Theory of shock waves and introduction to gas dynamics/Chapter 2

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Theory of shock waves and introduction to gas dynamics
by Yakov Zeldovich

Chapter 2 Principles of Acoustics. The Speed of Sound

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4464326Theory of shock waves and introduction to gas dynamics — Chapter 2 Principles of Acoustics. The Speed of SoundYakov Zeldovich

Chapter 2

Principles of Acoustics. The Speed of Sound

In the introduction as well as in the preceding Chapter we have several times referred to a characteristic value of velocity, namely, the speed of sound. As we study the propagation of small turbulences, we shall show how from the equations of gas dynamics we obtain, at the limit, the equations of acoustics, and how in the equations of gas dynamics is comprised the speed of sound.

We transform the equations of gas dynamics given above taking the rate of motion $u$ and the change in density to be small. The rate of motion is taken to be small as compared with the speed of sound, $u/c\ll 1$ , and the changes in density and pressure are taken to be small as compared with the mean values of density and pressure, $\Delta \rho /\rho \sim \Delta p/p\ll 1$ . The fluctuations of temperature in the wave in the gas are of the same order.

Furthermore, in the equations of motion we ignore the terms of an order higher than the first one in the expansion of the equation of state of matter by powers of $\Delta \rho$ or $\Delta p$ (they refer to the left out ones such as $\Delta p/p$ ); we also disregard $u^{2}$ as compared with $uc$ (the ratio of eliminated terms to the remaining ones is equal to $u/c$ ).

The values of the amplitude of pressure in a sound of a certain intensity, given below, show irrefutably that these omissions are fully permissible in acoustics.

Density is written as follows:

\rho =\rho _{0}+\varepsilon ,

(II-1)

where $\rho _{0}$ ,initial density, is taken to be a constant quantity, and the change in density $\varepsilon$ , connected with the propagation of sound or, generally. perturbations (turbulence) in the gas, we take to be a small quantity.

The equation of conservation of matter can be rewritten in the following form:

{\frac {\partial \varepsilon }{\partial t}}+u{\frac {\partial \varepsilon }{\partial x}}+(\rho _{0}+\varepsilon ){\frac {\partial u}{\partial x}}=0.

(II-2)

If we disregard quantities of a higher order of smallness, i.e., the products of two small quantities, we get

{\frac {\partial \varepsilon }{\partial t}}=-\rho _{0}{\frac {\partial u}{\partial x}}.

(II-3)

If we disregard, in the same fashion, terms of a higher order of smallness in the equation of motion, we get

\rho _{0}{\frac {\partial u}{\partial t}}=-{\frac {\partial p}{\partial x}}={\frac {\partial p}{\partial \rho }}{\frac {\partial \rho }{\partial x}}=-{\frac {\partial p}{\partial \rho }}{\frac {\partial \varepsilon }{\partial x}}.

(II-4)

By differentiating the equation of conservation of matter with respect to time, and the equation of motion with respect to the coordinate, we obtain a final fundamental acoustics equation:

{\frac {\partial ^{2}\varepsilon }{\partial t^{2}}}={\frac {\partial p}{\partial \rho }}{\frac {\partial ^{2}\varepsilon }{\partial x^{2}}}.

(II-5)

We write

{\frac {\partial p}{\partial \rho }}=c^{2},

(II-5a)

and see that this equation may have two groups of solutions: a first group

\varepsilon =\varepsilon (x-ct);\quad \rho =\rho (x-ct);\quad u=u(x-ct);\quad p=p(x-ct),

(II-6)

and a second group

\varepsilon =\varepsilon (x+ct);\quad \rho =\rho (x+ct);\quad u=u(x+ct);\quad p=p(x+ct),

(II-6a)

which differs from the first in that under the function sign there is $x+ct$ , instead of $x-ct$ , everywhere. We understand $c$ to be everywhere the positive root of $\partial p/\partial \rho$ , $c=+{\sqrt {\partial p/\partial \rho }}$ .

The first group of solutions in which all the quantities depend upon the combination $x-ct$ , represents disturbance which expands toward the right, i.e., in the direction of increasing values of the coordinate $x$ . In fact, if at an instant $t_{1}$ there occurred a certain state $(\rho _{1},p_{1},u_{1})$ at a point $x_{1}$ , then at the following instant $t_{2}$ this same state will occur at that point $x_{2}$ where the variable $x-ct$ (upon which depend all the quantities $\rho _{1},p_{1},u_{1}$ of the solution under investigation) has the same value

x_{2}-ct_{2}=x_{1}-ct_{1},

(II-7)

x_{2}=x_{1}+c(t_{2}-t_{1}).

(II-6)

The assigned state propagates in the direction of increasing $x$ at a velocity $c$ , q.e.d.

By substituting this type of solution into the fundamental equations, we can readily find for this wave from (II-3) $^{3a}$

-c\varepsilon '=-\rho _{0}u',

(II-6)

where the prime denotes the differentiation of function (11-6) with respect to the variable $x-ct$ . If we assume at high values of $x$ , i.e., way ahead in an unperturbed (undisturbed) gas, $u=0$ , $\varepsilon =0$ , and $\rho =\rho _{0}$ we find for a wave propagating to the right.

u=\varepsilon {\frac {c}{\rho _{0}}}=(\rho -\rho _{0}){\frac {c}{\rho _{0}}},

(II-10)

The instant pressure value is also linearly connected with density and velocity:

p-p_{0}={\frac {\partial p}{\partial \rho }}(\rho -\rho _{0})=\rho _{0}uc,

(II-11)

Let us point out specifically that pressure is proportional to the first degree of velocity in sound; according to Bernoulli's theorem, in a steady flow we should have a considerably smaller change in pressure:

p=p_{0}-{\frac {\rho _{0}u^{2}}{2}}.

(II-12)

Thus we draw extremely important conclusions from formulas (II-10) and (II-11): In a wave which propagates to the right, i.e., in the direction of increasing values of the coordinate $x$ , the mass rate of motion $u$ is positive where the substance is compressed, and is negative where the substance is diluted or rarefied and its density is less than normal.

Likewise, for the second wave in which all the quantities depend upon the combination $x+ct$ , that is, for a wave propagating to the left, in the direction of decreasing $x$ , we get

u=-\varepsilon {\frac {c}{\rho _{0}}}=-(\rho -\rho _{0}){\frac {c}{\rho _{0}}}.

(II-13)

In both cases the velocity of motion is directed towards the direction of wave propagation where the substance is compressed. If at an initial instant there is assigned an arbitrary distribution of density and an arbitrary distribution of velocity of motion in space

t=0;\quad \rho =\rho (x);\quad \varepsilon =\varepsilon (x)=\rho (x)-\rho _{0};\quad u=u(x),

(II-14)

then for the two waves looked for: the first $\varepsilon _{1}=\varepsilon _{1}(x-ct)$ , $u=u(x-ct)$ and the second $\varepsilon _{1}(x+ct)$ , $u_{2}=u_{2}(x+ct)$ , we obtain two equations

\varepsilon _{1}(x)+\varepsilon _{2}(x)=\rho (x)-\rho _{0}=\varepsilon (x),

(II-15)

u_{1}(x)+u_{2}(x)={\frac {\varepsilon _{1}(x)c}{\rho _{0}}}-{\frac {\varepsilon _{2}(x)c}{\rho _{0}}}=u(x).

(II-16)

The second equation, (II-16), is obtained by applying (II-10) to $\varepsilon _{1}$ and $u_{1}$ , and (II-13) to $\varepsilon _{2}$ and $u_{2}$ . Then we immediately obtain

{\begin{aligned}\varepsilon _{1}(x-ct)&={\frac {1}{2}}\varepsilon (x-ct)+{\frac {\rho _{0}}{2c}}u(x-ct);\\u_{1}(x-ct)&={\frac {c}{2\rho _{0}}}\varepsilon (x-ct)+{\frac {1}{2}}u(x-ct);\\\varepsilon _{2}(x-ct)&={\frac {1}{2}}\varepsilon (x+ct)-{\frac {\rho _{0}}{2c}}u(x+ct);\\u_{2}(x+ct)&={\frac {c}{2\rho _{0}}}\varepsilon (x+ct)+{\frac {1}{2}}u(x+ct).\end{aligned}}

(II-17)

Theory of shock waves and introduction to gas dynamics/Chapter 2

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