Need help understanding longitudinal and transversal string motion.

Hello Jake,

I would like to respond to your questions about transverse (flexure) vibrational modes, and longitudinal (back-and-forth rippling) vibrational modes of piano strings and how they relate to especially bass strings.

Here are your original questions:
Does the transverse vibration mainly function to create the fundamental, while the longitudinal vibration creates both the fundamental and the other partials? Is the transversal vibration responsible for the higher amplitude of the fundamental? We hear the freq from both the transverse and the longitudinal vibrations, so the amplitude is summed? Is that why the bass strings, unable to vibrate much transversely, have a fundamental at a lower amplitude?

In longitudinal modes of vibration, energy propagates lengthwise along the string (as periodic compressions of the string material) without sidewise (transverse) motion of the string. Longitudinal and transverse vibrations of a piano string can occur simultaneously. However, the lowest-frequency longitudinal mode of a piano string is always more than TEN TIMES THE FREQUENCY of the lowest-frequency transverse mode.

The longitudinal "ripple" is a property of the steel wire strung at a certain tension.  The reason why the frequency is up to ten times higher than the transverse vibrational frequency, has to do with the strength of the iron bonds in the steel wire.

Piano technicians are able to experience the importance of the longitudinal mode when they install new bass strings on a piano;  they sometimes notice that the string sounds better when it is tuned to the wrong fundamental frequency! After some study it became apparent that the reason had to do with the longitudinal mode. The first longitudinal mode of a piano string normally occurs at a frequency somewhere in the range between 3 octaves plus a fifth and 4 octaves plus a third above the "normal" fundamental transverse frequency of a string. This range is determined by certain design constraints related to the properties of piano wire that are common to all present-day pianos.

A piano tuner only tunes the transverse, or flexural, modes of the strings by changing the tension of the strings as he turns the tuning pins. A piano tuner can do nothing to affect the frequency of the longitudinal mode because turning the tuning pins doesn't change it. The longitudinal frequency of a plain steel string in a piano can be changed only by altering its speaking length. 

IN TERMS OF LONGITUDINAL FREQUENCY OF PIANO STINGS, THE LONGITUDINAL FREQUENCY OF A PLAIN, UN-WRAPPED STEEL STRING CAN BE CHANGED ONLY BY ALTERING ITS SPEAKING LENGTH.  In the case of WRAPPED PIANO STRINGS, the longitudinal mode can be tuned only in two ways: either by changing the speaking length or by changing the weight of the wrapping wire in relation to the weight of the core wire OR BY CHANGING THE WEIGHT OF THE COPPER WRAPPING WIRE IN RELATION TO THE MASS OF THE STEEL CORE WIRE.

So, the tuning of the longitudinal mode is established, either deliberately or accidentally, by the designer of the piano; and, as a practical matter, it cannot be changed after the piano has been built. IN TERMS OF PIANOTEQ SOFTWARE, IT IS THEORETICALLY POSSIBLE TO ALTER THE "WEIGHT" OF THE WRAPPINGS ON BASS STRINGS TO ALTER THE LONGITUDINAL FREQUENCIES.

In designing a piano nowadays, it is possible to tune the longitudinal modes of its strings to those frequencies that will make the piano sound best. The longitudinal mode is important in determining the tone color of the bass and tenor regions of the piano. The longitudinal mode creates a formant-like emphasis in the tone at its own frequency, with the result that some tunings of the longitudinal mode sound much better than others. In particular, it is desirable to have the longitudinal mode tuned so that it blends harmoniously with the tone from the transverse modes. This can be achieved by careful and deliberate choices in the design of the strings and scale of the instrument.

The above discussion was somewhat technical, but I hope it helps answer some of your questions.

Cheers,

Joe <jcfelice88keys>

Joe:

Thanks for the links - very interesting - and enlightening.

Now I know why some of the lowest notes on my Yamaha G2E sounded odd.

I wonder how many piano techs know of this phenomenon?

While the piano was on warranty, they suggested trying changing a few strings, but they couldn't guarantee it would solve the problem.

There's far more to piano design and construction than most are aware of.

Glenn

I would be interested in a thread that shares these kinds of articles.  They are quite interesting.

Hi Jake,

let me just add my 2c to this. It's an interesting discussion of a complex problem as you say yourself.

(It is really useful to look at the Five lectures on the acoustics of the piano found here: http://www.speech.kth.se/music/5_lectures/ if you're interested in these topics.)

Regarding the coupling of the strings to the soundboard: the main factor is the impedance of the coupling between the two, in other words how easy it is for the string to excite the motion of the soundboard.
The perpendicular motion has a low impedance, so it can easily excite the soundboard, but this excitation will decay quickly as the energy of that mode is transferred from the string to the soundboard.
The parallel mode has high impedance, it starts out with low amplitude but does not decay quickly.
This makes one of the characteristics of the piano tone: the loud early tone with fast decay plus the second tone, which takes over later and has a very long decay time.

About the localization of the oscillations on the soundboard: The way I understand it every excitation will essentially move the entire soundboard. This is most effective when you hit a resonance. Every resonance of the board has a very special distribution of it's oscillation maxima and minima - so it's more a soundboard property where a particular frequency can in turn be transmitted to air waves. This may be different for every tone.

These resonances also play a major role in the impedance for individual strings: If a string is coupled to the soundboard at a location, where it's own frequency has a maximum of the oscillation of the soundboard, it is very easily transmitted and may sound much louder than a neighboring string, which happens to sit at an oscillation node of its own frequency.

This thread was an ah-ha moment, in relation to tuning/intonating 12 string guitars. I had forgotten about longitudinal modes in strings, and have been puzzled why my 12 string sounded better with the lowest string detuned. (I use a software strobe tuner that analyzes the signal to 0.1 cent, and can also measure the 2nd and 3rd harmonic frequencies). I wonder how many guitar techs (and lutiers) know of this phenomenon?

I don't know about the availabilty of piano strings, but guitarists have a healthy choice, especially in wound strings - different thickness of core, different shape of core, different thickness of winding, different alloys, etc. Clearly, with so many different instruments around, there cannot be such as thing as a "best" (-sounding) guitar string, only best on a particular guitar.

More closely on topic, I prefer not to think of types of vibration being transmitted within the instrument. What is transmitted is energy, insofar as the arrangement and coupling of the parts allows. The instrument's parts have their own preferential modes of vibration. It is also possible for  energy to be transmitted "backwards", i.e. from soundboard to string. This is how sympathetic resonances arise.

The same reasoning applies by analogy to electronic audio equipment -- and with analogous effects: a loudspeaker will try to drive the amplifier thru back-emf; the output stages of an amplifier will try to drive the power supply. If interaction occurs, then the desired transfer of energy will suffer from defects - distortion of the audio signal. Fortunately, some amp techs know of this phenomenon!