Normal Modes and Resonance Frequencies Submit My Answers Give Up Learning Goal T

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Normal Modes and Resonance Frequencies Submit My Answers Give Up Learning Goal To understand the concept of normal modes of oscillation and to derive some properties of normal modes of waves on a string Correct A normal mode of a closed system is an oscillation of the system in which all parts oscillate at a single frequency. In general there are an infinite number of such modes, each one with a distinctive frequency fi and associated pattern of oscillation The frequencies are the only frequencies at which the system can oscillate. If the string is excited at one of these resonance frequencies it will respond by oscillating in the pattern given b n(z, t), that is, with wavelength , associated with the fi at which it is excited. In quantum mechanics these frequencies are called the eigenfrequencies, which are equal to the energy dof that mode divided by Planck's constant h. In SI units, Planck's constant has the value 6.63 × 10-34 J . s Consider an example of a system with normal modes: a string of length L held fixed at both ends located at z 0 and z L. Assume that waves on this string propagate with speed v. The string extends in the x direction, and the waves are transverse with displacement along the y direction. Part E Find the three lowest normal mode frequencies fi , , and f3 Express the frequencies in terms of L, v, and any constants. List them in increasing order separated by commas In this problem, you will investigate the shape of the normal modes and then their frequency The normal modes of this system are products of trigonometric functions. (For linear systems, the time dependance of a normal mode is always sinusoidal, but the spatial dependence need not be.) Specifically, for this system a normal mode is described by Submit My Answers Give Up y, (x, t) = A, sin (2n sin(2nf,t). Provide Feedback Continue

Explanation / Answer

f1, f2, f3 = (1/2) (v/L), (v/L), (3/2) (v/L) or 0.5 v/L , v/L, 1.5 v/L

For the string, these frequencies are multiples of the lowest frequency. For this reason the lowest frequency is called the fundamental and the higher frequencies are called harmonics of the fundamental. In an acoustic piano, the highest audible normal frequencies for a given string can be a significant fraction of a semitone sharper than a simple integer multiple of the fundamental. Consequently, the fundamental frequencies of the lower notes are deliberately tuned a bit flat so that their higher partials are closer in frequency to the higher notes.

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