188

G. Shailaja and N. Chari

5.

Ellington’s Theory (1999)

6.

Newton’s Theory

7.

Mass Flow Theory (1977, 2015)

8.

Deakin’s Theory (2010).

Theories

The mathematical basis for the above theories has been considered as below.

1.

GREENWALT’S THEORY (1962)

Greenwalt [2] in his Mechanical Oscillatory Theory considered biological flier as a

mechanical harmonic oscillator and calculated its wingbeat frequency. He framed a

differential equation and solved it in terms of frequency (νh) and wing length (l) of

the flier and accordingly obtained the following formula:

vhlⁿ = constant(k)

where the wing length (l) is measured in millimeters. In this analysis, wings of fliers

were considered as damped mechanical oscillators.

The exponent ‘n’ for ‘l’ is varying between 1 and 1.25. He showed the product to

be a constant with k = 3540 (obtained by trial and error method).

2.

CRAWFORD’S THEORY (1972)

Crawford [3] calculated wingbeat frequency of a flier by considering actual wing

swept area (Sw) and not the total wing disc area (Sd) of the flier. According to him

mass of the flier (M f ) and wing swept area (Sw) are the two factors which mainly

affect the frequency of the flier. His formula is as follows:

ϑ h = K ×

√M f

Sw

, Where K =

 g

4πρ ^{= 252.44}

M f = mass of the flier; Sw = wing swept area and is equal to

 π

3



∗radius²; ρ =

density of air and g = acceleration due to gravity.

This formula is applicable to small myogenic fliers such as mosquitoes where ϑ h

is high (500 and above).

3.

NORBERG’S THEORY (1990)

Norberg [4] in the book entitled “Zoophysiology” has considered upper and lower

side limits for calculating wing beat frequency. For geometrically similar animals,

wing beat frequency varies with the (−1/3)rd power of the body mass.

fw, max ∝M^−1/3