 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| G. G. Denisov, "On the pressure of waves on an obstacle in the case of transverse vibrations of a string," Mech. Solids. 36 (5), 151-156 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
5 |
Pages |
151-156 |
| Title |
On the pressure of waves on an obstacle in the case of transverse vibrations of a string |
| Author(s) |
G. G. Denisov (Nizhni Novgorod) |
| Abstract |
Interest in the action of waves on a reflecting obstacle has appeared
since long ago. This interest has been supported by the hypothesis that
waves of any physical nature interacting with an obstacle exert
on the latter nonzero mean pressure, as is the case for the pressure
exerted by electromagnetic waves on a reflecting surface. One of the first
publications on this topic is the paper [1] by Rayleigh. Rayleigh
has demonstrated this phenomenon on a simple mechanical model consisting
of a vibrating string and a point mass attached to it; the string is passed
through the hole of a ring that can be moved along the vertical. The average
pressure on the ring has been determined depending on the parameters
of the model and the vibration amplitude and frequency. Some general
observations concerning the influence of vibrations have been made.
In his other work [2], following Poynting's paper [3], Rayleigh
has assumed that to exert pressure on an obstacle when reflecting
from it, the wave package must possess a momentum. This hypothesis
has been confirmed for longitudinal plane waves in fluids [2].
The behavior observed for transverse vibrations of one-dimensional
elastic systems (e.g., beams or strings) is quite different. During
the propagation of transverse waves each particle, in the first
approximation, vibrates perpendicularly to the direction of the wave
propagation and does not contribute to the momentum of the wave.
In the second approximation, the particles vibrate
both transversely to and along the string,
but nevertheless, the momentum does not appear, since the translatory
motion along the elastic system on the average does not occur. However,
an obstacle interacting with transverse vibrations and waves is subject
to pressure under certain conditions [1, 4, 5], which contradicts
Poynting's and Rayleigh's observations. To resolve this paradox, the
authors of a number of recent publications [5-8] introduce
the concept of the wave momentum
and explain the appearance of the pressure (which is referred to
as the wave pressure) by the change of the wave
momentum during the interaction of the wave with an obstacle.
In these publications, the concepts of the wave momentum and pressure apply
to both longitudinal and transverse waves. These wave characteristics
are defined only for distributed systems and are quadratic in the amplitude.
The direction of the wave momentum coincides with that of the wave
propagation and the wave pressure is always positive, i.e., the incident
wave always pushes the obstacle ahead. The wave momentum is possessed
also by linear waves, in which each particle vibrates
harmonically, the translatory component of the motion is absent and, hence,
neither transverse nor longitudinal waves have a nonzero classical momentum.
The equations of motion of distributed systems are derived from Newton's equations.
Based on this observation, it is natural to suggest that the classical
concepts of momentum and pressure are quite sufficient for interpreting the results
of analysis of wave processes. Note also than the wave momentum and wave
pressure are not mentioned in the familiar courses of theoretical mechanics [9,10].
Therefore, one should seek the solution of this contradiction within the framework
of the classical mechanics.
It turners out that the solution of this problem is different for longitudinal
and transverse waves. For longitudinal waves, the pressure on the reflecting
obstacle is accounted for by the momentum carried by the incident wave,
thus confirming the point of view of Rayleigh and Pointing [2, 3]. For this
momentum to occur, the mass transfer (the motion of the center of mass of
the medium region occupied by the wave) is necessary. This mass transfer
follows from the initial conditions and nonlinear properties of the medium and
respective equations [11, 12]. For longitudinal vibrations, it is shown that
the wave reflected by an obstacle can exert either positive or negative pressure
on this obstacle and that the wave momentum does not have a physical meaning.
For transverse vibrations, Poynting's and Rayleigh's hypotheses are not
always satisfied, since the pressure of transverse waves on an obstacle
interacting with these waves can occur also in the case where the momentum
of the incident wave package is equal to zero. The present paper is devoted
to the solution of these contradictions. |
| References |
| 1. | Rayleigh, "On the pressure of vibrations," Phil. Mag., Vol. 3,
pp. 338-346, 1902. |
| 2. | Rayleigh, "On the momentum and pressure of gasseous vibrations and
on the connexion with virial theorem," Phil. Mag., Vol. 10, pp. 364-374,
1905. |
| 3. | T. H. Poynting, "Radiation pressure," Phil. Mag., Vol. 9,
pp. 393-407, 1905. |
| 4. | E. L. Nikolai, "To the issue of vibration pressure,"
Izv. St. Peterburg. Politekh. In-ta,
Vol. 18, No. 1, pp. 49-60, 1912. |
| 5. | A. I. Vesnitskii, L. E. Kaplan, and G. A. Utkin,
"Laws of change of energy and momentum for one-dimensional systems
with moving attachment," PMM [Applied Mathematics and Mechanics],
Vol. 47, No. 5, pp. 863-866, 1983. |
| 6. | A. I. Vesnitskii and G. A. Utkin, "Motion of a body along a string
under the action of wave pressure forces," Doklady AN SSSR, Vol. 302,
No. 2, pp. 278-280, 1988. |
| 7. | G. A. Utkin, "Statement of problems of dynamics of elastic
systems with objects moving along them,"
in Wave Mechanics of Machines [in Russian], pp. 4-14, Nauka, Moscow, 1991. |
| 8. | A. I. Vesnitskii and A. V. Metrikin, "Transient radiation in
mechanics," Uspekhi Fizicheskikh Nauk [Advances in Physics], Vol. 166,
No. 10, pp. 1043-1068, 1996. |
| 9. | L. D. Landau and E. M. Lifshits, Theoretical Physics. Volume 6.
Hydrodynamics [in Russian], Nauka, Moscow, 1986. |
| 10. | A. A. Il'yushin, Continuum Mechanics [in
Russian], Izd-vo MGU, Moscow, 1990. |
| 11. | G. G. Denisov, "On the wave momentum and forces appearing
on the boundary of a one-dimensional elastic system," Izv. AN. MTT
[Mechanics of Solids], No. 1, pp. 42-51, 1994. |
| 12. | G. G. Denisov, "To the issue of the momentum of a wave, radiation
pressure, and other quantities in the case of plane motions
of an ideal gas," PMM [Applied Mathematics and Mechanics], Vol. 63,
No. 3, pp. 390-402, 1999. |
| 13. | A. N. Tikhonov and A. A. Samarskii,
Equations of Mathematical Physics [in Russian], Gostekhizdat, Moscow, 1953. |
| 14. | O. A. Goroshko and G. N. Savin, Introduction to Mechanics
of Deformable One-dimensional Bodies of Variable Length [in Russian],
Naukova Dumka, Kiev, 1971. |
|
| Received |
08 July 1999 |
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