1. Dr. Jie ZouPHY 10711 Chapter 10 (Hall) Piano and Guitar Strings

2. Dr. Jie ZouPHY 10712 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?

3. Dr. Jie ZouPHY 10713 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.

4. Dr. Jie ZouPHY 10714 Outline Natural modes of a thin string Vibration recipes for plucked strings

5. Dr. Jie ZouPHY 10715 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

6. Dr. Jie ZouPHY 10716 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 f n = n(v t /2L), n = 1,2,3…, where v t = (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: f 1 = (1/2L)(T/ ) 1/2.

7. Dr. Jie ZouPHY 10717 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 f 1.

8. Dr. Jie ZouPHY 10718 Examples Suppose you have a string of length L = 0.5 m on which waves travel at speed v t = 150 m/s. What is the wavelength 6 and frequency f 6 of the 6 th 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.

9. Dr. Jie ZouPHY 10719 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.

10. Dr. Jie ZouPHY 107110 Example: plucking at a point located at L/5. Modes 5, 10, 15,… are absent because they have nodes at the plucking point.

11. Dr. Jie ZouPHY 107111 Homework Ch. 10 (Hall), P. 195, Exercises: #1, 2, 3, 4, 7, 20.