## Abstract

We have studied waves propagating in an infinite plate with a line of equally spaced point masses on it. In particular, we consider waves that are laterally decaying, i.e. waves that exist due to the point masses, not in spite of them. We begin with a simple continuum limit and then study the discrete case using a suitable Green function and a superposition method used earlier for other structures by Mead. The superposition involves a slowly decaying series that we transform to a rapidly converging one using the Poisson summation formula. The system has two non-dimensional parameters, and on this parameter plane we find a propagation zone and its boundary. The boundary involves successive point masses vibrating *π* radians out of phase, a situation far from the continuum limit. Outside the propagation zone, waves get attenuated; however, unlike common examples of linear periodic structures, and contrary to the usual assumption made in studies thereof, here the waves show *sub-exponential* attenuation. In particular, the eventual decay of the wave amplitude is like the 3/2 power of distance from the point of excitation. We conclude with a theoretical explanation of the 3/2 power in the decay.

## 1. Introduction

Propagating waves that remain localized near surfaces or interfaces are well known (Rayleigh waves, Love waves and Stonely waves; see Graff 1975 and references therein).

Here, we are interested in the propagation and attenuation of localized waves in infinite plates. Many papers on wave propagation in plates assume that the plate is an infinite *strip*, i.e. the structure has finite lateral extent (Massalas 1986; Mead 1986; Predoi & Rousseau 2005). Most papers on wave propagation in infinite plates consider waves of infinite lateral extent (i.e. not decaying in the lateral direction; e.g. Crighton 1979 and Sorokin & Ershova 2004). However, large plate structures involve joining of plates through, for example, riveting or welding. There are no solutions available for waves that propagate along, say, a line of rivets. We may mention here the recent work of Hull (2008), which considers series expansions for a plate of finite thickness with a line of point masses, but which assumes no variation in the lateral direction, i.e. perpendicular to the line of point masses.

We consider an infinite line of equally spaced point masses on an infinite plate and seek a solution for a wave that propagates along the line of point masses; further, we restrict our search to waves that decay in the lateral direction (i.e. in the direction normal to the line of masses). In other words, we are interested in waves that are localized near the line of point masses. The system is sketched in figure 1.

Our reason for considering waves that are localized near the line of point masses is that the waves that are *not* localized in this manner must, to match far-field conditions, have identical propagation characteristics to waves in the infinite plate *without* point masses. The question of how such far-field waves interact locally with the line of point masses, while interesting, is not considered here. We focus on waves that exist *due to*, not in spite of, the point masses.

Our strategy for constructing solutions will be to use superposition, as previously described for other structures by Mead (1986; see also Mace 1980 and Mead & Yaman 1991*a*,*b*). We will first find the axisymmetric response of a plate to a sinusoidal point force. We will then add a countably infinite number of these solutions, suitably shifted in space and lagged in time, to obtain the travelling wave (following Mead 1986). The infinite series thus obtained converges slowly, and we will use the well-known Poisson summation formula to speed up convergence. Investigation of the numerical sums thus obtained will show propagation and attenuation zones, the lateral decay in wave amplitude that we have posited in advance and some other interesting features. Most notably, inside the attenuation zones, we will demonstrate *sub-exponential* decay (specifically, power-law decay) in the amplitude of the propagating wave. Such attenuation, at odds with widely held perceptions about wave attenuation in linear periodic structures, will also be theoretically established.

## 2. Continuum approximation

Before considering the discrete case, we note that the problem has a continuum limit where the point masses merge into a line of concentrated mass (figure 2).

The line of concentrated mass is along the *y*-axis. The wave will decay in the positive and negative *x*-directions, far from the *y*-axis. The plate is governed by(2.1)where *ρ* is the density of the plate material; *h* is the plate thickness; and *D* is the bending rigidity, given by where, in turn, *E* and *ν* are Young's modulus and Poisson's ratio, respectively.

Let *m*_{c} be the mass per unit area of the original plate and be the added mass per unit length along the line of added mass. We then havewhere *δ*(*x*) is the Dirac delta function.

Equation (2.1) can be written as(2.2)We now assume, consistent with our search for a laterally decaying wave that propagates along the line of added mass,(2.3)where *ω* is the frequency of the wave and *k*_{y} is the (real) wavenumber in the *y*-direction. Substituting equation (2.3) in equation (2.2), we get(2.4)

We define the parameterAway from *x*=0 (say, *x*>0), we then have(2.5)The general solution of equation (2.5) is of the form(2.6)The possible values of the *λ*'s are and . However, to ensure a decaying solution for *x*>0, *λ* must have a negative real part, which means that components corresponding to two of the *λ*'s must be dropped, leaving a solution of the form(2.7)Now will be pure imaginary unless , giving an oscillatory (non-decaying) solution, and so we require . It follows that and .

In equation (2.7), there are now three undetermined constants *A*_{0}, *B*_{0} and *k*_{y}.

With no loss of generality, we can assume that *U*(0^{+})=1. Assuming symmetry in the solution for positive and negative *x*, we must have *U*′(0^{+})=0. For the third boundary condition, we integrate equation (2.4) w.r.t. *x* from 0^{−} to 0^{+}, getting(2.8)By symmetry, *U* is an even function whence is an odd function. Thus, equation (2.8) becomes(2.9)

Using the first two boundary conditions, we can obtain constants *A*_{0} and *B*_{0} in the solution. The solution then stands at(2.10)

Now thrice differentiating equation (2.10) w.r.t. *x* and substituting in equation (2.9), we find that *k*_{y} must satisfy (introducing a variable *α* for convenience)(2.11)Equation (2.11) can be used to solve for *k*_{y}, at which point the solution has been determined. Note that the solution depends on *m*_{c}, and *Ω*_{c}.

For and *Ω*_{c}=1, for example, the graph of *α* (from equation (2.11)) is shown in figure 3. The solution is seen to be *k*_{y}=1.022. We will check against this value later. We now move on to the discrete case.

## 3. Green's function

The axisymmetric solution for a (unit) point force e^{iωt} on an infinite plate is Green's function. A derivation of the same appears, for example, in Morse & Ingard (1986). We briefly present it here in a slightly different, but equivalent, form.

Using polar coordinates and with the point force at *r*=0,(3.1)Seeking a periodic solution, we assume *w*=*W*(*r*)e^{iωt}, where *ω* is the frequency of excitation, and substituting it in equation (3.1), we obtain(3.2)where *Ω* (same as *Ω*_{c}, with the ‘c’ here merely denoting a continuum) is defined as(3.3)where, in turn, *m* is the mass per unit area (*ρh*) of the plate.

The solution of equation (3.2) may be written as(3.4)To keep the solution bounded at infinity, we must let *E*=−i*C*, obtaining(3.5)Considering small *r* and eliminating the ln *r* term, we obtain *C*=−*B*. There remain two undetermined constants, *A* and *B*. We write the expression for *Q*, the shear force per unit length at a location *r*→0^{+}, as (e.g. Ugural 1999)(3.6)The amplitude of the point force being unity, we write 2*πrQ*=−1, obtaining

The solution obtained may further be shown to be identical to (for identities involving Bessel functions, see Abramowitz & Stegun 1972)(3.7)where *A* is still undetermined. We assume as usual that the far-field solution should be an outwards propagating wave. As *K*_{0} decays exponentially, in the far field, we getNow it is clear that setting(3.8)will givewhich represents an outwards travelling wave in the far field. With *A* as above, Green's function is (using |*r*| in *Y*_{0} and *K*_{0} to ensure symmetry)(3.9)

## 4. Superposition

Travelling wave solutions are constructed through superposition of space- and time-shifted versions of the basic solution obtained above (as carried out for beams and finite-width stiffened plates by Mead 1986).

Consider infinitely many solutions of the form of equation (3.9), multiplied in magnitude by some, as yet undetermined, *F* and successively centred about points *P*_{k} that lie equally spaced along a straight line. Let the distance between these points be *s*. Successive terms in this series also have a constant phase difference *sϕ* between them, so that *ϕ* is a discrete version of the wavenumber *k*_{y} used earlier.

The remaining condition is that the force *F* is itself a reaction force corresponding to the acceleration of the added-on point masses. This will give us *ϕ*.

Mathematically, the displacement *at the origin* is (with *ϕ* still undetermined)(4.1)

In the above, when then the terms containing the odd function sin *ksϕ* will sum to zero. We can therefore write(4.2)We must now specify what *F* is. If each point mass is *M*, then the instantaneous force on the mass is *Ma*=−*ω*^{2}*Mz*, whence the force from the mass acting on the plate is *ω*^{2}*Mz*. Noting that the time-varying *z* itself is of the form *z*_{0}e^{iωt}, with *z*_{0} a constant, we conclude thatSubstituting *z* from equation (4.2) into the above, we obtain(4.3)

The above equation gives a condition for the existence of a propagating wave solution and is to be viewed as an equation that must be solved for *ϕ*. Real values of the discrete wavenumber *ϕ* give a propagating wave. It remains to check that the infinite series wave solution thus obtained is in fact laterally decaying.

As a supporting check, we have verified that on setting and using small values of *s* (e.g. 0.05), we get *ϕ*≈1.022 (real value as required), matching the foregoing continuum approximation.

## 5. Non-dimensionalization

We define two non-dimensional parameters as follows:(5.1)For easy interpretation, we note that *σ* corresponds to spacing and *τ* to the added mass. The wavenumber *ϕ* is also viewed below in non-dimensional form aswhere *β* is the phase difference between successive masses.

In terms of symbols routinely used in the wave propagation literature (e.g. Mead 1986), we note that the usual propagation constant *μ* is here equal to i*β*. Real values of *β* correspond to propagation zones. Complex *β* would imply attenuation; however, the infinite series above would then diverge.

In terms of the non-dimensional parameters above, equation (4.3) becomes(5.2)For convenience, let the l.h.s. of equation (5.2) be called *Γ*, i.e.(5.3)Given *σ* and *τ*, we seek *β* such that *Γ*=0.

## 6. Poisson summation

As given, the series in *Γ* converges very slowly because its terms decay like for large *k*. To numerically find the sum more quickly, we use the Poisson summation formula (see Papoulis 1962), which iswhere the Fourier transformand the fundamental frequency

The slowly converging infinite series in *Γ* involves(6.1)

Here, *γ* can be written (putting back the previously dropped sine terms) as(6.2)as the terms containing the odd functions sin *kβ* would add to zero.

Applying the Poisson summation formula (for relevant transforms, see Magnus & Oberhettinger 1949), we find(6.3)where(6.4)Note that for large *x*, and so the series will converge rapidly. The series in equation (6.3) converges much faster than the one in equation (6.1) and is used for the numerics below. Propagation occurs if some real *β* satisfies(6.5)with given by equation (6.4). The l.h.s. above is real if the argument of exceeds unity in magnitude for all *m*.

## 7. Propagation

Since *γ* and *Γ* are 2*π*-periodic in *β*, we need to examine only *β*∈(−*π*, *π*). See figure 4, where we have taken *τ*=1 and plotted *Γ* (seen to be real) against *β* for different values of the parameter *σ*. We see that for small *σ* (e.g. 0.02), there are two equal and opposite zero crossings at *β* close to zero. Closeness to zero is expected because we expect *ϕ* to converge to the continuum limit when *s*→0 and hence *β*=*ϕs*→0. The symmetrical location of roots is also expected, because they represent the same wave, going right and left. As *σ* increases (with *τ*=1 fixed), the pair of zero crossings moves towards ±*π*. At *σ*≈2.93, the plot of *Γ* is tangential to the zero line, with *β*=±*π*, and represents a standing wave solution for which there is no continuum equivalent. This marks the end of the propagation zone. Further increases of *σ* will result in a complex root, i.e. attenuation, in as much as lack of a real root implies lack of propagation.

That *Γ* stays real for sufficiently small spacing *σ* is anticipated from the continuum approximation. In contrast, the fact that the propagation zone ends with *β*=±*π* is a useful insight not obtainable from the continuum approximation. It can be used to find the propagation zone boundary by putting *β*=*π* in equation (5.2), evaluating the sum therein for any chosen *σ*, and then solving for *τ*. Results obtained in this manner are shown in figure 5, where it is seen that *σ*→*π* as *τ*→0 and also that *σ*<*π* for all non-zero *τ*. Finally, noting that *σ*<*π*=*β*, we find that the argument of in equation (6.5) is always greater than 1 in magnitude in this region, showing again that *Γ* is real here.

## 8. Lateral localization

We now numerically examine the waveform corresponding to the propagating wave found above. Our focus remains on waves localized in the lateral, or *x*, direction. We replace *r* in equation (3.7) by . The wave shape along the line *y*=0 is given by(8.1)The series converges slowly and we are unable to use Poisson summation. For good convergence, we have used partial sums: for a convergent sequence *S*_{k},whereAlthough the above two limits are identical, the latter form works better. We have used this trick with *N*=10^{5}, and one sum each for positive *k* and negative *k*, to obtain the waveform. For the parameter values *τ*=1 and *σ*=2.5, along with *β* determined from equation (5.2), the waveform obtained is plotted in figure 6. In plotting the figure, we have taken *t*=0 and dropped the imaginary part, within the resolution achieved in our numerical summation, which was 0 (it was typically approx. 10^{−3} when, for verification, we used *N*=10^{6}). There is no doubt that the wave is indeed laterally decaying.

## 9. Attenuation

Equation (5.2) converges for most real values of *β*; however, if *β* is complex, then the series diverges. Interestingly, the series in equation (6.5) converges even for complex *β*. Can we use equation (6.5) to compute exponential attenuation rates outside the propagation zone? We do not think so, as discussed later, although for other periodic structures, Mead (1986) has done something similar.

However, it is interesting to ask what happens if we *do* use equation (6.5) outside the propagation zone and look numerically for complex roots. For example, with *τ*=1 and *σ*=2.94 just outside the propagation zone (figure 4*d*), we numerically obtain *β*=*π*−0.0242i. Is this number (0.0242) meaningful?

To investigate this question using a different calculation, we now conduct a direct analysis using Green's function obtained earlier (equation (3.9)), see figure 7. We consider an infinite plate with a *finite* (but reasonably large) number of point masses at *x*=0 and *y*=*ks*, for *k*=1, …, *N*. A force *F*_{0}e^{iωt} is applied at the origin. To obtain the response of this system, we first ignore the masses and consider that forces *F*_{k}e^{iωt} are applied at points (*x*, *y*)=(0, *ks*), for *k*=1, …, *N*. The complex displacements *X*_{k} at points *k*, (*k*=0, …, *N*), are then(9.1)In the above equation that represents *N*+1 equations, we substitute *N*+1 more conditions, namely *F*_{0}=1 and *F*_{m}=*ω*^{2}*MX*_{m}, *k*=1, …, *N*. Then, equation (9.1) becomes (retaining *F*_{0} to make units transparent)(9.2)The above equation involves no approximation (for the *N* value chosen) and can be solved numerically. For *τ*=1 and *σ*=2.94 as before (as may be achieved by setting *s*=2.94 and all remaining dimensional parameters to unity) and for *N*=100, we obtain the results as shown in figure 8. The straight line plotted for comparison in the figure, with slope −0.0242, indicates a reasonable match between equations (9.2) and (6.5). However, the match is misleading, as seen from further numerical results. The complex amplitudes *X* obtained for *N*=100 (with only 50 plotted) and parameter values indicated graphically in figure 9 are shown in figure 10.

In figure 10, we first examine plots (*a*) and (*f*) that correspond to point 1 in figure 9, which is inside the propagation zone. Accordingly *X*_{k} does not, on average, decrease with increasing *k*. The oscillatory nature of the plot may be thought of as due to reflection from the *N*th mass (the end of the chain), which sets up a standing wave. In an infinite chain, there would be no such reflection. These results are consistent with the theory developed so far.

We next examine plots (*b*) and (*g*) in figure 10, corresponding to point 2 in figure 9, which is outside the propagation zone. As expected, there is attenuation. The plot on figure 10*b*, on semilog axes, initially shows a roughly linear drop-off, corresponding to exponential decay as is common inside attenuation zones for linear periodic structures; the straight line plotted for comparison matches the questionable complex-*β* calculation described above in equation (6.5). More interestingly, the plot on figure 10*g*, on log–log axes, shows a transition to sub-exponential decay; the straight line plotted for comparison has slope −3/2. This unexpected observation (and ensuing explanation) of power-law attenuation, due to its novelty, is perhaps the main contribution of this paper.

Plots (*c*) and (*h*), and also (*d*) and (*i*) in figure 10, corresponding to points 3 and 4 in figure 9, are similar. The exponential decay rate is faster, and the number of masses over which this decay rate holds is smaller. Plots (*e*) and (*j*), corresponding to point 5, show no clear regime of exponential decay at all.

For points 2–5, the eventual power-law decay rate observed (based on the visual match in the figure above) is consistently −3/2.

Numerical results for the case where the excitation is given not at the end of a finite line of masses, but at the central mass in a long line of masses, show a similar power-law decay (again, the power is −3/2). For example, for point 5 above, results obtained are shown in figure 11.

As far as we know, there is no prior study that reports sub-exponential attenuation of waves in linear periodic structures. However, as we will show below, there is a theoretical explanation for this power law, which shows that such attenuation is, in principle, possible for other periodic structures as well.

## 10. Theoretical explanation

We consider an infinite number of collinear point masses *M* at coordinates *x*=0 and *y*=*ks*, for all integer values *k*. A force *F*_{0}e^{iωt} is applied at the origin. The complex displacements *X*_{k} at points *k* (*k* any integer) are then given by(10.1)where(10.2)In equation (10.1), the infinite sum is actually a convolution of the sequences *X*_{k} and *W*_{k}. Let us introduce new functions *x* and *w* of a new variable *u*, given by(10.3)and(10.4)i.e. *x* and *w* are the 2*π*-periodic functions whose complex Fourier coefficients are *X*_{k} and *W*_{k}, respectively. Then using the fact that the Fourier coefficients of the product of two 2*π*-periodic functions is the convolution of the individual Fourier coefficients, equation (10.1) becomes(10.5)or(10.6)Now *w*(*u*) can be evaluated using the Poisson summation formula. Substituting the expression of *W*(*ks*) from equation (3.9) in equation (10.4), we get (rewriting arguments in terms of the non-dimensional parameter *σ*),(10.7)The infinite series in equation (10.7) is of the same form as in equation (6.2). Adapting equation (6.3), we obtain(10.8)where is given by equation (6.4). Thus, *x*(*u*) is now determined.

Recall that *x*(*u*) has no direct physical interpretation. Its Fourier coefficients are *X*_{k}, the complex displacements that are of physical interest. Nevertheless, the singularities and jumps of *x*(*u*) and its derivatives determine the nature of the decay in its Fourier coefficients, and hence the decay rate of the wave in the structure.

When 1−*ω*^{2}*Mw*(*u*)=0 and *x*(*u*) has a pole and its Fourier coefficients do *not* decay in magnitude,1 then there is propagation. Note that this condition for propagation is exactly the same as equation (5.2), with *β* merely replaced by *u*.

When *x*(*u*) has no pole, there is attenuation or decay in the amplitudes *X*_{k} for large *k*. What sort of decay there is depends on the nature of *x*(*u*).

The −3/2 power-law decay observed above may now be formally established.

First, *w*(*u*) typically has two singularities in each period. To see this, note that *u* varies over an interval of length 2*π* and so 2*πn*−*u* covers the real line. Hence (2*πn*−*u*)/*σ* for *σ*>0 also covers the real line. Finally, each of the two singularities of , being at ±1, falls on this real line also and so enters the sum of equation (10.8) for some *n*.

Since *w*(*u*) appears both in the numerator and the denominator of *x*(*u*), however, these singularities do not contribute to a singularity in *x*(*u*) itself. In fact, as *w*(*u*)→±∞, *x*(*u*)→−1/(*ω*^{2}*M*).

However, a singularity of *w*(*u*) causes a singularity in the *derivative* of *x*(*u*) because(10.9)which is dominated by *w*′/*w*^{2} as *w*→∞.

Now, a singularity in *x*′(*u*) results in Fourier coefficients that decay like (since *w*(*u*), with an identical singularity, results in *W*_{k} that are known to decay like ). Since the Fourier coefficients of *x*′(*u*) are just *kX*_{k}, it follows that *X*_{k} decays like *k*^{−3/2}.

In conclusion, we remark that the demonstration of the decay of *X*_{k} like *k*^{−3/2} is crucially dependent on the decay of Green's function *W*(*r*) like . Accordingly, for other periodic structures such as, say, an infinite beam with equally spaced point masses attached, the non-decaying nature of the corresponding Green function precludes power-law decay in the corresponding response amplitudes, and the more familiar exponential decay is observed in the attenuation zones.

## 11. Conclusions

We have considered an infinite plate with a line of equally spaced point masses on it and examined laterally decaying waves that propagate along the line of point masses. We first studied the continuum approximation where the plate has a line of concentrated mass on it and found a closed form solution. Moving on to the discrete case, we found Green's function for an infinite plate excited by a point force that varies harmonically in time and then used suitable superposition to construct the travelling wave solution. The superposition-based propagation condition involves a slowly converging series, which we have transformed to an equivalent, faster converging series using the Poisson summation formula.

The problem has two non-dimensional parameters. On this parameter plane, we found a single propagation zone on whose boundary successive point masses are *π* radians out of phase; this boundary has no equivalent in the continuum limit.

Outside the propagation zone, we carried out numerical calculations for finitely many masses and observed roughly exponential attenuation over some distances for some parameter values. However, the initial exponential attenuation, if any, eventually gives way to slower, power-law attenuation. In the present case, the power is 3/2, as has been explained theoretically. Power-law decay in Green's function is the key to power-law attenuation observed here, and it seems clear that such power-law attenuation may be observed in other periodic structures as well.

## Acknowledgments

Pradeep Mahadevan read and commented on the manuscript. V. R. Sonti provided useful technical inputs. A.C. thanks Mythily Ramaswamy for her discussions.

## Footnotes

↵Consider, for example, the 2

*π*-periodically extended function given by 1/*u*for*u*∈(−*π*,*π*). In a Fourier sine series expansion, the*k*th coefficient is , which goes to 1 as*k*→∞.- Received February 5, 2008.
- Accepted March 19, 2008.

- © 2008 The Royal Society