Equations of motion for rays in a Snell's law medium Open Access

We obtain equations of motion for rays in a Snell's law medium with a variable index of refraction. Our main results are the explicit equations for ray velocity and acceleration as functions of time, explicitly given by Eqs. (11), (12) and rederived in (20), (21), (23), (26). The velocity equation for the horizontal direction has been previously given by Watson.¹ In this letter, we consider a stratified medium where the index of refraction is a function of position in the stratification direction. There are a number of ways that our equations can be derived and originally our derivation followed directly from Snell's law. This is presented in Sec. 3. However, after deriving the equations, we realized that they can be cast as Newtonian-like equations of motion but with a variable mass. This is presented in Sec. 2.

Before proceeding, we briefly mention some historical points pertaining to rays and Snell's law. In arguing for the particle theory of light, Newton showed that he could obtain Snell's law by assuming that light consists of particles.² Of course, subsequently it was accepted that light is a wave, and Maxwell's equations lead to Snell's law when light passes form one medium to another with a different index of refraction. Similar considerations apply to acoustic wave propagation. Currently wave propagation in materials with variable index of refraction has become of importance in many fields including sound propagation in the ocean and metamaterials.³ Perhaps the first to study propagation with variable index of refraction was Rayleigh, who addressed the propagation of light in the atmosphere.⁴ Of particular relevance to our considerations are references.^1,5–8 Traditionally, ray paths have been obtained by applying Fermat's principle of least time. Our method allows the explicit calculation of the velocity, acceleration and position of a ray as a function of time.

We consider a stratified medium where stratification is in the z direction (with positive pointing down) and the range is signified by x (with positive pointing to the right). Units are normalized appropriately. For the speed we use c(z) and for its derivative with respect to z we use c′(z). For this case, we take the forces to be

and we take the mass to be speed dependent

We derive the equations of motion in a number of ways and subsequently, in Sec. 6, we show that they lead to Snell's law. For variable mass, the equations motion have to be written in terms of momentum.^9,10 In particular for the x and z directions, we have

where (p_x, p_z), (v_x, v_z), and (F_x, F_z) are the momenta, velocities, and forces in the corresponding directions. Consider first the equation of motion in the x direction for the mass and forces as given by Eqs. (1) and (2),

Therefore, the equation of motion for the horizontal direction is

For the z direction we have, from Eqs. (1), (2), and (4),

which leads to

Equations (6) and (8) are the equations of motion. Equation (5) shows that momentum in the x direction is conserved and is a constant of the motion since

For emphasis we call this constant of the motion η,

We note that the equations of motion, Eqs. (6) and (8), may then be written, respectively,

From the equations of motion, Eqs. (11), one can verify that

In Sec. 5, we discuss the properties of the motion and prove that $1−4η2c2(z)$ never becomes imaginary.

We now show that the equations of motion, Eqs. (11) and (12), follow directly from Snell's law. Snell's law as usually written,

is equivalent to

for a stratified medium. The velocities in the x and z direction v_x and v_z are generally

From Eqs. (14) and (15), we have that

as defined in Eq. (10).

Using

it follows that

which upon substitution in Eq. (15) become

and hence,

which are our velocity equations, Eqs. (12), but here derived directly from Snell's law. The accelerations, Eqs. (11), are obtained by differentiating.

Alternatively, we now show that one can obtain the equations of motion from the Snell's law constraint η, as given by Eq. (16). Differentiating Eq. (16) we have

and hence,

which is the acceleration equation in the horizontal direction [Eq. (6)]. Rewriting η as

differentiation gives

which leads to the acceleration equation in the vertical direction [Eq. (8)],

Lagrange's equation involves positions and velocities; however, the standard forms of Lagrange's equations have to be modified when we have a variable mass. This has been done in many fields, and in particular, we use the formulation given in references.^11–14 For a two dimensional case with position dependent mass, Lagrange's equations are

The Lagrangian is the kinetic energy minus the potential energy,

where φ(x, z) and T are the potential and kinetic energy, respectively. We point out that if the mass and potential φ(x, z) are only functions of position we also have

which are the equations given in Refs. 11 and 12. We apply these equations to our case where the mass is a function of position and where the potential is

This potential yields the forces in Eqs. (1). The Lagrangian is then

We can see from Eq. (2) that the mass varies with position as

which shows that the right hand side of Eq. (30) is zero. Furthermore,

which shows that the right hand side of Eq. (31) is also zero. Therefore, Lagrange's equations, Eqs. (30) and (31), become

These equations now lead to the equations of motion as given by Eqs. (6), (8), and the constraint Eq. (10).

The Hamiltonian formulation involves positions and canonical momenta. As with the Lagrangian formulation, Hamilton's equations have to be modified to take variable mass into account. For the general two dimensional case with position-dependent mass, Hamilton's equations become

where p_x and p_z are the canonical momenta and are given by p_i = $(∂/∂vi)$L. For our case, the canonical momenta are

and hence, the Hamiltonian is

with the same potential [Eq. (32)] used for our Lagrangian. We now derive the equations of motion. From the first pair of Hamilton's equations [Eqs. (37)], we obtain

and from the second pair of the modified Hamilton's equations [Eqs. (38)], we obtain

Using Eqs. (32) and (34), these lead straightforwardly to

which lead to the equations of motion as given by Eqs. (6), (8), and (10).

From Eqs. (12) we see that the sign of $dx/dt$ is constant and given by the sign of the constant of the motion η, which is obtained from the initial conditions, η = $vx(z0)/2c2(z0)$⁠. Hence the particle always moves to the right or to the left depending on the sign of η. We point out that the equation for $dx/dt$ in Eqs. (12) has been previously given in Ref. 1.

Notice that for the $dz/dt$ equation in Eqs. (12) one would run into difficulty if the square root became imaginary. We now show that indeed it will never become imaginary. For convenience, we define Δ(z) = 1 − 4η²c²(z) and define the point z_* where Δ(z_*) = 0. At z_*, we have that 4η²c²(z_*) = 1 and from Eq. (12) we have that $dz/dt| z=z*$ = 0, yielding $d/dt Δ| z=z*$ = 0. Hence, we have to look at the second derivative of z at z = z_*. We obtain

which shows that if Δ(z) = 0 at a point, then it must increase at that point and hence can never go negative.

We obtain the equation for the path in the usual way, wherein one eliminates time to obtain a relationship between x and z.^9,10 We have $dz/dx=(dz/dt)(dt/dx)$ and using Eqs. (12) we obtain

Using Eq. (40) and taking into account variable mass we obtain

Equation (47) is general. For our case, we have

and substituting into Eq. (47), we obtain

where F_zv_z can be interpreted as the power. This shows that energy is not conserved although it may be conserved over a time interval.

We now show that Snell's law follows from conservation of momentum in the horizontal direction. Consider the usual horizontal two-medium situation with respective velocities c₁ and c₂. As we have shown, momentum in the horizontal direction is conserved and hence for the two medium case we have $px(1)$ = $px(2)$⁠, where $px(1)$ and $px(2)$ are the horizontal components of the momenta in media 1 and 2, respectively. Using Eqs. (2) and (39), we have

But because we are considering a horizontally stratified medium where the speed is only a function of z we have that

where the angle θ is measured from the horizontal. Hence, from Eq. (50),

giving

which is Snell's law for angles θ measured from the horizontal.

As usual with mechanics problems, the formulation that is used depends on the circumstances of the particular problem. We have found that the use of the velocity equations [Eqs. (12)] is the simplest from an analytic point of view. In Eqs. (12), η is obtained from the initial conditions, $η=vx(z0)/2c2(z0)$⁠. The general approach is to first solve the $dz/dt$ equation in Eq. (12),

This gives z(t) which is then substituted into the $dx/dt$ equation

to give

As an example consider the case where speed is a linear function of depth, c(z) = a + bz. The velocity equations are then

These equations can be solved exactly, and we give here only the final result,

and

where β is a constant determined by the initial conditions

We note that it follows that

This is a useful relation in manipulating the above equations. For the energy and its rate of change, one obtains

We have presented equations of motion for the velocity and acceleration of a ray in a medium with variable index of refraction. One can use these equations to not only calculate the ray path, which could alternatively be calculated by other means, such as the Bellhop algorithm¹⁵ and Fermat's principle, but also to obtain other information of interest. For example, one could obtain the response of a time-varying source as a function of delay time along the ray path. We have shown that Snell's law follows from conservation of momentum in the direction parallel to the interface, which is contrary to the usual historical particle view derivation of Snell's law.² Although we have considered the important case of a horizontally stratified medium, the general case is important, that is, a medium described by an arbitrary speed c = c(x, y, z). We note that the forces given in Eqs. (1) do not suggest any possible symmetry between the x and z directions which would be required for such a case. However, in our equations for the Lagrangian and Hamiltonian, there is an indication of such a symmetry. One possibility is to take the mass and potential to be a functions of x,y and z, m(x,y,z) = $1/[2c2(x,y,z)]$ and $φ(x,y,z)=(1/2)lnc(x,y,z)$⁠. This suggests that our Lagrangian and Hamiltonian formulations might be appropriate for describing a general medium. These issues are currently being studied.

Work supported by the Office of Naval Research.