Fluid Mechanics by L.D. Landau & E.M. Lifshitz

Author:L.D. Landau & E.M. Lifshitz
Language: eng
Format: epub
Tags: Physics; Ideal Fluids; Viscous; Turbulence; Boundary Layers; Diffusion; Shock Waves; Superfluids

S2

Mm

As t -> oo, this expression tends to zero as

0 = Vt j* 1 ® 6 **

Si

i.e. inversely as the time.

Thus the potential in an outgoing cylindrical wave, due to a source which operates only for a finite time, vanishes, though slowly, as t -> oo. This means that, as in the spherical case, the integral of p' over all time is zero:

00

fp'dt = 0. (70.9)

—oo

Hence a cylindrical wave, like a spherical wave, must necessarily include both condensations and rarefactions.

§71. The general solution of the wave equation

We shall now derive a general formula giving the solution of the wave equation in an infinite fluid for any initial conditions, i.e. giving the velocity and pressure distribution in the fluid at any instant in terms of their initial distribution.

We first obtain some auxiliary formulae. Let <f>(x, y, z, t) and i[j(x, y, z, t) be any two solutions of the wave equation which vanish at infinity. We consider the integral

/=/ («^-#)dF,

taken over all space, and calculate its time derivative. Since <f> and ifj satisfy the equations A<f>-$lc 2 = 0 and A«A~$/c 2 = 0, we have

dl/dt = j {<f4-^)dV = c* j (M«A-M<£)dF

= c 2 J div(<£ grad ifj — if/ grad <f>)dV.

The last integral can be transformed into an integral over an infinitely distant surface, and is therefore zero. Thus we conclude that dljdt = 0, i.e. J is independent of time :

f {<j4 - #)d V = constant. (71.1)

Next, let us consider the following particular solution of the wave equation :

4> = 8[r-c{to-t)]lr (71.2)

(where r is the distance from some given point O, to is some definite instant,

and 8 denotes the delta function), and calculate the integral of ifj over all space. We have

00 00

j l fsdV = J* if*-4irr 2 dr = 4tt j rS[r-c(t 0 -t)]dr. o o

The argument of the delta function is zero for r = c(to — t) (we assume that *o > *)• Hence, from the properties of the delta function, we find

jt/,dV = +nc(to-t). (71.3)

Differentiating this equation with respect to time, we obtain

ffdV = -4ttc. (71.4)

We now substitute for «/r, in the integral (71.1), the function (71.2), and take <f> to be the required general solution of the wave equation. According to (71.1), J is a constant; using this, we write down the expressions for / at the instants t = 0 and t = to, and equate the two. For t = t o the two functions t[t and if, are each different from zero only for r = 0. Hence, on integrating, we can put r = 0 in (f> and (f> (i.e. take their values at the point O), and take <f> and <f> outside the integral:

I = <f>{x,y, z y t 0 ) j 4 &V-<f>{x,y, z, t 0 ) j tfidV,

where x t y, z are the co-ordinates of O. According to (71.3) and (71.4), the second term is zero for t — to, and the first term gives

I = - 47TC(j}(x > y, z y to).

Let us now calculate /for t = 0.

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