Solution Found!
By applying Newton’s laws to the oscillations of a
Chapter 1, Problem 39P(choose chapter or problem)
Problem 39P
By applying Newton’s laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by
where ρ is the density of the medium (mass per unit volume) and B is the bulk modulus, a measure of the medium’s stiffness. More precisely, if we imagine applying an increase in pressure ΔP to a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:
This definition is still ambiguous, however, because I haven’t said whether the compression is to take place isothermally or adiabatically (or in some other way).
(a) Compute the bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions.
(b) Argue that for purposes of computing the speed of a sound wave, the adiabatic B is the one we should use.
(c) Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the rms speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature.
(d) When Scotland’s Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?
Questions & Answers
QUESTION:
Problem 39P
By applying Newton’s laws to the oscillations of a continuous medium, one can show that the speed of a sound wave is given by
where ρ is the density of the medium (mass per unit volume) and B is the bulk modulus, a measure of the medium’s stiffness. More precisely, if we imagine applying an increase in pressure ΔP to a chunk of the material, and this increase results in a (negative) change in volume ΔV, then B is defined as the change in pressure divided by the magnitude of the fractional change in volume:
This definition is still ambiguous, however, because I haven’t said whether the compression is to take place isothermally or adiabatically (or in some other way).
(a) Compute the bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions.
(b) Argue that for purposes of computing the speed of a sound wave, the adiabatic B is the one we should use.
(c) Derive an expression for the speed of sound in an ideal gas, in terms of its temperature and average molecular mass. Compare your result to the formula for the rms speed of the molecules in the gas. Evaluate the speed of sound numerically for air at room temperature.
(d) When Scotland’s Battlefield Band played in Utah, one musician remarked that the high altitude threw their bagpipes out of tune. Would you expect altitude to affect the speed of sound (and hence the frequencies of the standing waves in the pipes)? If so, in which direction? If not, why not?
ANSWER:
Solution 39P
The bulk modulus of a substance is a measure of how compressible that substance is. It compares the fractional change in volume to the pressure applied to create that change. It is defined as
where the minus sign in the denominator is there because an increase in pressure causes a decrease in volume.
However, the change in volume due to a change in pressure depends on the nature of the compression.
Step 1 of 4
The bulk modulus of an ideal gas, in terms of its pressure P, for both isothermal and adiabatic compressions can be computed as follows.
For an isothermal compression
=
=
=
=
For an adiabatic compression
=
=
=
=