Comment on “What happens to energy and momentum when two oppositely-moving wave pulses overlap?,” by N. Gauthier [Am. J. Phys. 71 (8), 787–790 (2003)]

N.

Gauthier

, “,” , – ().

P. M. Morse and K. U. Ingard, Theoretical Acoustics (McGraw–Hill, New York, 1968), Sec. 4.1.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw–Hill, New York, 1953), Sec. 2.1.

R.

Benumof

, “,” , – ().

W. C. Elmore and M. A. Heald, Physics of Waves (McGraw–Hill, New York, 1969).

D. R.

Rowland

and

C.

Pask

, “,” , – ().

In continuum mechanics, it is important to distinguish between material coordinates and spatial coordinates. In this case, the spatial coordinates of the ith element of the string are $(xi(t),yi(t),zi(t))=(Xi+ξi,ηi,0),$ where $(Xi,0,0)$ are the material coordinates of this element. We can either treat t and X as the independent variables and x as a dependent variable (as is done in this paper), or treat t and x as the independent variables and X as a dependent variable (see Ref. 8).

A. G.

Herrmann

, “,” , – ().

D. F.

Nelson

, “,” , – ();

I. H.

Gilbert

and

B. R.

Mollow

, “,” , – ().

Another way of seeing that $−ρ(∂η/∂t)(∂η/∂X)$ is an expression for pseudomomentum rather than real momentum is as follows. Reference 8 shows how conservation laws (more generally balance laws) can be derived by differentiating the Lagrangian density of a system with respect to the independent variables. The differentiation of the Lagrangian density with respect to the time leads to a balance law for energy. If the Lagrangian density has been written so that the spatial coordinates are treated as independent variables, then differentiating with respect to the spatial coordinates leads to a balance law for real momentum. However, if the Lagrangian density is written so that material coordinates are treated as the independent variables (as is implicitly the case in this paper), then differentiating with respect to material coordinates leads to a balance law for material momentum/pseudomomentum.

Generally speaking, if a physical quantity obeys an equation of the form, ∂(density of physical quantity)/∂t=−∇⋅(current density of physical quantity), then it follows from the divergence theorem of vector calculus that the quantity is conserved and the given equation is called a “continuity equation.” If, however, the time rate of change of the density of a physical quantity cannot be written purely as the divergence of some quantity (interpreted as a current density), then it follows from the divergence theorem that the physical quantity is not conserved and the equation is then called a “balance” rather than a continuity equation.

Reference 2, Sec. 14.3.

Juenker (Ref. 18) has noted that purely transverse motion is only possible in strings for which $cT=cL,$ which requires the string to have a zero relaxed length, a situation approximated by slinky springs.

N.

Giordano

and

A. J.

Korty

, “,” , – ().

D. R.

Rowland

, “,” , – (), Sec. VI.

Actually, this procedure produces a cusp in the longitudinal velocity of the end supports at the peak of the Gaussian and because this cusp could not be modeled precisely numerically, small L waves were in fact generated.

If it seems counterintuitive that parts of a L wave can have momenta opposite to the direction of propagation, consider a stretched slinky spring, one of whose ends is given a sharp impulse that first moves the end one way and then the other, with the end being brought back to rest where it started. As this disturbance propagates, elements of the spring are likewise first moved one way and then the other and are left where they started after the disturbance has passed. Obviously, in this case, parts of the disturbance move in a direction opposite to the direction of propagation of the disturbance.

D. W.

Juenker

, “,” , – ();

K. U. Ingard, Fundamentals of Waves and Oscillations (Cambridge University Press, Cambridge, UK, 1988), Sec. 7.3.

By symmetry, the total potential energy due to longitudinal motion would be zero for all normal modes of transverse vibration when the end supports do not move longitudinally.

The reason this is only approximate is because of the small kinetic energy associated with longitudinal motion.

Walstad has shown that for an isolated T wave, the correct longitudinal momentum density can be obtained by neglecting longitudinal motion if one calculates the longitudinal force density arising from a constant tension $τ0$ and the curvature of the string. However, such an analysis is of no use for determining what happens if the T wave interacts with anything.

See,

A.

Walstad

, “,” , – ().

D. R.

Rowland

, “,” , – ().