Phy 315 Homework 6 Wave Motion and Optics

PHY 315 Homework 6 Wave Motion and Optics

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PHY 315 Homework 6 Wave Motion and Optics

Problem 1: Sound in gases – Newton vs. Laplace

Background: Isaac Newton’s calculation of the speed of sound in a gas was found to produce results about 15% less than measured values in air. Laplace corrected Newton’s formula to give better results. We will explore the subtletly involved in this problem, and “derive” both formulas.

From fluid mechanics, one can derive the formula for the speed of sound waves in a fluid. When one linearizes the equations of fluid mechanics, supplemented with what is known as an equation of state, one obtains a wave equation with the speed

v= sqrt(dp/dρ) The way to obtain this for a specific fluid is to know the equation of state, which gives the dependence of pressure on density, p(ρ) for the fluid under consideration. It is worth noting that the result makes some sense because the speed of a wave usually has the form

sqrt(Restoring property/Inertial property)

Setup: The one equation of state we know very well is that of an ideal gas, pV = nRT. This may also be re-written as pM = ρRT , where M is the molar mass and ρ is the density. To see how, note that n moles of gas have mass nM and density ρ is nM/V .

Part 1 – Newton: Assuming that the process of compressions and rarefactions is isothermal (i.e. T remains constant), use the ideal gas law in equation (1) to show that

v = sqrt(RT/M) = sqrt(p/ρ)

This is Newton’s formula, which was shown to not capture the physics under ordinary conditions.

Some more background: Recall that there is a special thermodynamic process that does not exchange heat with the environment, which we call an adiabatic process. This usually occurs when there is insufficient time to facilitate heat exchange. Such a process for an ideal gas is characterized by the property that pV γ = constant, where the quantity γ = CP /CV is called the...