How and in which direction the piano sound is projected or radiated from the instrument is an indication of how the soundboard is responding to the vibrations from the string and bridge.

The soundboard transforms the mechanical vibrations into radiated sound. As a first approximation, the soundboard acts like a large diaphragm clamped around its edge. Like all diaphragms, the soundboard exhibits a series of resonances, the individual intensities being determined by the point of excitation.²²⁵

Many instruments have an additional resonator whose function is to convert more efficiently the oscillation of the primary vibrating element into sound vibrations of the surrounding air and to giver the tone its final timbre. ²²⁶

The resonating system of a piano transforms the energy of the vibrating string into a composite sound wave that radiates the qualities of tone into the atmosphere. The treble and bass bridges transmit the initial string vibrations to the plate and the ribs disperse the vibrational energy throughout the broader area which radiates the sound out of the instrument. The impedance of the wood in the bridges, plate and ribs influences the efficiency and quality of the transmission of vibrational energy.

The soundboard is formed in such a way that the grain follows the general direction of the treble bridge. Sound travels about twice as fast with the grain as against it, and the modulus of elasticity of spruce is twenty times greater with the grain than against it. This results in the sound not being delivered uniformly to the entire soundboard and to compensate for the even delivery ribs of the same material are attached at fixed intervals to the underside of the soundboard.²²⁷

The sound radiation pattern of a piano is largely determined by the shape of the soundboard and the modal shapes of the various modes in which it vibrates. ²²⁸

String vibration and the soundboard.

….. the initial impact on the string occurs in a direction perpendicular to the sound board; in this direction the sound board is in a position to extract energy from the string in relatively strong measure…..
In addition, string vibrations parallel to the sound board are formed, though muchweaker. Since the sound board presents a much higher impedance for transmitting such vibrations, this energy transmission process is much slower. The radiated sound field includes a superposition of these two different forms of vibration.
²²⁹

We have only to think of each tiny moving patch of the soundboard as a small pump to realise that the complete board acts as a vast multitude of simple sources that run not necessarily in step and have many different amplitudes. ²³⁰

This research investigated the vibrational characteristics of the soundboards of each piano playing the same sound (frequency) at precisely the same dynamic (volume).

²²⁵ Klaus Wogram, “Five Lectures on The Acoustics Of The Piano”, The Strings and Soundboard Chapter 5 (Royal Swedish Academy of Music, 1990), 2.
^{226     6}Roederer,The Physics and Psychophysics of Music (2008) p.2
^{227     2}Foulcher, T. Fundamental Notes for Piano Technicians (1981) pp.24-25
^{228     6}Fletcher, The Physics Of Musical Instruments2nd Ed.(1999)p.392
^{229     9}Weinreich, G. Coupled piano strings. source: 7Meyer Chpt 3:”Tonal Characteristics of Musical Instruments: The Piano: Time Structure”in Acoustics and the Performance of Music,107-108.
^{230     4}Benade,Fundamentals of Musical Acoustics, (1990) p.347