wavelength= wave speed". That is, the wavelength of a soundwave is the speed of sound divided by the frequency. Of course, the frequency of the soundwave is the same as the frequency of the wave on the string that produced it and the length of the string must be an integer multiple of the 1/2 the wave length (One wave is a full "sine curve". That is 0 at both ends AND the middle. Since the string is fastened at both ends, its motion MUST be 0 at each end- it must be either at an end or in the middle of a wave).
    i.e length= (n/2)*wavelength= (n/2)* speed of sound/frequency so, solving for frequency: frequency= ((n/2)*speed of sound)/length of string. You don't need to calculate all that. Just note that if you place your finger in the middle of the string, you reduce its length to 1/2 and so
    ((n/2)*speed of sound)/length becomes ((n/2)* speed of sound)/((1/2) length)= 2((n/2)*speed of sound/length)- halving the length of the string doubles the frequency: If the full string gives a frequency of 280hz, then placing your finger at the midpoint will give a frequency of 560hz as you say.

    You are given that the wave speed (on the string) is given by
    v= √(T/μ). Once again: frequency times wavelength= wave speed and L is an integer multiple of 1/2 of wavelength. In the case of the "fundamental frequency", that multiple is 1 (that's why it's fundamental!). Here you have (1/2)wavelength= 1*L or wavelength= 2L and so 2*L*frequency= v= √(T/μ). Solving for frequency will give you the formula you want.

    Replace T by (1/2)T in the formula above.

    Okay, if the tension halves (is reduced) the string contracts (is shorter) while the mass stays the same. "mass per unit length" is a fraction. The numerator stays the same while the denominator gets smaller: you are right- the fraction gets larger: then "MPUL" increases.
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