A string of length L and mass M is under a tension F. One end of it is fixed in place at x = 0, while the other end is free to move up and down at x = L. (a) Starting from the standard form of y(x, t) for a harmonic standing wave, derive the wavelength of the normal modes on this string: λn = 4L/n. (b) State clearly, what values of n are allowed. (c) Obtain the normal mode frequencies v in terms of L, M, F and n and write the full wave functions in these terms (for arbitrary amplitude, A). (d) Sketch the first two allowed harmonics, indicating the positions of
A string of length L and mass M is under a tension F. One end of it is fixed in place at x = 0, while the other end is free to move up and down at x = L. (a) Starting from the standard form of y(x, t) for a harmonic standing wave, derive the wavelength of the normal modes on this string: λn = 4L/n. (b) State clearly, what values of n are allowed. (c) Obtain the normal mode frequencies v in terms of L, M, F and n and write the full wave functions in these terms (for arbitrary amplitude, A). (d) Sketch the first two allowed harmonics, indicating the positions of
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A string of length L and mass M is under a tension F. One end of it is fixed in place at x = 0,
while the other end is free to move up and down at x = L. (a) Starting from the standard form
of y(x, t) for a harmonic standing wave, derive the wavelength of the normal modes on this
string: 2n :4L/n. (b) State clearly, what values of n are allowed. (c) Obtain the normal mode
frequencies v in terms of L, M, F and n and write the full wave functions in these terms (for
arbitrary amplitude, A). (d) Sketch the first two allowed harmonics, indicating the positions of
all modes and antinodes. (e) With L = 2 m and M = 8 g, the string supports the standing waves
0.03 sin(3.25x) cos(162.5πt) for x and y in metres and t in seconds. Find (i) the
value of n for this harmonic; (ii) the tension in the string.
y(x, t)
=
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