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The physics of musical instruments I Neville H. Fletcher : Thomas D. Rossing. -2nd ed.p. em. Includes bibliographical references (p.

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The admittance of the piano bridge has a crucial effect on piano tone by coupling together the strings belonging to one note into a single dynamical system. In this paper, we first develop theoretical expressions that show how the rate of energy transmission to the bridge as a function of time (including the phenomena of beats and "aftersound") depends on bridge admittance, hammer irregularities, and the exact state in which the piano is tuned. We then present experimental data showing the effects of mutual string coupling on beats and aftersound, as well as the great importance of the two polarizations of the string motion. The function of the una corda pedal in controlling the aftersound is explained, and the stylistic possibilities of a split damper are pointed out. The way in which an excellent tuner can use fine tuning of the unisons to make the aftersound more uniform is discussed.PACS numbers: 43.75.Mn, 43.40. Cw INTRODUCTIONWith very few exceptions, and starting with the earliest instruments known, pianos have been built with more than one string per note; yet relatively little attention has been paid to this feature by acousticians. It was noted first by Martin and Ward, t and later in more detail by Kirk, •' that the strings comprising a single "triplet"--that is, the group struck by a single hammer-were not tuned in perfect unison by excellent tuners; and that, indeed, listeners often preferred the sound produced when there was a small discrepancy among the individual frequencies.Setting aside the subjective "preferences," whose cause and nature--and, hence, significance--is difficult to assess, the practice of artist tuners must be taken with extreme seriousness. We note that in this context "discrepancies from unison" refers to frequency differences observed when the strings are sounded individually, with the other members of the same triplet damped. These differences are not equal to the beat frequencies heard when the strings sound together, due to the coupled nature of the string vibrations.A number of years earlier, Martin s had reported on the characteristic "double decay" of piano tones, whereby the sound amplitude dies away with two distinct rates, breaking from an original "fast" decay to a later "slow" decay, or "aftersound." This feature is important in producing the characteristic piano tone. Martin did not at that time suggest any mechanism for the double decay, but in later work found that the phenomenon is connected with the presence of more than one string per note, and l{hat the amount of aftersound is affected by the exact manner in which the unisons are tuned. 4The suggestion that the phase relations among the ß strings play an important role appears to have been made first by Hundley, Martin, and Benioff, s and in more explicit form by Benade. 6 Benade points out that when three strings vibrate in phase, the motion of the bridge is three times what it would be if one string were vibrating alone; hence, the rate of energy loss of each string is tripled. He suggests that, after some ...

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Theory and experiment are presented for the drag exerted on electrons in a solid by a traveling ultrasonic wave. After a discussion of the reasons why the effect is generally very small, it is shown that under certain conditions in w-type germanium it may be quite appreciable, and that its size directly indicates the intervalley scattering rate. Experimental data are given for arsenicdoped germanium ranging in impurity content from 10 14 to 10 16 cm -3 , at temperatures from 20 to 160°K. These yield the absolute value of the "uniaxial" deformation potential constant (16 ev) and the intervalley scattering rate as a function of temperature and doping. The interpretation of the results ascribes intervalley scattering action both to phonons and to impurities. The phonon contribution yields the frequency of the (100) longitudinal phonon (6.6X 10 12 sec" 1 ) and the size of the appropriate coupling parameter ; the impurity contribution yields intervalley scattering cross sections for neutral and ionized donors as a function of temperature. The two cross sections are explained as, respectively, being due to exchange scattering events and to compound capture-reemission processes. Further analysis gives a lower limit to the valley-orbit splitting of the arsenic donor ground state (1.7X10 -3 ev). The experiment verifies that the conduction-band valleys lie on (111) axes at the Brillouin zone edge. Ultrasonic attenuation due to intervalley scattering is discussed and is shown to be too small to be easily measurable.

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