Supplements for Chapter 19
Highly recommended: Physics of the Piano, by Nicholas J. Giordano of Purdue University
This paper, by Prof. Giordano, is a concise and fairly non-techical (except for repeated reference to mechanical impedance, which we know about!) scholarly article on soundboard physics. Several impedance graphs are given. The paper would be an excellent read for anyone planning to look into soundboards for a project.
Excellent and quite well organized, accessible, comprehensive web book on the physics of the piano:
Another valuable resource is a free PDF file, Acoustics of the Piano, by Juan Jose Burred, that among other things contains a lucid account of the issue of double decay of sound from piano strings. After a strike with the hammer, the sound from most notes decays rapidly at first ("immediate sound"), then more slowly ("resonance sound"). The explanation was apparently not coffered until 1973, with Gabriel Weinreich’s article “Coupled piano strings,” in the Journal of the Acoustical Society of America. In chapter 19 we state: "As the three strings drift out of phase (that they do by design), they no longer add amplitudes constructively at the bridge, reducing power flow to the bridge and leading to a longer sustained sound. " We continue the issue from there. The strings are acting in phase at first, since the hammer hits them all the same way. At first, they pull on the bridge like so many sailors raising an anchor. Then they drift out of phase, husbanding their energy since sometimes one string is "pushing" while the other is "pulling"; they cancel and thus fail to dump much energy at that moment.
See the hammer action animated gif by Wayne Land here.
This site, although partly commercial, has some gems, and is well worth rummaging through if you love the piano.
Piano soundboard physics
Very interesting animations of a grand piano soundboard responding to various piano keys being struck are seen in the animation below. Unfortunately not very well labeled, it nontheless shows that the motion of the soundboard is nothing like that of a single mode, as befits the drive and its partials, and the strongly overlapping resonances of the soundboard. Notice it is clamped everywhere along the edge. Short transients give way to other oscillations after the key strike,
See this site for image below (a laser interferometry image showing motion of the soundboard in two different modes, analogous to Chladni patterns (see chapter 15). and other interesting research of Prof. Moore of Rollins College:
Below, grab the hammer with your mouse and move it over to see how the partials are excited by a strike there. Delete the web page when you are done, it seems the applet uses cycles even when you stop adjusting it - a bug in CDF Player?