The sound created by a vibrating string, clamped at both ends, is governed by a partial differential equation that dictates how the string is allowed to move over space and time:
∂2u / ∂t2 = c2 ∂2u / ∂x2
where u is the deflection of the string, t is time, x is the linear distance along the string, and c is speed of the wave.
Following these assumptions:
1. The string has constant mass along its length.
2. The string is infinitescimally thin.
3. The string is perfectly elastic.
4. The ends of the string are immobile where clamped.
5. Gravity can be ignored (this is justifiable, because the force due to tension should be much much higher than the force due to weight)
With these assumptions and boundary conditions, we can solve the equation using Fourier Series:
u ( x, t ) = Σ An cos ( c n π t / L ) sin ( n π x / L )
This is a simple harmonic oscillator with frequencies given by
f0 = ( 1 / 2 L ) √ ( T / μ )
fn = n f0
where f0 is the fundamental frequency (or lowest frequency), fn is the nth harmonic, L is the length of the string, T is the force of tension, and μ is the mass per unit length.
If you measure all quantities in SI units, there should be no conversion necessary. Measure the frequency in hertz, the length in meters, the mass per unit length in kilograms per meter, the tension in newtons, etc. If you use D'Addario strings, you can get a chart that will tell you the mass per unit length of the string by product here.
While you may see this as a pedagogical exercise in classroom physics, there is another application of this that I find intriguing - inharmonicity.
So, you made five assumptions, what if you start to take back those assumptions?
Real strings have some imperfections, which cause the harmonic overtones to be "off" by a tine amount. This "offness" is called inharmonicity. Inharmonicity is defined as the measure of departure of an overtone frequency from that of an ideal harmonic. Real overtones are given by the equation
fn = n f0 ( 1 + G )
where G is the coefficient of inharmonicity.
It may also be convenient for some to express inharmonicity in terms of "cents" or hundreths of a semitone by the conversion
1 + G = exp ( cents / 1731 )
although a typically acceptable approximation is that the cents are 1731 times G.
Obviously, not all strings have constant mass along their length. Many strings are one type of strong wire wrapped with a heavier wire (for instance bronze wound or nickel wound strings). I will make the argument that the average mass per unit length of these strings is close enough, because the windings are very small in comparison to the length of the string; however, you may note that if the string length becomes exceptionally short (or if the windings become very big, and the string becomes very short), this may begin to become an issue. Some bass strings are tapered on the gear-end to accomodate installation. As long as the taper does not occur anywhere between the bridge and the nut of the instrument, there is no use considering this a problem; however, some strings do taper on the ball end, which will cause an issue. When that is the case, there is an inharmonicity, which can be calculated as:
G = ∫L .. 0 (μ/μ0 - 1) sin2 ( n π x / L ) dx / L
Every string has a thickness and an elastic limit, so assumptions two and three are not very good assumptions. This leads to some inharmonicity of the overtones. Unlike variable mass, this inharmonicity is always present, although usually the amount is so incredibly small that it makes no difference in audible tone. However, with guitarists the last decade or so tuning very low, using very heavy gauge strings, tuning lower and lower, and using heavier and heavier strings, this inharmonicity plays a very observable and measurable role in the tone of the instrument, which may be a driving factor in the desire to tune low. The inharmonicity, which depends on the diameter of the string "d," and Young's modulus of elasticity "Y," is given by
G = ( n2 π3 d4 Y ) / ( 128 L2 T )
Clamping the string to a yielding support will cause inharmonicity. Again, this should be very small, particularly with fixed-bridge instruments, because the flexibility of the bridge and nut are so small in comparison with the size of the string; however, with very loose tremolo bridges, like those used to create "flutter" effects, there will be some possibly noticeable inharmonicity. For a support yielding with elastic spring constant k,
G = ( 4 μ L ) / ( 4 π2 n2 M - k / f02 )
As for the fifth assumption, again, I believe that this is justifiable in any circumstances. If strings of an extremely dense material were made, then the assumption may cause some trouble for our model.