The Lagrangian approach to simple waves - several common waves that lack momentum

summary	outline	1.0 mass-spring	2.0 waves on a spring	2.1 energy and momentum	2.2 computer simulation
2.3 adding tension	3.0 mixed waves	4.1 transverse waves with tension	5.0 sound	5.1 shocking	5.2 computer simulation

 

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This posting includes flash animations showing the physics discussed. Most computers have a flash player already installed, but if yours does not, download the free Adobe flash player here. Flash animations: Fig. 1.1. Flash animation of mass on a spring Fig. 2.1. Flash animation of longitudinal waves Fig. 2.5 Flash animation of ring resonator Fig. 2.8 Flash animation of waves in linear simulation setup Fig. 2.9 Flash animation of waves in circular simulation setup Fig. 3.1. Flash animation of mixed longitudinal and transverse waves Fig. 5.1. Flash animation of sound propagation

The Lagrangian approach to simple waves - several common waves that lack momentum

revised June 2018

Summary

In this article we demonstrate use of the Lagrangian for a mass on a spring and then for wave propagation in one dimension in several different media. For most of the wave media presented, the generalized coordinate s is used, which is simply the x position of each mass element in its equilibrium position. In spite of the fact that the mass elements move around during the passage of a wave, we identify each by its equilibrium position s. We also use the variable ξ which is the displacement of a mass element from its equilibrium position. When a wave is present ξ is a function of s and time, t.

For waves on an elastic cord and for sound waves, we found that the average momentum transported by the waves is zero, in contrast to the usual belief that these waves possess net momentum. We verified our results with computer simulations which relied on extremely basic equations of physics: F = ma for motion of the mass elements and F = −k Δℓ for the springs (Newton's second law and Hooke's law). This article has several Flash animations as listed above.

Outline

1. Lagrangian for a mass on a spring Fig. 1.1. Flash animation of mass on a spring 2. Lagrangian for linear waves Fig. 2.1. Flash animation of longitudinal waves Fig. 2.2. Lumped parameter model for longitudinal waves Eqn. 2.1. Lagrange equation for one-dimensional waves Kinetic energy in longitudinal waves Potential energy in longitudinal waves Fig. 2.3. Changing from generalized coordinate to Cartesian coordinate Octave simulation Fig. 2.4. Linear simulation setup Fig. 2.5. Flash animation of ring resonator Fig. 2.6 Waves in linear simulation setup Fig. 2.7 Waves in circular simulation setup Fig. 2.8 Flash animation of waves in linear simulation setup Fig. 2.9 Flash animation of waves in circular simulation setup Table 2.1. Results of octave simulation in linear setup Table 2.2. Results of octave simulation in circular setup The addition of static tension 3. Mixed longitudinal and transverse waves in linear media Fig. 3.1. Flash animation of mixed longitudinal and transverse waves References 4. Unstretchable string 5. Sound Fig. 5.1. Flash animation of sound propagation Fig. 5.2. Comparison of adiabatic ideal gas and linear spring Eqn. 5.4. Potential energy in a gas parcel Eqn. 5.8a Wave equation for sound Fig. 5.3. Development of shocking in a sound wave Eqn. 5.24. Momentum in a sound wave Table. 5.1. Octave simulation results for sound in a circular setup

The Lagrangian is a mathematical construct used to solve classical mechanics problems in a very general way. It allows use of odd coordinate systems chosen for convenience for a particular problem at hand. We demonstrate its use here for several wave media.

Flash animation of mass on a spring
Fig. 1.1. Mass on a spring. Mouse over to see the action. Mouse off it to suspend it and click on it to restart it. If your computer does not support Flash click on the video to the left.

1. Lagrangian for a mass on a spring

The Lagrangian is defined to be the difference between the kinetic energy and potential energy in a system. For example for a mass supported on the end of a spring (a classical resonator - see Fig. 1.1) we write:

    ,   (1.1)

where η is the vertical displacement from equilibrium of the mass and k is the spring constant. Note the dot over the η in the first term, standing for a time derivative of η ,  i.e.  .  We insert our L into the standard Lagrange equation:

    .   (1.2)

One oddity of using the Lagrange equation is that we are to treat η and as separate, distinct and unrelated variables for the purpose of differentiation.

Another oddity of this equation is that it specifies that we take derivatives with respect to the unknowns η and . Most differential equations used in physics are written in terms of derivatives with respect to time and position, not in terms of derivatives with respect to the unknowns such as η and  .  Plugging ahead and inserting L from (1.1) we get:

    ,   (1.3)

which is the standard differential equation governing the motion of a mass on a spring. It has standard oscillatory sinusoidal solutions. For example see this reference.

2. Lagrangian for one-dimensional waves in a medium of masses and linear springs

Flash animation of longitudinal waves - mouse over to activate. Mouse off it to suspend it and click on it to restart it.
Fig. 2.1. Animation of longitudinal waves in an array of masses interconnected by linear springs. This is a good model for a continuous stretchy string or spring as long as there are many masses in a wavelength of the wave being observed. Color is used to enhance the visibility of the regions of spring compression and stretching.

Video of longitudinal waves - click on to start

Lumped parameter model for longitudinal waves
Fig. 2.2. Mass and spring model for waves on a slinky (the brand name of a long spring sold as a toy). The springs are considered massless and linear, i.e. spring force = −k Δx where k is the spring constant which does not change with stretching of the spring. The variable Δx is the amount of stretching of the spring.    The s - ruler shows the x coordinates that the masses will rest above when there are no waves. In this illustration we show the effects of a wave passing through the system of masses and springs. The dashed lines connecting the masses to numbers on the s - ruler are purely to show which mass would rest on which number when at rest. There are no forces associated with the dashed lines.    We use the coordinate s to be our generalized coordinate. For a particular mass, s does not change with time even when waves are present. The displacement ξ is the horizontal distance a mass is from its resting place on the s axis. This displacement will change due to waves as a function of s and time t, i.e. ξ(s, t). The x position of a mass also changes with time and is given by x = s + ξ.    The distinction in mass locations is similar to the distinction between Lagrangian and Eulerian coordinates encountered in fluid dynamics where use of our s would be considered Lagrangian.

The first type of wave we discuss is a longitudinal wave on a stretchy cord or slinky (see Figs. 2.1 and 2.2 above as well as a nice video of longitudinal and transverse waves). As we start out, the math below assumes that there is zero tension on the string of masses and springs when they are in equilibrium. Observing this with a normal slinky would require permanently prestretching it so that it is not collapsed on itself at zero tension. How to support the slinky to counter the effects of gravity can also be a problem. We will ignore the gravity problem in this analysis.

We assume that this system can be modeled as masses connected by linear springs as shown in Fig. 2.2. We need to solve for the displacement from equilibrium of each mass.

We use as our ‘‘generalized'' coordinate system a number s equal to the equilibrium x position that a particular mass has when no waves are present (as illustrated in Fig. 2.2). Furthermore, note that the distortion, ξ, of a particular mass is the distance that mass is from its equilibrium position, from its position when no waves are present. In general, ξ will be a function of generalized coordinate s and time t, so we can write ξ(s,t). One of the big advantages of using this generalized coordinate system is that the mass per unit s remains unchanged by passing waves. It is also true that the spring constant for a unit s is unaffected by wave motion.

Note that the variable η for the previously solved mass-on-a-spring problem was only a function of the time coordinate t. We wrote the solution for that as η(t). Because the linear wave system we are now considering has two independent coordinates, s and t, a solution for this is written as ξ(s,t). We need to broaden the Lagrange equation eqn (1.2) to include derivatives with respect to both coordinates s and t [reference]:

    ,      (2.1)

where is defined to be (i.e with respect to time) and ξ ' is similarly defined to be the other derivative,  ,  i.e. the derivative with respect to the generalized coordinate s. Another change we've made is to replace the simple Lagrangian L with the Lagrangian density L₁ which is just the Lagrangian per unit length, as is necessary for a continuum wave medium.

In this wave medium of masses and linear springs the Lagrangian density is:

    ,      (2.2)

where m₁ is the mass per unit length (in coordinate s) of the string assembly, and k₁ is the spring constant of a unit length (in s) of the assembly.

Also Δℓ(s,t)/Δs is the change in length of the spring at position s at time t from its non-stretched length x₀ , per non-stretched length (i.e. per unit s). We can relate this to the distortion in an incremental system as:

    ,      (2.3)

or in a continuum system we can write this as:

    ,      (2.4)

where, again, ξ ' is defined to be  . This makes our Lagrangian density become:

    .      (2.5)

The Lagrange equation now becomes:

    .      (2.6)

Note that we could have derived (2.6) using F = ma in terms of the s coordinate.

Also note that Eq. (2.6) is the simple wave equation having d'Alembert's solutions:

    .      (2.7)

Equation (2.7) is the standard (i.e. well accepted) solution to the wave equation with one exception: it is written in terms of s, the coordinate specifying the resting positions of the masses, instead of the more usual dynamic positions of the masses x .  The best known solutions of (2.7) are sinusoidal, such as:

    .        (2.8)

where κ is the wavenumber (radians per meter of s) and ω is the angular frequency (radians per second) of the wave. We can relate κ and ω to m₁ and k₁ with the simple relation:

      ,      (2.8')

which gives the phase velocity v_p, 2nd ref. This equation is derivable from a simple but careful examination of (2.6) and (2.8).

Note that (2.8) is an exact solution, not just a small amplitude, linearized solution. In contrast the versions of (2.7) and (2.8) written in terms of x instead of s are only valid in the small amplitude regime.

Energy and momentum

Using (2.8) and the first term of (2.5) we get the kinetic energy in the wave:

    ,      (2.8a)

which we see has an average value of  . We have used the trig identity sin²x = ½ − ½cos2x.

Similarly, using (2.8) with the second term of (2.5), the potential energy becomes:

    ,      (2.9)

which has an average value of  . A little algebra will show that the kinetic energy equals the potential energy (use v = √k₁/m₁ = ω/κ  for the wave velocity with respect to the coordinate s). The generalized momentum density in our sinusoidal wave (eqn. (2.8) ) is given by:

    .      (2.10)

Because the coordinate s is attached to the masses themselves, the s density, the mass per unit s, does not change due to a passing wave. The generalized momentum is thus sinusoidal and averages to zero. That is  .

We might ask if this means that the regular momentum averages to zero.

1. The regular momentum density is the product of mass density m1 times velocity v of the masses. 2. First, is the change in displacement of a mass with respect to time and thus is just the regular velocity. 3. The mass density is mass per length of string. 4. The average momentum density is given by:       ,    (2.10a) because the average s-momentum density is zero. So we conclude: yes, the regular momentum density averages to zero. We verified this finding numerically in an Octave simulation. The results are shown in Table 2.1 below.

Effect of changing from s generalized coordinate to x Cartesian coordinate
Fig. 2.3. Graph showing velocities of the masses as a function of the two horizontal coordinate systems: of the generalized coordinate s (in blue) and of the Cartesian coordinate x (shown in green). While the blue trace is a normal sinusoid, the green trace is a distorted sinusoid. The green trace is narrower in the positive half cycle (where the density is greater) and broader in the negative half cycle where the density is less. With respect to average momentum the density variations cancel the distortion effect and produce zero average momentum in these waves. The density in terms of x equals m1/[1 - dξ/dx] which is plotted in red. While the density versus x varies over the wave cycle, the density versus s is constant and equals m1.

We might take a moment to ponder the difference between a solution in terms of the generalized coordinate s and one in terms of the regular spatial coordinate x. If we have a nice sinusoidal solution in terms of s, a graph of it will appear sinusoidal as shown in blue in Fig. 2.3. However because s is, in general, displaced from x by a passing wave, we might expect the wave, when plotted versus x, to appear as plotted in green in the same graph, i.e. somewhat warped by horizontal displacements. The equation that relates x to s and ξ is:

x = s + ξ     ,     (2.11)

where for a particular mass, s equals the unstretched x position of that mass and ξ is the displacement from equilibrium.

In terms of momentum, we have equal masses in the half cycles labeled a and b. In terms of the graph having x as its axis (the blue one), we see that the upward half cycle is compacted and has a width of "b". This decreased width is compensated for by an increase in mass density during this half cycle shown by the red line. The opposite is true for the lower half cycle: it has increased width and decreased density. Thus the two effects, the width and the density, cancel leaving equal and opposite momentums in the two half cycles.

A reader might also note that a given physical mass, at a specific time, has a displacement ξ and velocity v = dξ/dt independent of which coordinate system (s or x) we work in.

Computer simulations … energy and momentum The figures below show the setups for two Octave simulations. The springs between the masses (black rectangles) were assumed to be linear. In both Octave simulations we use Newton's second law and Hooke's law for a linear spring to calculate the dynamics of wave propagation for the particular setup. Because the simulations rely on extremely basic equations of physics, we believe the simulations are good alternate checks on the results of the Lagrange algebraic methods above. In order that we test for sinusoidal waves it is important that we try to achieve single frequency results in the simulations. After all, a wave that is a pulse will transport net momentum (and mass) since it only has a positive part and no compensating negative half cycle. We minimize the non-sinsuoidal waves by exciting the setups using a pure sinusoidal excitation. The ring resonator is somewhat better in this regard because the excitation used there is an extremely long sinusoid. On the other hand, use of the ring resonator requires us to determine and use the exact resonant frequency of the ring resonator for the excitation signal.

Computer simulation results Progression of longitudinal waves in a linear array Slow build up of longitudinal rotating waves in a ring resonator ⇑ Fig. 2.4. In the first simulation, a burst of waves were introduced on the left side and observed to propagate towards the right. The simulation ceased when the waves reached the right side. The calculations showed that the potential and kinetic energies were equal. They also demonstrated that the momentum was orders of magnitude smaller than these energies divided by their velocity (the commonly held relationship) and certainly consistent with zero as derived above. Some numbers from this simulation are reported below in Table 2.1. ⇑ Fig. 2.5. In the second simulation, we used a ring resonator setup, where waves traveled round and round the ring, slowly building up. This simulation also showed the potential and kinetic energies to be equal while the momentum was orders of magnitude smaller than the energies divided by the wave velocity. The actual simulation used hundreds of masses around the ring. Some of the results from this simulation are reported below in Table 2.2. This is a flash animation. To see the action mouse over the image, off to suspend it and click on it to restart it. This animation does not show the slow build up of rotating waves but only the propagation of constant amplitude rotating waves around the ring resonator. ⇑ Fig. 2.6. Progression of small amplitude longitudinal waves in linear simulation setup (shown above). This graph shows the wave at a progression of times, during and after its launch. The length of the simulation medium is 10. An animation of this is shown below in Fig. 2.8. ⇑ Fig. 2.7. Results for the ring resonator. The above graph shows snap shots of longitudinal waves traveling around the ring taken at equal time intervals after a long time interval of build up. The horizontal axis is labeled in terms of distance around the circumference (total circumference equals 10) while the y axis is the wave amplitude. These waves propagate similar to traveling waves and should have similar momentum. The advantage of this simulation setup is that it avoids the effect of short sinusoidal bursts and instead uses continuous sinusoidal excitation. An animation of this is shown below in Fig. 2.9. Fig. 2.8.⇑ Flash animation of a wave burst propagating on a linear medium, such as shown in Fig. 2.4 above. The ξ displacement is plotted on the y axis and does not mean to imply that the displacement is in the y direction. Mouse over the graph to see the wave propagation. Mouse off it to suspend it and click on it to restart it. Fig. 2.9.⇑ Flash animation of a rotating wave mode in a ring resonator, shown in Fig. 2.5 above. The ξ displacement (longitudinal - in the direction around the ring) is plotted on the y axis. Mouse over the graph to see the wave propagation. Mouse off it to suspend it and click on it to restart it. Note that the wave at 2π radians has the same value as that at 0 radians as is required for a circular resonator (since 2π radians is physically the same place as 0 radians). Note also that the waves propagate in the positive direction around the resonator, much the same as traveling waves would. Note also the slow steady build up of the waves due to the exciting source. In the actual Octave simulation, the build up took even more cycles of the resonance. Table 2.1. Octave calculated values of energy and momentum at four different times during the wave propagation in a linear array. Wave velocity was set equal to 1. These results show that the average momentum is much less than the wave energy divided by the wave velocity, consistent with zero momentum. As in many numerical simulations, the results are unitless, i.e. normalized to some arbitrary value. Energy in the wave 207, 213, 213, 213 Energy over wave velocity 207, 213, 213, 213 Momentum in the wave -6.1172e-001, -3.2102e-004, -7.6408e-007, 5.3848e-005 Table 2.2. Octave calculated values of energy and momentum at four different times during the wave propagation in a circular array. Wave velocity was set equal to 1. These results show that the average momentum is much less than the wave energy divided by the wave velocity. The wave amplitude was different than in Table 2.1. Energy in the wave 5.53, 5.59, 5.61, 5.77 Energy over wave velocity 5.53, 5.59, 5.61, 5.77 Momentum in the wave 1.40e-014, 1.40e-014, 1.40e-014, 1.40e-014