Misplaced Pages

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95-432: The basic idea of the jump conditions is to consider what happens to a fluid when it undergoes a rapid change. Consider, for example, driving a piston into a tube filled with non-reacting gas. A disturbance is propagated through the fluid somewhat faster than the speed of sound . Because the disturbance propagates supersonically , it is a shock wave , and the fluid downstream of the shock has no advance information of it. In

190-444: A l = γ ⋅ p ρ = γ ⋅ R ⋅ T M = γ ⋅ k ⋅ T m , {\displaystyle c_{\mathrm {ideal} }={\sqrt {\gamma \cdot {p \over \rho }}}={\sqrt {\gamma \cdot R\cdot T \over M}}={\sqrt {\gamma \cdot k\cdot T \over m}},} where This equation applies only when

285-402: A dispersive medium , the speed of sound is a function of sound frequency, through the dispersion relation . Each frequency component propagates at its own speed, called the phase velocity , while the energy of the disturbance propagates at the group velocity . The same phenomenon occurs with light waves; see optical dispersion for a description. The speed of sound is variable and depends on

380-407: A calorically ideal gas γ {\displaystyle \gamma } is a constant and for a thermally ideal gas γ {\displaystyle \gamma } is a function of temperature. In the latter case, the dependence of pressure on mass density and internal energy might differ from that given by equation ( 4 ). Before proceeding further it is necessary to introduce

475-453: A compression wave in a fluid is determined by the medium's compressibility and density . In solids, the compression waves are analogous to those in fluids, depending on compressibility and density, but with the additional factor of shear modulus which affects compression waves due to off-axis elastic energies which are able to influence effective tension and relaxation in a compression. The speed of shear waves, which can occur only in solids,

570-410: A computation of the speed of sound in air as 979 feet per second (298 m/s). This is too low by about 15%. The discrepancy is due primarily to neglecting the (then unknown) effect of rapidly fluctuating temperature in a sound wave (in modern terms, sound wave compression and expansion of air is an adiabatic process , not an isothermal process ). This error was later rectified by Laplace . During

665-698: A final volume of 100   mL of solution would be labeled as "1%" or "1% m/v" (mass/volume). This is incorrect because the unit "%" can only be used for dimensionless quantities. Instead, the concentration should simply be given in units of g/mL. Percent solution or percentage solution are thus terms best reserved for mass percent solutions (m/m, m%, or mass solute/mass total solution after mixing), or volume percent solutions (v/v, v%, or volume solute per volume of total solution after mixing). The very ambiguous terms percent solution and percentage solutions with no other qualifiers continue to occasionally be encountered. In thermal engineering , vapor quality

760-421: A frame of reference moving with the wave, atoms or molecules in front of the wave slam into the wave supersonically. On a microscopic level, they undergo collisions on the scale of the mean free path length until they come to rest in the post-shock flow (but moving in the frame of reference of the wave or of the tube). The bulk transfer of kinetic energy heats the post-shock flow. Because the mean free path length

855-467: A negative slope (since m 2 {\displaystyle m^{2}} is always positive) in the p − ρ − 1 {\displaystyle p-\rho ^{-1}} plane. Using the Rankine–Hugoniot equations for the conservation of mass and momentum to eliminate u 1 and u 2 , the equation for the conservation of energy can be expressed as

950-481: A one-dimensional container (e.g., a long thin tube). Assume that the fluid is inviscid (i.e., it shows no viscosity effects as for example friction with the tube walls). Furthermore, assume that there is no heat transfer by conduction or radiation and that gravitational acceleration can be neglected. Such a system can be described by the following system of conservation laws , known as the 1D Euler equations , that in conservation form is: where Assume further that

1045-486: A pipe aligned with the x {\displaystyle x} axis and with a cross-sectional area of A {\displaystyle A} . In time interval d t {\displaystyle dt} it moves length d x = v d t {\displaystyle dx=v\,dt} . In steady state , the mass flow rate m ˙ = ρ v A {\displaystyle {\dot {m}}=\rho vA} must be

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1140-618: A requirement for a unique single-valued solution is that the solution should satisfy the admissibility condition or entropy condition . For physically real applications this means that the solution should satisfy the Lax entropy condition where f ′ ( w 1 ) {\displaystyle f'\left(w_{1}\right)} and f ′ ( w 2 ) {\displaystyle f'\left(w_{2}\right)} represent characteristic speeds at upstream and downstream conditions respectively. In

1235-426: A shock wave also generalizes to reacting flows, where a combustion front (either a detonation or a deflagration) can be modeled as a discontinuity in a first approximation. In a coordinate system that is moving with the discontinuity, the Rankine–Hugoniot conditions can be expressed as: where m is the mass flow rate per unit area, ρ 1 and ρ 2 are the mass density of the fluid upstream and downstream of

1330-456: A single given gas (assuming the molecular weight does not change) and over a small temperature range (for which the heat capacity is relatively constant), the speed of sound becomes dependent on only the temperature of the gas. In non-ideal gas behavior regimen, for which the Van der Waals gas equation would be used, the proportionality is not exact, and there is a slight dependence of sound velocity on

1425-509: Is a parameter (the slope of the shock Hugoniot) obtained from fits to experimental data, and u p = u 2 is the particle velocity inside the compressed region behind the shock front. The above relation, when combined with the Hugoniot equations for the conservation of mass and momentum, can be used to determine the shock Hugoniot in the p - v plane, where v is the specific volume (per unit mass): Alternative equations of state, such as

1520-426: Is also constant across the wave, the change in enthalpies (calorific equation of state) can be simply written as where the first term in the above expression represents the amount of heat released per unit mass of the upstream mixture by the wave and the second term represents the sensible heating. Eliminating temperature using the equation of state and substituting the above expression for the change in enthalpies into

1615-419: Is associated with compression and decompression in the direction of travel, and is the same process in gases and liquids, with an analogous compression-type wave in solids. Only compression waves are supported in gases and liquids. An additional type of wave, the transverse wave , also called a shear wave , occurs only in solids because only solids support elastic deformations. It is due to elastic deformation of

1710-427: Is assumed to be negligible in comparison to all other length scales in a hydrodynamic treatment, the shock front is essentially a hydrodynamic discontinuity . The jump conditions then establish the transition between the pre- and post-shock flow, based solely upon the conservation of mass, momentum, and energy. The conditions are correct even though the shock actually has a positive thickness. This non-reacting example of

1805-417: Is called the object's Mach number . Objects moving at speeds greater than the speed of sound ( Mach 1 ) are said to be traveling at supersonic speeds . In Earth's atmosphere, the speed of sound varies greatly from about 295 m/s (1,060 km/h; 660 mph) at high altitudes to about 355 m/s (1,280 km/h; 790 mph) at high temperatures. Sir Isaac Newton 's 1687 Principia includes

1900-561: Is deemed that the isentrope is close enough to the Hugoniot that the same assumption can be made. If the Hugoniot is approximately the loading path between states for an "equivalent" compression wave, then the jump conditions for the shock loading path can be determined by drawing a straight line between the initial and final states. This line is called the Rayleigh line and has the following equation: Most solid materials undergo plastic deformations when subjected to strong shocks. The point on

1995-529: Is derived for the case p ~ → 0 {\displaystyle {\tilde {p}}\rightarrow 0} because pressure cannot take negative values). If γ = 1.4 {\displaystyle \gamma =1.4} (diatomic gas without the vibrational mode excitation), the interval is 1 / 6 ≤ v ~ ≤ 2 α + 6 {\displaystyle 1/6\leq {\tilde {v}}\leq 2\alpha +6} , in other words,

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2090-412: Is determined by the medium's compressibility , shear modulus , and density. The speed of shear waves is determined only by the solid material's shear modulus and density. In fluid dynamics , the speed of sound in a fluid medium (gas or liquid) is used as a relative measure for the speed of an object moving through the medium. The ratio of the speed of an object to the speed of sound (in the same medium)

2185-807: Is determined simply by the solid material's shear modulus and density. The speed of sound in mathematical notation is conventionally represented by c , from the Latin celeritas meaning "swiftness". For fluids in general, the speed of sound c is given by the Newton–Laplace equation: c = K s ρ , {\displaystyle c={\sqrt {\frac {K_{s}}{\rho }}},} where K s = ρ ( ∂ P ∂ ρ ) s {\displaystyle K_{s}=\rho \left({\frac {\partial P}{\partial \rho }}\right)_{s}} , where P {\displaystyle P}

2280-577: Is fully excited (i.e., molecular rotation is fully used as a heat energy "partition" or reservoir); but at the same time the temperature must be low enough that molecular vibrational modes contribute no heat capacity (i.e., insignificant heat goes into vibration, as all vibrational quantum modes above the minimum-energy-mode have energies that are too high to be populated by a significant number of molecules at this temperature). For air, these conditions are fulfilled at room temperature, and also temperatures considerably below room temperature (see tables below). See

2375-513: Is like that from above substituting the relation between mass and molar concentration: where c i {\displaystyle c_{i}} is the molar concentration, and M i {\displaystyle M_{i}} is the molar mass of the component i {\displaystyle i} . Mass percentage is defined as the mass fraction multiplied by 100. The mole fraction x i {\displaystyle x_{i}} can be calculated using

2470-399: Is one way of expressing the composition of a mixture in a dimensionless size ; mole fraction (percentage by moles , mol%) and volume fraction ( percentage by volume , vol%) are others. When the prevalences of interest are those of individual chemical elements , rather than of compounds or other substances, the term mass fraction can also refer to the ratio of the mass of an element to

2565-1109: Is the therefore sign . Note, to arrive at equation ( 8 ) we have used the fact that d x 1 / d t = 0 {\displaystyle dx_{1}/dt=0} and d x 2 / d t = 0 {\displaystyle dx_{2}/dt=0} . Now, let x 1 → x s ( t ) − ϵ {\displaystyle x_{1}\to x_{s}(t)-\epsilon } and x 2 → x s ( t ) + ϵ {\displaystyle x_{2}\to x_{s}(t)+\epsilon } , when we have ∫ x 1 x s ( t ) − ϵ w t d x → 0 {\textstyle \int _{x_{1}}^{x_{s}(t)-\epsilon }w_{t}\,dx\to 0} and ∫ x s ( t ) + ϵ x 2 w t d x → 0 {\textstyle \int _{x_{s}(t)+\epsilon }^{x_{2}}w_{t}\,dx\to 0} , and in

2660-434: Is the universal gas constant and the mean molecular weight W ¯ {\displaystyle {\overline {W}}} is assumed to be constant (otherwise, W ¯ {\displaystyle {\overline {W}}} would depend on the mass fraction of the all species). If one assumes that the specific heat at constant pressure c p {\displaystyle c_{p}}

2755-1716: Is the difference between the values of any physical quantity on the two sides of the discontinuity. The remaining conditions are given by These conditions are general in the sense that they include contact discontinuities ( j = 0 , H n ≠ 0 , [ [ u ] ] = [ [ p ] ] = [ [ H ] ] = 0 , [ [ ρ ] ] ≠ 0 {\displaystyle j=0,\,H_{n}\neq 0,\,[\![\mathbf {u} ]\!]=[\![p]\!]=[\![\mathbf {H} ]\!]=0,\,[\![\rho ]\!]\neq 0} ) tangential discontinuities ( j = H n = 0 , [ [ u t ρ ] ] ≠ 0 , [ [ H t ] ] ≠ 0 , [ [ ρ ] ] ≠ 0 {\displaystyle j=H_{n}=0,\,[\![\mathbf {u} _{t}\rho ]\!]\neq 0,\,[\![\mathbf {H} _{t}]\!]\neq 0,\,[\![\rho ]\!]\neq 0} ), rotational or Alfvén discontinuities ( j = H n ρ / 4 π ≠ 0 , [ [ ρ ] ] = [ [ u n ] ] = [ [ p ] ] = [ [ H t ] ] = 0 {\textstyle j=H_{n}{\sqrt {\rho /4\pi }}\neq 0,\,[\![\rho ]\!]=[\![u_{n}]\!]=[\![p]\!]=[\![\mathbf {H} _{t}]\!]=0} ) and shock waves ( j ≠ 0 , [ [ ρ ] ] ≠ 0 {\displaystyle j\neq 0,\,[\![\rho ]\!]\neq 0} ). Speed of sound The speed of sound

2850-430: Is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At 20 °C (68 °F), the speed of sound in air is about 343  m/s (1,125  ft/s ; 1,235  km/h ; 767  mph ; 667  kn ), or 1  km in 2.91 s or one mile in 4.69 s . It depends strongly on temperature as well as

2945-472: Is the pressure and the derivative is taken isentropically, that is, at constant entropy s . This is because a sound wave travels so fast that its propagation can be approximated as an adiabatic process , meaning that there isn't enough time, during a pressure cycle of the sound, for significant heat conduction and radiation to occur. Thus, the speed of sound increases with the stiffness (the resistance of an elastic body to deformation by an applied force) of

Rankine–Hugoniot conditions - Misplaced Pages Continue

3040-434: Is to say, the pressure decreases and the specific volume increases across the wave; the pressure decrease a flame is typically very small which is seldom considered when studying deflagrations. For shock waves and detonations, the pressure increase across the wave can take any values between 0 ≤ p ~ < ∞ {\displaystyle 0\leq {\tilde {p}}<\infty } ;

3135-542: Is to say, the pressure increases and the specific volume decreases across the wave (the Chapman–Jouguet condition for detonation is where Rayleigh line is tangent to the Hugoniot curve). Deflagrations, on the other hand, correspond to the bottom-right white region wherein p ~ < 1 {\displaystyle {\tilde {p}}<1} and v ~ > 1 {\displaystyle {\tilde {v}}>1} , that

3230-501: Is used for the mass fraction of vapor in the steam. In alloys, especially those of noble metals, the term fineness is used for the mass fraction of the noble metal in the alloy. The mass fraction is independent of temperature until phase change occurs. The mixing of two pure components can be expressed introducing the (mass) mixing ratio of them r m = m 2 m 1 {\displaystyle r_{m}={\frac {m_{2}}{m_{1}}}} . Then

3325-493: The Mie–Grüneisen equation of state may also be used instead of the above equation. The shock Hugoniot describes the locus of all possible thermodynamic states a material can exist in behind a shock, projected onto a two dimensional state-state plane. It is therefore a set of equilibrium states and does not specifically represent the path through which a material undergoes transformation. Weak shocks are isentropic and that

3420-403: The mass fraction of a substance within a mixture is the ratio w i {\displaystyle w_{i}} (alternatively denoted Y i {\displaystyle Y_{i}} ) of the mass m i {\displaystyle m_{i}} of that substance to the total mass m tot {\displaystyle m_{\text{tot}}} of

3515-446: The ozone layer . This produces a positive speed of sound gradient in this region. Still another region of positive gradient occurs at very high altitudes, in the thermosphere above 90 km . For an ideal gas, K (the bulk modulus in equations above, equivalent to C , the coefficient of stiffness in solids) is given by K = γ ⋅ p . {\displaystyle K=\gamma \cdot p.} Thus, from

3610-548: The springs , and the mass of the spheres. As long as the spacing of the spheres remains constant, stiffer springs/bonds transmit energy more quickly, while more massive spheres transmit energy more slowly. In a real material, the stiffness of the springs is known as the " elastic modulus ", and the mass corresponds to the material density . Sound will travel more slowly in spongy materials and faster in stiffer ones. Effects like dispersion and reflection can also be understood using this model. Some textbooks mistakenly state that

3705-605: The "One o'Clock Gun" is fired at the eastern end of Edinburgh Castle. Standing at the base of the western end of the Castle Rock, the sound of the Gun can be heard through the rock, slightly before it arrives by the air route, partly delayed by the slightly longer route. It is particularly effective if a multi-gun salute such as for "The Queen's Birthday" is being fired. In a gas or liquid, sound consists of compression waves. In solids, waves propagate as two different types. A longitudinal wave

3800-575: The 17th century there were several attempts to measure the speed of sound accurately, including attempts by Marin Mersenne in 1630 (1,380 Parisian feet per second), Pierre Gassendi in 1635 (1,473 Parisian feet per second) and Robert Boyle (1,125 Parisian feet per second). In 1709, the Reverend William Derham , Rector of Upminster, published a more accurate measure of the speed of sound, at 1,072 Parisian feet per second. (The Parisian foot

3895-438: The Hugoniot equation, one obtains an Hugoniot equation expressed only in terms of pressure and densities, where γ {\displaystyle \gamma } is the specific heat ratio , which for ordinary room temperature air (298 KELVIN) = 1.40. An Hugoniot curve without heat release ( q = 0 {\displaystyle q=0} ) is often called a "shock Hugoniot", or simply a(n) "Hugoniot". Along with

Rankine–Hugoniot conditions - Misplaced Pages Continue

3990-409: The Hugoniot equation: The inverse of the density can also be expressed as the specific volume , v = 1 / ρ {\displaystyle v=1/\rho } . Along with these, one has to specify the relation between the upstream and downstream equation of state where Y i {\displaystyle Y_{i}} is the mass fraction of the species. Finally,

4085-405: The Newton–Laplace equation above, the speed of sound in an ideal gas is given by c = γ ⋅ p ρ , {\displaystyle c={\sqrt {\gamma \cdot {p \over \rho }}},} where Using the ideal gas law to replace p with nRT / V , and replacing ρ with nM / V , the equation for an ideal gas becomes c i d e

4180-525: The Rayleigh line equation, the above equation completely determines the state of the system. These two equations can be written compactly by introducing the following non-dimensional scales, The Rayleigh line equation and the Hugoniot equation then simplifies to Given the upstream conditions, the intersection of above two equations in the v ~ {\displaystyle {\tilde {v}}} - p ~ {\displaystyle {\tilde {p}}} plane determine

4275-467: The calorific equation of state h = h ( p , ρ , Y i ) {\displaystyle h=h(p,\rho ,Y_{i})} is assumed to be known, i.e., The following assumptions are made in order to simplify the Rankine–Hugoniot equations. The mixture is assumed to obey the ideal gas law , so that relation between the downstream and upstream equation of state can be written as where R {\displaystyle R}

4370-423: The case of the hyperbolic conservation law ( 6 ), we have seen that the shock speed can be obtained by simple division. However, for the 1D Euler equations ( 1 ), ( 2 ) and ( 3 ), we have the vector state variable [ ρ ρ u E ] T {\displaystyle {\begin{bmatrix}\rho &\rho u&E\end{bmatrix}}^{\mathsf {T}}} and

4465-596: The concept of a jump condition – a condition that holds at a discontinuity or abrupt change. Consider a 1D situation where there is a jump in the scalar conserved physical quantity w {\displaystyle w} , which is governed by integral conservation law for any x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , x 1 < x 2 {\displaystyle x_{1}<x_{2}} , and, therefore, by partial differential equation for smooth solutions. Let

4560-456: The denser materials. An illustrative example of the two effects is that sound travels only 4.3 times faster in water than air, despite enormous differences in compressibility of the two media. The reason is that the greater density of water, which works to slow sound in water relative to the air, nearly makes up for the compressibility differences in the two media. For instance, sound will travel 1.59 times faster in nickel than in bronze, due to

4655-626: The discontinuity the normal component H n {\displaystyle H_{n}} of the magnetic field H {\displaystyle \mathbf {H} } and the tangential component E t {\displaystyle \mathbf {E} _{t}} of the electric field E = − u × H / c {\displaystyle \mathbf {E} =-\mathbf {u} \times \mathbf {H} /c} (infinite conductivity limit) must be continuous. We thus have where [ [ ⋅ ] ] {\displaystyle [\![\cdot ]\!]}

4750-540: The discontinuity. Here, ω {\displaystyle \omega } is the mass production rate of the i -th species of total N species involved in the reaction. Combining conservation of mass and momentum gives us which defines a straight line known as the Michelson–Rayleigh line , named after the Russian physicist Vladimir A. Mikhelson (usually anglicized as Michelson) and Lord Rayleigh , that has

4845-620: The downstream conditions; in the v ~ {\displaystyle {\tilde {v}}} - p ~ {\displaystyle {\tilde {p}}} plane, the upstream condition correspond to the point ( v ~ , p ~ ) = ( 1 , 1 ) {\displaystyle ({\tilde {v}},{\tilde {p}})=(1,1)} . If no heat release occurs, for example, shock waves without chemical reaction, then α = 0 {\displaystyle \alpha =0} . The Hugoniot curves asymptote to

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4940-413: The elimination of u 2 ′ {\displaystyle u'_{2}} from the transformed equation ( 13 ) using the transformed equation ( 12 )), that the shock speed is given by where c 1 = γ p 1 / ρ 1 {\textstyle c_{1}={\sqrt {\gamma p_{1}/\rho _{1}}}} is the speed of sound in

5035-445: The fastest it can travel under normal conditions. In theory, the speed of sound is actually the speed of vibrations. Sound waves in solids are composed of compression waves (just as in gases and liquids) and a different type of sound wave called a shear wave , which occurs only in solids. Shear waves in solids usually travel at different speeds than compression waves, as exhibited in seismology . The speed of compression waves in solids

5130-469: The figure. As mentioned in the figure, only the white region bounded by these two asymptotes are allowed so that μ {\displaystyle \mu } is positive. Shock waves and detonations correspond to the top-left white region wherein p ~ > 1 {\displaystyle {\tilde {p}}>1} and v ~ < 1 {\displaystyle {\tilde {v}}<1} , that

5225-403: The fluid at upstream conditions. For shocks in solids, a closed form expression such as equation ( 15 ) cannot be derived from first principles. Instead, experimental observations indicate that a linear relation can be used instead (called the shock Hugoniot in the u s - u p plane) that has the form where c 0 is the bulk speed of sound in the material (in uniaxial compression), s

5320-430: The formula where M i {\displaystyle M_{i}} is the molar mass of the component i {\displaystyle i} , and M ¯ {\displaystyle {\bar {M}}} is the average molar mass of the mixture. Replacing the expression of the molar-mass products, In a spatially non-uniform mixture, the mass fraction gradient gives rise to

5415-520: The gas is calorically ideal and that therefore a polytropic equation-of-state of the simple form is valid, where γ {\displaystyle \gamma } is the constant ratio of specific heats c p / c v {\displaystyle c_{p}/c_{v}} . This quantity also appears as the polytropic exponent of the polytropic process described by For an extensive list of compressible flow equations, etc., refer to NACA Report 1135 (1953). Note: For

5510-473: The gas pressure. Humidity has a small but measurable effect on the speed of sound (causing it to increase by about 0.1%–0.6%), because oxygen and nitrogen molecules of the air are replaced by lighter molecules of water . This is a simple mixing effect. In the Earth's atmosphere , the chief factor affecting the speed of sound is the temperature . For a given ideal gas with constant heat capacity and composition,

5605-610: The greater stiffness of nickel at about the same density. Similarly, sound travels about 1.41 times faster in light hydrogen ( protium ) gas than in heavy hydrogen ( deuterium ) gas, since deuterium has similar properties but twice the density. At the same time, "compression-type" sound will travel faster in solids than in liquids, and faster in liquids than in gases, because the solids are more difficult to compress than liquids, while liquids, in turn, are more difficult to compress than gases. A practical example can be observed in Edinburgh when

5700-401: The ground, creating an acoustic shadow at some distance from the source. The decrease of the speed of sound with height is referred to as a negative sound speed gradient . However, there are variations in this trend above 11 km . In particular, in the stratosphere above about 20 km , the speed of sound increases with height, due to an increase in temperature from heating within

5795-413: The gunshot with a half-second pendulum. Measurements were made of gunshots from a number of local landmarks, including North Ockendon church. The distance was known by triangulation , and thus the speed that the sound had travelled was calculated. The transmission of sound can be illustrated by using a model consisting of an array of spherical objects interconnected by springs. In real material terms,

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5890-466: The important factors, since fluids do not transmit shear stresses. In heterogeneous fluids, such as a liquid filled with gas bubbles, the density of the liquid and the compressibility of the gas affect the speed of sound in an additive manner, as demonstrated in the hot chocolate effect . In gases, adiabatic compressibility is directly related to pressure through the heat capacity ratio (adiabatic index), while pressure and density are inversely related to

5985-416: The interval 1 / 8 ≤ v ~ ≤ 2 α + 8 {\displaystyle 1/8\leq {\tilde {v}}\leq 2\alpha +8} . In reality, the specific heat ratio is not constant in the shock wave due to molecular dissociation and ionization, but even in these cases, density ratio in general do not exceed a factor of about 11–13 . Consider gas in

6080-420: The isentrope represents the path through which the material is loaded from the initial to final states by a compression wave with converging characteristics. In the case of weak shocks, the Hugoniot will therefore fall directly on the isentrope and can be used directly as the equivalent path. In the case of a strong shock we can no longer make that simplification directly. However, for engineering calculations, it

6175-524: The jump conditions become Equations ( 12 ), ( 13 ) and ( 14 ) are known as the Rankine–Hugoniot conditions for the Euler equations and are derived by enforcing the conservation laws in integral form over a control volume that includes the shock. For this situation u s {\displaystyle u_{s}} cannot be obtained by simple division. However, it can be shown by transforming

6270-575: The jump respectively, i.e. w 1 = lim ϵ → 0 + w ( x s − ϵ ) {\textstyle w_{1}=\lim _{\epsilon \to 0^{+}}w\left(x_{s}-\epsilon \right)} and w 2 = lim ϵ → 0 + w ( x s + ϵ ) {\textstyle w_{2}=\lim _{\epsilon \to 0^{+}}w\left(x_{s}+\epsilon \right)} . ∴ {\displaystyle \therefore }

6365-439: The limit where we have defined u s = d x s ( t ) / d t {\displaystyle u_{s}=dx_{s}(t)/dt} (the system characteristic or shock speed ), which by simple division is given by Equation ( 9 ) represents the jump condition for conservation law ( 6 ). A shock situation arises in a system where its characteristics intersect, and under these conditions

6460-454: The lines v ~ = ( γ − 1 ) / ( γ + 1 ) {\displaystyle {\tilde {v}}=(\gamma -1)/(\gamma +1)} and p ~ = − ( γ − 1 ) / ( γ + 1 ) {\displaystyle {\tilde {p}}=-(\gamma -1)/(\gamma +1)} , which are depicted as dashed lines in

6555-485: The mass fractions of the components will be The mass ratio equals the ratio of mass fractions of components: due to division of both numerator and denominator by the sum of masses of components. The mass fraction of a component in a solution is the ratio of the mass concentration of that component ρ i (density of that component in the mixture) to the density of solution ρ {\displaystyle \rho } . The relation to molar concentration

6650-473: The material and decreases with an increase in density. For ideal gases, the bulk modulus K is simply the gas pressure multiplied by the dimensionless adiabatic index , which is about 1.4 for air under normal conditions of pressure and temperature. For general equations of state , if classical mechanics is used, the speed of sound c can be derived as follows: Consider the sound wave propagating at speed v {\displaystyle v} through

6745-563: The medium perpendicular to the direction of wave travel; the direction of shear-deformation is called the " polarization " of this type of wave. In general, transverse waves occur as a pair of orthogonal polarizations. These different waves (compression waves and the different polarizations of shear waves) may have different speeds at the same frequency. Therefore, they arrive at an observer at different times, an extreme example being an earthquake , where sharp compression waves arrive first and rocking transverse waves seconds later. The speed of

6840-451: The medium through which a sound wave is propagating. At 0 °C (32 °F), the speed of sound in air is about 331 m/s (1,086 ft/s; 1,192 km/h; 740 mph; 643 kn). The speed of sound in an ideal gas depends only on its temperature and composition. The speed has a weak dependence on frequency and pressure in ordinary air, deviating slightly from ideal behavior. In colloquial speech, speed of sound refers to

6935-454: The mixture. Expressed as a formula, the mass fraction is: Because the individual masses of the ingredients of a mixture sum to m tot {\displaystyle m_{\text{tot}}} , their mass fractions sum to unity: Mass fraction can also be expressed, with a denominator of 100, as percentage by mass (in commercial contexts often called percentage by weight , abbreviated wt.% or % w/w ; see mass versus weight ). It

7030-548: The problem to a moving co-ordinate system (setting u s ′ := u s − u 1 {\displaystyle u_{s}':=u_{s}-u_{1}} , u 1 ′ := 0 {\displaystyle u'_{1}:=0} , u 2 ′ := u 2 − u 1 {\displaystyle u'_{2}:=u_{2}-u_{1}} to remove u 1 {\displaystyle u_{1}} ) and some algebraic manipulation (involving

7125-432: The properties of the substance through which the wave is travelling. In solids, the speed of transverse (or shear) waves depends on the shear deformation under shear stress (called the shear modulus ), and the density of the medium. Longitudinal (or compression) waves in solids depend on the same two factors with the addition of a dependence on compressibility . In fluids, only the medium's compressibility and density are

7220-947: The region near 0 °C ( 273 K ). Then, for dry air, c a i r = γ ⋅ R ∗ ⋅ T = γ ⋅ R ∗ ⋅ ( θ + 273.15 K ) , c a i r = γ ⋅ R ∗ ⋅ 273.15 K ⋅ 1 + θ 273.15 K . {\displaystyle {\begin{aligned}c_{\mathrm {air} }&={\sqrt {\gamma \cdot R_{*}\cdot T}}={\sqrt {\gamma \cdot R_{*}\cdot (\theta +273.15\,\mathrm {K} )}},\\c_{\mathrm {air} }&={\sqrt {\gamma \cdot R_{*}\cdot 273.15\,\mathrm {K} }}\cdot {\sqrt {1+{\frac {\theta }{273.15\,\mathrm {K} }}}}.\end{aligned}}} Mass fraction (chemistry) In chemistry ,

7315-1421: The same at the two ends of the tube, therefore the mass flux j = ρ v {\displaystyle j=\rho v} is constant and v d ρ = − ρ d v {\displaystyle v\,d\rho =-\rho \,dv} . Per Newton's second law , the pressure-gradient force provides the acceleration: d v d t = − 1 ρ d P d x → d P = ( − ρ d v ) d x d t = ( v d ρ ) v → v 2 ≡ c 2 = d P d ρ {\displaystyle {\begin{aligned}{\frac {dv}{dt}}&=-{\frac {1}{\rho }}{\frac {dP}{dx}}\\[1ex]\rightarrow dP&=(-\rho \,dv){\frac {dx}{dt}}=(v\,d\rho )v\\[1ex]\rightarrow v^{2}&\equiv c^{2}={\frac {dP}{d\rho }}\end{aligned}}} And therefore: c = ( ∂ P ∂ ρ ) s = K s ρ , {\displaystyle c={\sqrt {\left({\frac {\partial P}{\partial \rho }}\right)_{s}}}={\sqrt {\frac {K_{s}}{\rho }}},} If relativistic effects are important,

7410-461: The section on gases in specific heat capacity for a more complete discussion of this phenomenon. For air, we introduce the shorthand R ∗ = R / M a i r . {\displaystyle R_{*}=R/M_{\mathrm {air} }.} In addition, we switch to the Celsius temperature θ = T − 273.15 K , which is useful to calculate air speed in

7505-577: The shock Hugoniot at which a material transitions from a purely elastic state to an elastic-plastic state is called the Hugoniot elastic limit (HEL) and the pressure at which this transition takes place is denoted p HEL . Values of p HEL can range from 0.2 GPa to 20 GPa. Above the HEL, the material loses much of its shear strength and starts behaving like a fluid. Rankine–Hugoniot conditions in magnetohydrodynamics are interesting to consider since they are very relevant to astrophysical applications. Across

7600-533: The shock wave can increase the density at most by a factor of 6. For monatomic gas, γ = 5 / 3 {\displaystyle \gamma =5/3} , the allowed interval is 1 / 4 ≤ v ~ ≤ 2 α + 4 {\displaystyle 1/4\leq {\tilde {v}}\leq 2\alpha +4} . For diatomic gases with vibrational mode excited, we have γ = 9 / 7 {\displaystyle \gamma =9/7} leading to

7695-470: The solution exhibit a jump (or shock) at x = x s ( t ) {\displaystyle x=x_{s}(t)} , where x 1 < x s ( t ) {\displaystyle x_{1}<x_{s}(t)} and x s ( t ) < x 2 {\displaystyle x_{s}(t)<x_{2}} , then The subscripts 1 and 2 indicate conditions just upstream and just downstream of

7790-426: The sound wave is a small perturbation on the ambient condition, and the certain other noted conditions are fulfilled, as noted below. Calculated values for c air have been found to vary slightly from experimentally determined values. Newton famously considered the speed of sound before most of the development of thermodynamics and so incorrectly used isothermal calculations instead of adiabatic . His result

7885-404: The speed of sound increases with density. This notion is illustrated by presenting data for three materials, such as air, water, and steel and noting that the speed of sound is higher in the denser materials. But the example fails to take into account that the materials have vastly different compressibility, which more than makes up for the differences in density, which would slow wave speeds in

7980-423: The speed of sound is about 75% of the mean speed that the atoms move in that gas. For a given ideal gas the molecular composition is fixed, and thus the speed of sound depends only on its temperature . At a constant temperature, the gas pressure has no effect on the speed of sound, since the density will increase, and since pressure and density (also proportional to pressure) have equal but opposite effects on

8075-506: The speed of sound is calculated from the relativistic Euler equations . In a non-dispersive medium , the speed of sound is independent of sound frequency , so the speeds of energy transport and sound propagation are the same for all frequencies. Air, a mixture of oxygen and nitrogen, constitutes a non-dispersive medium. However, air does contain a small amount of CO 2 which is a dispersive medium, and causes dispersion to air at ultrasonic frequencies (greater than 28  kHz ). In

8170-404: The speed of sound is dependent solely upon temperature; see § Details below. In such an ideal case, the effects of decreased density and decreased pressure of altitude cancel each other out, save for the residual effect of temperature. Since temperature (and thus the speed of sound) decreases with increasing altitude up to 11 km , sound is refracted upward, away from listeners on

8265-539: The speed of sound waves in air . However, the speed of sound varies from substance to substance: typically, sound travels most slowly in gases , faster in liquids , and fastest in solids . For example, while sound travels at 343 m/s in air, it travels at 1481 m/s in water (almost 4.3 times as fast) and at 5120 m/s in iron (almost 15 times as fast). In an exceptionally stiff material such as diamond, sound travels at 12,000 m/s (39,370 ft/s),  – about 35 times its speed in air and about

8360-490: The speed of sound, and the two contributions cancel out exactly. In a similar way, compression waves in solids depend both on compressibility and density—just as in liquids—but in gases the density contributes to the compressibility in such a way that some part of each attribute factors out, leaving only a dependence on temperature, molecular weight, and heat capacity ratio which can be independently derived from temperature and molecular composition (see derivations below). Thus, for

8455-402: The spheres represent the material's molecules and the springs represent the bonds between them. Sound passes through the system by compressing and expanding the springs, transmitting the acoustic energy to neighboring spheres. This helps transmit the energy in-turn to the neighboring sphere's springs (bonds), and so on. The speed of sound through the model depends on the stiffness /rigidity of

8550-533: The steeper the slope of the Rayleigh line, the stronger is the wave. On the contrary, here the specific volume ratio is restricted to the finite interval ( γ − 1 ) / ( γ + 1 ) ≤ v ~ ≤ 2 α + ( γ + 1 ) / ( γ − 1 ) {\displaystyle (\gamma -1)/(\gamma +1)\leq {\tilde {v}}\leq 2\alpha +(\gamma +1)/(\gamma -1)} (the upper bound

8645-429: The temperature and molecular weight, thus making only the completely independent properties of temperature and molecular structure important (heat capacity ratio may be determined by temperature and molecular structure, but simple molecular weight is not sufficient to determine it). Sound propagates faster in low molecular weight gases such as helium than it does in heavier gases such as xenon . For monatomic gases,

8740-528: The total mass of a sample. In these contexts an alternative term is mass percent composition . The mass fraction of an element in a compound can be calculated from the compound's empirical formula or its chemical formula . Percent concentration does not refer to this quantity. This improper name persists, especially in elementary textbooks. In biology, the unit "%" is sometimes (incorrectly) used to denote mass concentration, also called mass/volume percentage . A solution with 1   g of solute dissolved in

8835-400: The wave, u 1 and u 2 are the fluid velocity upstream and downstream of the wave, p 1 and p 2 are the pressures in the two regions, and h 1 and h 2 are the specific (with the sense of per unit mass ) enthalpies in the two regions. If in addition, the flow is reactive, then the species conservation equations demands that to vanish both upstream and downstream of

8930-422: Was 325 mm . This is longer than the standard "international foot" in common use today, which was officially defined in 1959 as 304.8 mm , making the speed of sound at 20 °C (68 °F) 1,055 Parisian feet per second). Derham used a telescope from the tower of the church of St. Laurence, Upminster to observe the flash of a distant shotgun being fired, and then measured the time until he heard

9025-435: Was missing the factor of γ but was otherwise correct. Numerical substitution of the above values gives the ideal gas approximation of sound velocity for gases, which is accurate at relatively low gas pressures and densities (for air, this includes standard Earth sea-level conditions). Also, for diatomic gases the use of γ = 1.4000 requires that the gas exists in a temperature range high enough that rotational heat capacity

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