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How would one go about finding the sound field produced by a vibrating string? I know how to find the vibrations of the string itself, by solving the wave equation, with boundary conditions that correspond to a string thats held fixed at both ends, and an initial condition that corresponds to some stimulus of the string. This question is not about how to solve this PDE, but concerns what an observer would hear at an arbitrary position close to the string.

Let's assume the string is a length L, along the y axis, and v is the speed of sound at STP. The observer is at a distance d, perpendicular to the center of the string, along the x axis. Can I divide the string up into small segments dl, and consider sound as spherical waves radiating from each of these positions?

Thanks in advance for help with this question!

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    $\begingroup$ "Can I divide the string up into small segments dl, and consider sound as spherical waves radiating from each of these positions?" - No. an infinitely thin string would not engage the air and would produce no sound. Please note that sound in musical string instruments is not produced by strings, as is evident by comparing acoustic and electric guitars (with no amplification). $\endgroup$
    – safesphere
    Nov 14, 2019 at 19:11
  • $\begingroup$ The movement of the string still produces a weak field of its own. It is a fair question. But you are correct about the construction of the guitar and that the body of an acoustic is where the sound is generated. $\endgroup$
    – user196418
    Nov 14, 2019 at 21:33

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You are asking about solving a PDE just not the one you think (I think). You know the solution to the vibrating string. Not the string acts as a vibrating source pushing elements of air back and forth creating a pressure wave in the air (or other fluid). You absolutely can consider each tiny element of the string as a source and connect them all by integrating over the entire string. However they are not mono-pole sources, not really a spherical wave. This would be appropriate if you had an expanding volume of air as a source (a true mono-pole). A moving point source acts as a dipole source and this couples to the derivative of the green's function (if I recall correctly, I am not really checking my work here). So you need to choose the correct form of the green's function and integrate over the string and in theory that should give you the acoustic field created by the vibrating string. This is in fact how to solve the wave equation, a PDE, hence the nature of my introductory comment. I'd recommend looking in an acoustics text like the one by Allan Pierce, a true classic. It should have an approach to the type of problem you are trying to solve.

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  • $\begingroup$ Awesome, thank you for this explanation! I'm going to look up the sound field of a dipole source in the book you mentioned and then use that to construct the string. $\endgroup$
    – Evan Diehl
    Nov 14, 2019 at 20:22
  • $\begingroup$ @EvanDiehl, I hope it helps. Are you really interested in the sound field produced by the string or that of a stringed instrument? As the comment above correctly indicates the string usually initiate vibration of some other component which causes sound. $\endgroup$
    – user196418
    Nov 14, 2019 at 21:35
  • $\begingroup$ I am ultimately interested in learning about the sound fields produced by a variety of acoustic instruments, but to achieve that goal I am working towards it one step at a time. I'm in a course on PDE's and Fourier series, but I'm also a musician so I'm sitting in class thinking to myself, okay solving the wave equation is interesting, but I what to make some sounds with this new information! $\endgroup$
    – Evan Diehl
    Nov 15, 2019 at 5:04
  • $\begingroup$ I started with a vibrating string because I thought if I can get that down, I'll then eventually move on to objects like plates, circular membranes, and beams. Although based on your answer I'm realizing I should start with even more primitive sound generators like monopole and dipole sources. I don't expect the results to be very interesting until I consider the body of the instrument but I'll get to that point eventually. That book you recommended is exactly what I needed! $\endgroup$
    – Evan Diehl
    Nov 15, 2019 at 5:04
  • $\begingroup$ You should look into text books by fletcher and rossing. They have texts on the physics of musical instruments and cover plates with boundaries $\endgroup$
    – user196418
    Nov 15, 2019 at 11:16

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