where:

vca is the RMS colliding speed of particles.

For air at 15 C, the RMS colliding speed of particles can be calculated backward from the formula above as:

Note that the RMS colliding speed of air particles is faster than the speed of sound and is related to the ratio, γ, of specific heats. Also, the ratio, γ, of specific heats is related to the ratio, α, as:

γ 3α þ 2 3α

The corresponding value of α of air is:

α ¼ 53

where α is a ratio between the total energy (rotational energy + translational energy) and the translational energy of molecules as:

12 Iaθ2ca þ 12 mav2ca ¼ α 12 mav2ca

2.6Homework Exercises

Exercise 2.1 The bulk modulus B and the density ρo of certain seawater are:

B ¼ 2:307 109 ½Pa&

hKgi

m3

Determine the speed of sound in this certain seawater.

m s

Exercise 2.2 Sound can propagate through gaseous He which has a molar mass:

h g i

M ¼ 4:003 mol

Determine the speed of sound in He at 15 C.

Hint: The ideal gas constant is the same for all gases. The ratio of specific heats

for He is 1.67.

(Answer) 1000 ms

Exercise 2.3 The average molar mass of the molecules in the air is given as:

h g i

M ¼ 28:95 mol

What is the speed of sound at 30 C?

m s

In Building Block 1, basic complex solutions (BCSs) are constructed based on the acoustic wave equation under the requirement ωk ¼ c. BCSs are the most direct solutions of the acoustic wave equation. But because BCSs are complex numbers, they have no physical attributes by themselves.

In Building Block 2, basic traveling waves (BTWs) are constructed using the BCSs under the requirement that the BCSs must be a complex conjugate pair. BTWs are the addition of a complex conjugate pair. Unlike BCSs, BTWs are real numbers and have physical attributes of traveling waves. Any traveling wave can be constructed by using the four BTWs.

In Building Block 3, basic standing waves (BSWs) are constructed using the BTWs under the requirement that two traveling waves are traveling in the opposite direction and have exactly the same amplitude. Any standing wave can be constructed by using the four BSWs.

These “basic” solutions and waves will be used as building blocks for acoustic analysis throughout this course. Basic solutions and waves allow for a more in-depth analysis of acoustics and noise control.

Based on the traveling waves, this chapter will demonstrate how to construct standing waves by adding two traveling waves that are moving in the opposite direction and having the same amplitude: