When side-walls of a rectangular tank, which is filled with water, are vibrating, the inside water will also make a vibratory motion. This motion of water lowers considerably the natural frequency of vibration of side-walls of the tank. This effect is conveniently expressed by "virtual mass" of water. In the previous reports, I to VII, of the same title, the author has made theoretical studies on the value of "virtual mass" of water, and examined various factors affecting it. Especially, in the report VI, an approximate formula for the virtual mass was given for the case in which two opposite (rectangular) side-walls are vibrating in a mode which correspond to the case of "rectangular plate with clamped four-edges." The calculation was made, by assuming tentatively, the mode of vibration of the rectangular plate.
The question of degree of accuracy of this approximate formula will naturally be raised. In the present report VIII, this question of degree of accuracy is taken up. The treatment may be said to be a case of hydro-elasticity. A set of normalized orthogonal functions (which correspond to the mode of free vibration of elastic bar with fixed ends) is used. The transverse displacement w of the rectangular plate in vibration is expressed as a double infinite series of these set of functions. Putting this expression into the equation of vibratory motion of rectangular elastic plate (wherein, the effect of vibratory water pressure is taken into account), a system of linear equations about the component amplitudes Aaβ (a,β= 1, 2, 3, ...... ) is obtained. And, thence, equations for Aaβ is made, by means of which the values of Aaβ can be obtained by successive (iterative) approximations. We start with A11= 1, and all the others (Aaβ)=0, which we regard as the zeroth approximation. And, we are to calculate first, second, ...... , approximate values.
Numerical examples for the case of (A). B : H : L=1: 1: 2, and (B). B : H : L=1: 1: 1, are shown. It is seen that the values of A12/A11, A31/A11, ...... , are comparatively small, but not so small enough to say that we may neglect them at all. It is concluded that, we may use the approximate formula given in our report VI, for practical purposes, on the undererstanding that they give olny approximate values.