ABSTRACT
A simple way to simulate an explosion is to view it as a spherical piston motion, pushing out undisturbed air/gas ahead of it. To motivate stronger disturbances it is convenient to view small changes that a ‘small’ motion of the piston will bring about (see Taylor (1946)). In the spherically symmetric case, small motions are governed by the linear wave equation
( φrr +
2φr r
) , (2.1.1)
where φ is the velocity potential. The forward moving wave as solution of (2.1.1) is
φ = r−1f(r − a0t), (2.1.2) where a0 is the (constant) speed of sound. The velocity and pressure disturbances from (2.1.2), therefore, are found to be
u = ∂φ
∂r = r−2f(r − a0t)− r−1f ′(r − a0t), (2.1.3)
p− p0 = −ρ0∂φ ∂t
= ρ0a0r −1f ′(r − a0t). (2.1.4)
If the piston motion is given by R = R(t), then using the kinematic condition at the piston u(R(t), t) = dR/dt, we find from (2.1.3) that
R˙ = R−2f(R− a0t)−R−1f ′(R − a0t). (2.1.5)
If we know the piston motion R = R(t), we may find the form of the function f by solving the first order ODE (2.1.5). This can be done easily for the
and
special case for which the spherical piston moves with a constant speed, starting from the position R = 0 at t = 0. Thus,
R = αa0t, (2.1.6)
where α is a nondimensional constant. Since we consider small motions, α << 1. Writing w = R− a0t = (α− 1)a0t < 0 in (2.1.5), we get an ODE for f(w):
α− 1 αw
f ′(w) − ( α− 1 αw
)2 f(w) + a0α = 0. (2.1.7)
The solution of (2.1.7) for negative values of w is
f(w) = a0α
1− α2w 2 + c(−w)(α−1)/α, (2.1.8)
where c is the constant of integration. Since 0 < α << 1, f(w) in (2.1.8) is finite only if we choose c equal to zero. Thus, the solution φ = r−1f(r− at) and hence other physical variables become
φ = a0α
1− α2 (r − a0t)2
r =
1− α2 r ( 1− a0t
r
)2 , (2.1.9)
u = a0α
1− α2 ( a20t
r2 − 1
) , (2.1.10)
p− p0 = 2ρ0 a 2 0α
1− α2 ( a0t
r − 1
) . (2.1.11)
The undisturbed medium (u = 0, p = p0), outside the moving sphere r = at, remains undisturbed for subsequent time while the motion inside this sphere is governed by (2.1.9)–(2.1.11). The form (2.1.9)–(2.1.11) of the linear solution, namely, the spherically symmetric sound wave, suggests that all physical variables are functions of the nondimensional combination r/a0t of the independent variables. The air wave produced by a uniformly expanding sphere expands at a uniform rate and the velocity and pressure disturbances are constant along the lines r/a0t = constant. We may observe that, although we started with a damped travelling wave (2.1.2), the actual solution dictated by the boundary condition is a ‘progressive’ wave or a similarity solution (2.1.9)–(2.1.11). We shall see that the same form holds even when we analyse the full nonlinear problem.
