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he wave equation of a stretched string can be written as, J²y μ J²y дх2 Tat² (a) Define each variable and identify what the wave speed is on the string? (b) The most general solution, = y(x, t) = A cos(kx − wt) + B sin(kx — wt) show that is is a solution and that for a point of fixed x on the string that that particle experiences simple harmonic motion. Sketch for this particle how y, y and y vary with time. = (c) The two ends of the string are now fixed such that y 0 when x = 0 and x = L. With these boundary conditions, what are the allowed vibrations on the string? Sketch the first three modes and list their \, v, k and w. (d) The highest and lowest frequencies on a piano are 4186.01 Hz and 27.5Hz, given that piano string has a typical tension of 700N and assuming the length of the string is 0.5m, what is the range of the in the piano string? μ

he wave equation of a stretched string can be written as, J²y μ J²y дх2 Tat² (a) Define each variable and identify what the wave speed is on the string? (b) The most general solution, = y(x, t) = A cos(kx − wt) + B sin(kx — wt) show that is is a solution and that for a point of fixed x on the string that that particle experiences simple harmonic motion. Sketch for this particle how y, y and y vary with time. = (c) The two ends of the string are now fixed such that y 0 when x = 0 and x = L. With these boundary conditions, what are the allowed vibrations on the string? Sketch the first three modes and list their \, v, k and w. (d) The highest and lowest frequencies on a piano are 4186.01 Hz and 27.5Hz, given that piano string has a typical tension of 700N and assuming the length of the string is 0.5m, what is the range of the in the piano string? μ

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The wave equation of a stretched string can be written as, a²y μου əx² Tat² (a) Define each variable and identify what the wave speed is on the string? (b) The most general solution, y(x, t) = = A cos(kx - wt) + B sin(kx − wt) show that is is a solution and that for a point of fixed x on the string that that particle experiences simple harmonic motion. Sketch for this particle how y, y and y vary with time. 0 and x = L. (c) The two ends of the string are now fixed such that y With these boundary conditions, what are the allowed vibrations on the string? Sketch the first three modes and list their X, v, k and w. = 0 when x = (d) The highest and lowest frequencies on a piano are 4186.01 Hz and 27.5Hz, given that piano string has a typical tension of 700N and assuming the length of the string is 0.5m, what is the range of the н in the piano string? (e) Are these values physically reasonable? Is our approximation of constant tension and string length for all piano strings a good one?
Transcribed Image Text:The wave equation of a stretched string can be written as, a²y μου əx² Tat² (a) Define each variable and identify what the wave speed is on the string? (b) The most general solution, y(x, t) = = A cos(kx - wt) + B sin(kx − wt) show that is is a solution and that for a point of fixed x on the string that that particle experiences simple harmonic motion. Sketch for this particle how y, y and y vary with time. 0 and x = L. (c) The two ends of the string are now fixed such that y With these boundary conditions, what are the allowed vibrations on the string? Sketch the first three modes and list their X, v, k and w. = 0 when x = (d) The highest and lowest frequencies on a piano are 4186.01 Hz and 27.5Hz, given that piano string has a typical tension of 700N and assuming the length of the string is 0.5m, what is the range of the н in the piano string? (e) Are these values physically reasonable? Is our approximation of constant tension and string length for all piano strings a good one?
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