0.02μm. The weakest sound that can be heard by the human ear is dependent on the
frequency; at about 400 Hz the threshold of audibility is approximately equal to
10
À6
μWm
À2 , hence, the displacement amplitude is given by
a m ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2I
ω 2 ρ 0 c 0
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2 Â 1 Â 10
À12
2π Â 400
ð
Þ
2 Â 1:25 Â 340
s
¼ 2:73 Â 10
À11 m,
which is about 0.25 Angstrom units!
1.9.2 Loud Sounds
A sound wave can be considered loud when the intensity approaches about
1Wm
À2 and sound waves can become quite painful when the intensity reaches
about 10Wm
À2 . The excess pressure in this latter extreme case is above 90Nm
À2 ,
which is about 0.1% of the atmospheric pressure and the maximum amplitude a m for
molecular displacement at a frequency of 400 Hz is approximately 0.08 mm. The
velocity amplitude is about 0.22ms
À1 or approximately 0.06% of speed of sound.
References
1. J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic
shocks. J. Appl. Phys. 21, 232–237 (1950)
2. M. A. Saad, Compressible Fluid Flow, (Prentice-Hall, Inc., Englewood Cliffs, New Jersey
1985)
3. G. Falkovich, Fluid Mechanics: A Short Course for Physicists, (Cambridge University Press,
New York, 2011)
4. V. Babu, Fundamentals of Gas Dynamics, 2nd edn. (John Wiley & Sons Ltd., Chichester,
United Kingdom, 2015)
5. O. Regev, O.M. Umurhan, P.A. Yecko, Modern Fluid Dynamics for Physics and Astrophysics
(Springer, New York, 2016)
6. J.H.S. Lee, The Gas Dynamics of Explosions (Cambridge University Press, New York, 2016)
Chapter 1
7. J.D. Anderson Jr., Computational Fluid Dynamics: The Basics with Applications (McGrawHill, New York, 1995) Chapter 2
8. J.D. Logan, An Introduction to Nonlinear Partial Differential Equations (John Wiley & Sons,
New York, 1984) Chapter 5
9. J. von Neumann, Theory of Shock Waves, Collected Works, vol 6 (Pergamon Press, New York,
1976), p. 178
10. H. A. Bethe et al LA-165 Report (Los Alamos Scientific Laboratory of the University of
California, Los Alamos, New Mexico, October 1944), Section 1
References
41
frequency; at about 400 Hz the threshold of audibility is approximately equal to
10
À6
μWm
À2 , hence, the displacement amplitude is given by
a m ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2I
ω 2 ρ 0 c 0
r
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2 Â 1 Â 10
À12
2π Â 400
ð
Þ
2 Â 1:25 Â 340
s
¼ 2:73 Â 10
À11 m,
which is about 0.25 Angstrom units!
1.9.2 Loud Sounds
A sound wave can be considered loud when the intensity approaches about
1Wm
À2 and sound waves can become quite painful when the intensity reaches
about 10Wm
À2 . The excess pressure in this latter extreme case is above 90Nm
À2 ,
which is about 0.1% of the atmospheric pressure and the maximum amplitude a m for
molecular displacement at a frequency of 400 Hz is approximately 0.08 mm. The
velocity amplitude is about 0.22ms
À1 or approximately 0.06% of speed of sound.
References
1. J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic
shocks. J. Appl. Phys. 21, 232–237 (1950)
2. M. A. Saad, Compressible Fluid Flow, (Prentice-Hall, Inc., Englewood Cliffs, New Jersey
1985)
3. G. Falkovich, Fluid Mechanics: A Short Course for Physicists, (Cambridge University Press,
New York, 2011)
4. V. Babu, Fundamentals of Gas Dynamics, 2nd edn. (John Wiley & Sons Ltd., Chichester,
United Kingdom, 2015)
5. O. Regev, O.M. Umurhan, P.A. Yecko, Modern Fluid Dynamics for Physics and Astrophysics
(Springer, New York, 2016)
6. J.H.S. Lee, The Gas Dynamics of Explosions (Cambridge University Press, New York, 2016)
Chapter 1
7. J.D. Anderson Jr., Computational Fluid Dynamics: The Basics with Applications (McGrawHill, New York, 1995) Chapter 2
8. J.D. Logan, An Introduction to Nonlinear Partial Differential Equations (John Wiley & Sons,
New York, 1984) Chapter 5
9. J. von Neumann, Theory of Shock Waves, Collected Works, vol 6 (Pergamon Press, New York,
1976), p. 178
10. H. A. Bethe et al LA-165 Report (Los Alamos Scientific Laboratory of the University of
California, Los Alamos, New Mexico, October 1944), Section 1
References
41