Volume of sound

[DOUG RICHARDS]

Jim,

Sorry about the delay in my response, but I'm in the middle of transferring 
to a new computer.  I thought I sent this to pianotech but sent it to 
myself instead...  Sometimes hightech is a headache.

Although I hate interspersing comments with previous posts, it is the only 
way to make any sense to all the questions. 

You wrote :
"The longer a given structure is the more potential energy can be stored in 
it (when the only parameter that changes is the length).  If you put a 
length of wire on the edge of a bench and secure it down right at the edge, 
with some varying length protruding. Two things will happen.
1. A deflection of equal measured distance at the end of the wire will 
cause wires of the same size, but different lengths, to vibrate at rates 
(pitches) but for varying lengths of time. 
2. A shorter wire will vibrate faster and for a shorter period of time than
will a longer wire."

My reply:
To make sure I understand you,  first assume the structure is a string 
clamped in a vice and the only change is the length of the "string beam".  
If you push on the end of the string so that it deflects .020", it will 
push back to show the potential energy it has stored.  If you let go of the 
string beam, it will vibrate at some frequency.
Now lets say we double the length of the string beam.  In item #1 of your 
post you say to deflect the same amount.  OK, done.  Realize that the force 
required to do this will be less because the beam stiffness is related to 
the length.  Since less force is required and deflection is the same, the 
potential energy of string beam#2 is less than the first case.  Also, when 
released it will vibrate at a new frequency because of the longer beam 
length.

1. If the same energy is imparted to each length of wire is the same, why 
will the longer wire vibrate longer ?
The energy is not the same for the case described above, so the energy 
release is not the same.  

2. Doesn't the longer wire have more potential energy in it than the 
shorter wire ?
Not if beam deflection is the controlling factor.  If you would have said 
to apply the same force to each string beam, then the potential energy 
would be the same, but the now you would see the deflection of the beams 
would be much different.

3.  Does the shorter wire stop vibrating quicker because it uses up energy 
in vibrating faster or because there was less potential energy there to 
begin with. ?
The same deflection for two different beams will give different potential 
energies for the two cases.

4. Would the same factors that apply to open ended wires apply to wire that 
is captive at both ends ?
Generally speaking, yes.  

5. Would this captive wire, say size #17, produce more energy at  15 inches
of length than it would at would at 16 inches of length?  Or vice versa? 
Same force, similar energy, but different deflections. 

6. Isn't the amount of energy potential in a wire determined by the length,
weight, and tension of the wire ?
Lets change the string beam/vise experiment.  Now, we have vices to clamp 
the string at both ends!  If the string has no tension when the vises are 
tightened, the string beam clamped at both ends will require some force to 
deflect a distance.  If the string has tension in it when clamped, the 
force - deflection rate will also depend on the amount of tension when 
clamped. 
BUT, back to the energy stuff.  If the force is the same in the tensioned 
and un-tensioned cases, the energy stored by the wire is the same but the 
string deflection will be different.  The different tensions will also mean 
different frequencies when released.  Also, the weight of the string is 
important, but does not mean more potential energy is stored. It will give 
the string inertia and will be one of the controlling factors in how fast 
the energy is released (steel vs copper wire). 

7.  Given the two lengths above,  15 and 16 inches, when either of these 
wires, equally tensioned, are struck by the same hammer with the same 
amount of force, say 65 grams, which will deliver the greatest amount of 
energy to the bridge?
Theoretically, it should be the same energy if the hammer velocity (read 
momentum transfer) is the same.  Remember though that since they are 
different lengths and were pre-tensioned the same, the frequency will be 
different.  Also, the energy transfer is related to structural dynamics, 
but up till now we were talking about static deflections and forces. 

8. Last Question !  If a 15 inch length of #17 wire  were pulled to 181 lbs 
of tension in a nine foot grand and in a  5' 6", grand would there not be 
the same amount of potential energy in each wire?  Where does the 
difference in volume as perceived and the sustain as perceived come from if 
not the sounding board? Let's not confuse the potential energy of the 
pre-tensioned string with the energy the hammer puts into the string.  The 
only way to get any of the pre-tensioned potential energy out of the string 
is to cut it. Then, the string flies!  Otherwise, the energy into the 
string from the hammer deflects the string and excites the bridge.  As far 
as the difference in a 9' grand and a 5' grand,  the longer string would 
have more potential energy if cut because, although the force is the same 
the "spring" is longer, providing more reserves to store energy.  But 
again, that potential energy is only available if the string is cut. 
The same hammer momentum should provide the same energy in the string 
whether it is short or long.  With that is structural dynamics, and now we 
have a real complex system.
I need to go for now but in parting, don't forget we haven't even gotten to 
soundboard sizes, bridge pin angles, downbearing....

Latter,

doug richards
drichard@qntm.com
San Jose, Ca.