give newtons formula for speed of sound in air hence explain laplace correction
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In order to determine the speed of sound in a particular medium, we need to know the medium’s elastic properties and its density. A medium’s elastic properties determine whether or not the medium will deform or lose its shape due to external forces. Sound travels faster in mediums with high elasticity and minimal deformity – like steel. Sound travels more slowly in less rigid mediums that deform easily – like rubber. Sound travels more slowly in mediums with greater density.
Sir Isaac Newton proposed a simple formula for calculating the speed of sound. Years later Pierre-Simon Laplace would revise Newton’s formula and the new formula would be called the Newton-Laplace Equation.
Developing the Equation
In the latter half of the 17thcentury, Sir Isaac Newton published his famous work Principia Mathematica. Newton thought that he correctly predicted the speed of sound though a medium: solid, liquid, or gas. If he knew the density of the medium and pressure acting on the sound wave, he believed he could ascertain the speed of sound by calculating the square root of the pressure divided by the medium’s density: c=√P
----
p
c = speed of sound P = pressure acting on the sound
p = density of the medium
On page 301 in The Principia Mathematica Newton states “in mediums of equal density and elastic force (the medium’s pressure in Pascals, or Newtons over area measured in meters squared), all pulses are equally swift”. In the earth’s atmosphere at sea level all sound will consistently travel at the same velocity. The same goes for other mediums like homogenius liquids and solids.
Pierre-Simon Laplace
Newton further states that “if density or the elastic force of the medium were increased, then, because the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of density, the time which is necessary for producing the same motion as before will be increased in the subduplicate ratio of the density, and will be diminished in the in the subduplicate ratio of the elastic force.” Ultimately, Newton is saying that if the medium’s density increases, the sound velocity slows and inversely if the medium’s pressure increases the sound velocity accelerates.
Later in the same century, French mathematician Pierre-Simon Laplace saw the flaw in Newton’s thinking and ultimately corrected Newton’s formula. He expanded Newton’s equation to include the idea that the process is not isothermic as Newton had thought, but it is adiabatic. Laplace slightly revised Newton’s formula by adding gamma to Newton’s pressure component. Laplace correction: c=√Gama P
----------
p
Newton’s formula neglected the influence of heat on the speed of sound. Laplace made this correction. Laplace multiplied the gamma (heat component) x the pressure. Experimentation proved that Newton’s results were wrong. Results from Newton’s equations fell short of what really took place. Laplace’s fix hit the mark. A new equation was born: The Newton-Laplace Equations. Laplace shorted the equation by having K = gamma × pressure. K refers to the elastic bulk modulus. The formula is called the Newton-Laplace equation:
c=√k
---
p
c = speed of sound
K = elastic bulk modulus
p = density of the medium
Sir Isaac Newton proposed a simple formula for calculating the speed of sound. Years later Pierre-Simon Laplace would revise Newton’s formula and the new formula would be called the Newton-Laplace Equation.
Developing the Equation
In the latter half of the 17thcentury, Sir Isaac Newton published his famous work Principia Mathematica. Newton thought that he correctly predicted the speed of sound though a medium: solid, liquid, or gas. If he knew the density of the medium and pressure acting on the sound wave, he believed he could ascertain the speed of sound by calculating the square root of the pressure divided by the medium’s density: c=√P
----
p
c = speed of sound P = pressure acting on the sound
p = density of the medium
On page 301 in The Principia Mathematica Newton states “in mediums of equal density and elastic force (the medium’s pressure in Pascals, or Newtons over area measured in meters squared), all pulses are equally swift”. In the earth’s atmosphere at sea level all sound will consistently travel at the same velocity. The same goes for other mediums like homogenius liquids and solids.
Pierre-Simon Laplace
Newton further states that “if density or the elastic force of the medium were increased, then, because the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of density, the time which is necessary for producing the same motion as before will be increased in the subduplicate ratio of the density, and will be diminished in the in the subduplicate ratio of the elastic force.” Ultimately, Newton is saying that if the medium’s density increases, the sound velocity slows and inversely if the medium’s pressure increases the sound velocity accelerates.
Later in the same century, French mathematician Pierre-Simon Laplace saw the flaw in Newton’s thinking and ultimately corrected Newton’s formula. He expanded Newton’s equation to include the idea that the process is not isothermic as Newton had thought, but it is adiabatic. Laplace slightly revised Newton’s formula by adding gamma to Newton’s pressure component. Laplace correction: c=√Gama P
----------
p
Newton’s formula neglected the influence of heat on the speed of sound. Laplace made this correction. Laplace multiplied the gamma (heat component) x the pressure. Experimentation proved that Newton’s results were wrong. Results from Newton’s equations fell short of what really took place. Laplace’s fix hit the mark. A new equation was born: The Newton-Laplace Equations. Laplace shorted the equation by having K = gamma × pressure. K refers to the elastic bulk modulus. The formula is called the Newton-Laplace equation:
c=√k
---
p
c = speed of sound
K = elastic bulk modulus
p = density of the medium
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