### Input impedance

An example of measured input impedance and radiated sound level curves for an upright piano assembly consisting of the wooden frame, soundboard and metal plate, strung and tuned to normal pitch, is shown in Fig. 5. It can be seen that the impedance curve is characterized by prominent tall peaks and valleys in the lower part, superimposed on a continual decrease from 100 Hz to the highest frequencies. For those readers who are particularly interested in engineering units we may add that the impedance reaches values around 1000 kg/s at 100 Hz and 10 kg/s at 10 000 Hz.

Fig. 5. Acoustic measurements on an assembled unit for an upright piano consisting of the wooden frame, soundboard, metal plate and strings, tuned to normal pitch (MP 7). Input impedance (top), phase angle (middle), and sound level (bottom). The 0 dB level for impedance corresponds to 1000 kg/s.

All measuring points are characterized by a uniform decrease in impedance above 1000 Hz at a rate of about 6 dB per octave, without any prominent resonance peaks. The phase angle is close to -90º, which means that the input impedance is dominated in this region by the resiliency (springiness) of the soundboard. We find a similar situation in the extreme low frequency region (below 100 Hz); here too the phase angle is almost -90º and no resonance peaks show up (the peaks below 50 Hz are not generated by the soundboard but by the stands and crossbeam holding the electrodynamic shaker).

The corresponding sound level curve (Fig. 5, bottom) demonstrates that the soundboard is incapable of radiating sound below 100 Hz. Above this frequency, individual resonances, which may reinforce the sound radiation, become noticeable. At approximately 1000 Hz, the upper limit of the range of favored sound radiation is reached. Above this frequency the sound level decreases steadily.

The connection between a soundboard resonance and the corresponding sound radiation can be demonstrated in a simple way. By knocking on the soundboard with a finger (preferably close to measuring point 9), we hear a thump sound with a definite pitch. The spectrum of this thump sound shows a strong peak at 102 Hz (see Fig. 6, top). This peak, which indicates the pitch of the thump, is due to the fundamental resonance of the soundboard. The same resonance can be seen as a minimum in the input impedance curve at the corresponding frequency (Fig. 6, middle). Also the following maxima (resonances) at 135, 150 and 165 Hz in the sound level curve correspond to minima in the impedance curve.

Fig. 6. Tap tone and influence of mass loading. The soundboard is tapped at the treble bridge (MP 9). Sound spectrum (top), input impedance (middle), and input impedance with a mass load (550 g) close to the measuring point (bottom). The vertical lines indicate the frequency of the fundamental resonance of the soundboard.

The input impedance curves can also be used to demonstrate changes in the properties of a soundboard. The resonances of the soundboard are controlled by the distribution of mass, stiffness, and damping. By changing the mass distribution of the soundboard in Fig. 6 with an additional mass at the treble bridge, we obtain a modified impedance curve (Fig. 6, bottom). The resonances below 100 Hz have not changed (properties of the shaker and beams), whereas the fundamental soundboard resonance has been shifted down from 102 Hz to 95 Hz. This is also easily heard by knocking on the soundboard.

This lecture is one of Five lectures on the Acoustics of the piano