This formula for the quarter-wave resonance frequency of a bare (unloaded) secondary in grounded-base configuration over a good ground plane, is valid for coils in which the start of the winding is less than half the coil length above the ground plane, and is accurate for coils greater than 10cm in diameter and h/d greater than 1.5. Accuracy is around 2% average, with a peak error of around 4%.

Starting with:

 turns;
 h = length of secondary winding, metres;
 d = diameter of secondary - metres;
 b = height of winding start above ground - metres;
 awg = wire gauge, AWG;

Compute:

 x = h/d                                  (form factor)
 wd = 7.348e-3/pow(1.122932, awg-1)       (wire diameter - metres)
 sr = turns * wd/h                        (spacing ratio)

 fa = -94.6683*awg*awg*awg + 9000.55*awg*awg - 301175*awg + 3.64056e+6
 fs = 3.50662*sr*sr - 7.90171*sr + 5.83019
 fx = -0.000211179*x*x*x + 0.00557568*x*x + 0.0664809*x - 0.0153254
 t = fa * fs * fx/h/h
 s = -3.85188e-15*t*t*t + 1.17176e-8*t*t + 0.631829*t + 482.463

and finally,

 fb = log( b/h/0.2)                       (use the natural logarithm)
 Fres = s * (1.02 + fb/98.9065);          (Hertz)

The prediction is more accurate than estimates based on Nagaoka/Medhurst, particularly at large h/d ratios. The resulting frequency prediction should be considered an upper limit. The presence of nearby walls, ceiling, primary winding, strike rings, etc, will all reduce the actual frequency somewhat below this prediction. To take these factors into account, a program such as Terry Fritz's E-Tesla6 is required.

The inductance of a solenoid effective at its lowest self-resonant frequency differs from the low frequency (Nagaoka) inductance due to the non-uniform current distribution at resonance. The following two formulae may be used to calculate the effective series inductance,

Bare coil:

  Les =  Ldc * (pow( 8.0724 + 4.5129 * h/d, -0.8016) *
                pow( 0.0079 + 1.384 * b/h, 0.2623) *
                pow( 338.89 + 18.9111 * awg, 0.1493) *
                pow( 0.346 + 4.23 * h, 0.0232) *
                pow( 115.768 + 7.1 * sr, 0.0241) + 0.658311)

    average error = 1.3%, peak error = 2.8%

Toroided:

   Les =  Ldc * (pow( 4.6675 + 5.5509 * h/d, -0.5983) *
                 pow( -0.0147 + 0.9557 * b/h, 0.131) *
                 pow( 5.2204 + 0.145 * awg, 0.0703) *
                 pow( 3.94 + 0.9199 * h, 0.0487) *
                 pow( 5.41 + 80.798 * sr, 0.0114) *
                 pow( -4.1441 + 14.3397 * td/h, 0.1537) *
                 pow( 0.312 + 0.58 * tb/h, 0.088) + 0.686997)

    average error = 1.6%, peak error = 4.5%

where

 h = length of secondary winding, metres;
 d = diameter of secondary - metres;
 b = height of winding start above ground - metres;
 awg = wire gauge, AWG;
 wd = 7.348e-3/pow(1.122932, awg-1)       (wire diameter - metres)
 sr = spacing ratio = turns * wd/h
 td = toroid outer diameter, metres.
 tb = height of toroid plane above the top of the winding, metres.

An approximation to the Nagaoka inductance, accurate to 5 decimal places, was given by Lundin [1].

 a = radius, metres. 
 b = length, metres.
 N = number of turns

 MU = 4*3.14159*1.0E-7

 x = 4 * a*a/(b*b)
 y = 1/x
 c = ln(8*a/b)-0.5       (where Ln is the natural logarithm)

 f(t) = (1 + 0.383901*t + 0.017108*t*t)/(1+0.258952*t)
 g(t) = 0.093842*t + 0.002029*t*t - 0.000801*t*t*t

 If 2*a < b then
    Ldc = MU*N*N*a*a*PI/b * (f(x) - 8*a/(3*PI*b))   ...Henries.

 otherwise when 2*a > b

    Ldc = MU*N*N*a * (c*f(y) + g(y))   ...Henries.

Accurate is rather better than 1%.

[1] R. Lundin,
'A Handbook formula for the Inductance of a Single-Layer Circular Coil',
Proc IEEE, Vol 73, 1428-1429.

[2] R.G. Medhurst,
"HF Resistance and Self-Capacitance of Single-Layer Solenoids",
Wireless Engineer, Feb and March 1947.