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Chapter 10 (Hall)

Chapter 10 (Hall). Piano and Guitar Strings. Comparing sounds from a piano and a drum. The piano, with its struck strings, sounds very different from drums.

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Chapter 10 (Hall)

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  1. Chapter 10 (Hall) Piano and Guitar Strings PHY 1071

  2. Comparing sounds from a piano and a drum • The piano, with its struck strings, sounds very different from drums. • Piano sounds have clear, sustained pitch and provide all the necessary melodic, harmonic, and rhythmic elements to make a complete piece of music. • What underlying acoustical characteristics of the piano set it apart in this way? PHY 1071

  3. Answer: Special property of long, thin strings • Long, thin strings have a special property: Their natural mode frequencies form a harmonic series. • This is what makes their tone more musical than that of other percussion instruments. PHY 1071

  4. Outline • Natural modes of a thin string • Vibration recipes for plucked strings PHY 1071

  5. Natural modes of a thin string • Natural modes: Each of the special patterns of vibration that gives simple harmonic motion is called a natural mode. • These natural mode vibrations are sometimes called standing waves. • Natural mode frequency: Each natural mode has its own characteristic frequency, called the natural mode frequency. • Nodes N and antinodes A PHY 1071

  6. Natural mode frequencies • About each natural mode: the nth natural mode fits n “loops” (vibrating sections between adjacent nodes) into the available string length L. • About nodes: The distance between two adjacent nodes (or each loop) is half a wavelength. • So, we have n(1/2)n = L and n =2L/n. • Natural mode frequency fn = n(vt/2L), n = 1,2,3…, where • vt = (T/)1/2 is the velocity of transverse waves on the string. • T: the tension in the string; : the string's mass per unit length. • Fundamental frequency of the string: f1 = (1/2L)(T/)1/2. PHY 1071

  7. Most remarkable thing about the natural modes of a thin string • The natural mode frequencies of a thin string form a harmonic series. • All these modes have intimate musical relationships and cooperate in establishing a sense of pitch – the presence of definite pitch for a piano. • The pitch corresponds to the fundamental frequency f1. PHY 1071

  8. Examples • Suppose you have a string of length L = 0.5 m on which waves travel at speed vt = 150 m/s. What is the wavelength 6 and frequency f6 of the 6th mode of this string? • Suppose that this string vibrates in mode 4. Sketch its appearance at several successive moments during a cycle, also indicating direction of motion. PHY 1071

  9. Vibration recipes for plucked strings • The recipe of vibrations (Fourier spectrum) is determined by the place and manner in which we first excite (strike or pluck) the string. • The excitation of any mode is proportional to how much motion that mode has at the plucking point, meaning in particular that any mode with a node at the plucking point is omitted from the recipe. PHY 1071

  10. Example: plucking at a point located at L/5. Modes 5, 10, 15,… are absent because they have nodes at the plucking point. PHY 1071

  11. Homework • Ch. 10 (Hall), P. 195, Exercises: #1, 2, 3, 4, 7, 20. PHY 1071

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