Поверхностная электромагнитная волна по самому своему смыслу может распространяться только вдоль поверхности, например, вдоль земной поверхности. Причем поверхность должна быть как минимум двухслойной. Первый слой является диэлектриком, например, лед. А вторая среда - проводником, например, соленая вода.

Thank you for your response. However, I have not asked a question about general ideas about surface waves, but my question was about a particular equation, which describes surface plasmons propagating between dielectric and layered media (1D metal/dielectric superlattice).

Решение уравнений обычно происходит следующим образом. Сначала задается система координат, в Вашем случае одну из координат надо направить вдоль слоя раздела Ваших сред. Затем в этой системе координат надо расписать уравнения Максвелла, причем с учетом диэлектрической проницаемости для обеих сред. Затем ищется такое решение, чтобы в металле волна экспоненциально затухала. Для обычных металлов диэлектрическая проницаемость будет постоянной в любом частотном диапазоне. Чтобы учесть возникновение плазмонов, диэлектрической проницаемости необходимо приписать частотную зависимость согласно известной формуле, где присутствует дебаевская частота.

Dear Vasiliy Karlovich, again thank you for your kind and profound enough response. However, again I should say that I have not asked about a general idea, but about particular equation. I know the exact equation, which describes plasmon dispersion ralation in this particular situation. The only problem I have is solving this equation numerically. That's why I have attached this equation in Mathematica, paper, where this equation may be found.

Are you considering a propagation that is parallel or normal to the plane interfaces between those media?

I consider a parallel propagation.

Then you can solve your problem with a closed form, instead of running an algorithm.

I am sorry, what do you mean? What is a closed form?

Aclosed form is self-sufficient, to the extent that there is no need for an algorithm, for computing the number that is sought after.

Dear, Arezou Rashidi. I can demonstrate to you how I do this for metal-dielectric, metal-dielectric-metal, diel-metal-diel structure. I have managed to do this in Mathematica. I have also plotted dispersion relations for plasmons in infinite superlattice. But what I cannot do (I have already spent about 3 months on it) is to plot these relations for a semiinfinite superlattice (for waves which propagates through the surface between continious media and superlattice).

Dear Artem,

Dispersion equations can be written in the form $f(h,k,v) = 0$, where $h$ is a thickness of a layer (or a vector consisting of thicknesses in your particular case, if I understand the problem correctly), $k$ is a propagation constant, and $v$ is a frequency. Usually, one plots either $k = g(h)$ with fixed $v$ or $k = g(v)$ with fixed $h$. One of the way to plot these things is to use the shooting method. In particular, in the case of plotting $k = g(v)$ with fixed $h$ or in a more general case $g(k,v) = 0$, you split the segments $[k_1, k_2]$ (for $k$) and $[v_1,v_2]$ (for $v$) into smaller pieces. Then choosing particular value $v^j \in [v_1, v_2]$ (or $k^i \in [k_1, k_2]$ in accordance with what is more convenient in your case) and then you calculate the value $g_{i,j} = g(k^i, v^j)$ for each $i$. If you find $i = s$ such that $g_{s,j}*g_{s+1,j} < 0$, then there is $k^* \in (k_s, k_{s+1})$ such that $g(k^*, v^j) = 0$. In other words, the value   $k = k^*$ is a root of equation $g(k, v) = 0$ for particular frequency $v = v^j$. Then making these calculations for all possible $v^j$ you find all necessary pairs of solutions, then you plot these pairs and get a curve (a dispersion curve). 

However, for this algorithm there is one exception case in which it does not work. I have not met such a case yet but it possibly exist.

If I understood you problem in a right way (I also plotted many times dispersion curves for layered dielectric waveguides in different problems), then I can give you more detailed explanations. You may write email to me in case of necessary. 

I don't understand why you people go on about "algorithms", whereas one can use a closed form (a formula if you will) which one derives on the basis of an accurate approximation, to turn a transcendental equation into a second degree polynomial equation.

Dear Dmitry, thank you so much for a very detailed explanation. it is very clear (and TeX simbols are also convenient). I will try to do this. If I don't manage to do this, I will write to you.

Following my remark from 3 years ago, I am bringing to your attention a paper* where closed formulae for the number of a wave propagated by a metal film a light frequencies are derived by hand. One can perform derivations along the same line, more or less, in the case of a wave propagated by a substrate at a lower frequecy (publication pending)...

* Article Simple Closed Formulae for the Number of the Short-Range Sur...

This other paper might be of interest to you, too, as it is about a closed formula for the fundamental surface-wave mode of a dielectric-coated metal plane: Article Closed formulae for the wave number of the TM0 mode guided b...