# What a Surface Wave Is

A surface wave is a wave that occurs at the interface between two different media (material layers) and is localized to that interface. Typically, the amplitude of the surface wave decreases exponentially with increasing distance from the interface. The rates of exponential decrease in the two media will generally differ.

Sometimes the amplitude of the surface wave in one of the media is zero. For example, with sound waves at an interface between a solid and a vacuum, the sound wave obviously does not penetrate into the vacuum. In such cases, the amplitude of the surface wave will typically still decrease exponentially in the other medium.

# Some examples of Surface waves

The most commonly seen surface waves are water waves. The amplitude of gravity waves [1] on the surface of water falls off as e-d/L, where L is the wavelength and d is the depth below the surface. [2] So water waves atop a body of water many times deeper than their wavelength are surface waves. [3] Tsunamis, on the other hand, have a wavelength that is large compared to the depth of the ocean, and are not surface waves.

Suppose you have an interface between a vacuum and an ideal elastic solid which is isotropic and homogeneous. [4] A kind of surface wave called Rayleigh waves can occur there. [5] If the elastic solid is covered with a thin layer in which s-waves travel more slowly than in the medium below, there a kind of transverse wave called Love waves can occur. [5] Love waves are often called surface waves although they require this additional layer to occur. Love waves and Rayleigh waves can be considered to be sound waves.

Earthquake waves are sound waves in the earth [7], and Rayleigh waves and Love waves are the two types of earthquake waves that travel along the surface of the earth.

Surface acoustic wave (SAW) devices use sound waves at the surface of piezoelectric materials. Many of them use Rayleigh waves; however there are a variety of other waves that can occur at the surface of an electromagnetic medium, and some use those. The most common kind of SAW device is a low power filter for signals at microwave or radio frequencies. A cell phone might have as many as four of these SAW filters.

Surface plasmons, sometimes called "surface plasmon-polaritions", are electromagnetic surface waves. The ground wave in radio is mathematically the same as a surface plasmon. [8]

# References, Notes, and Random Pedantry

1. What I am calling “gravity waves” are not the gravitational waves predicted by general relativity. Suppose you have one or more fluids in a gravitational field with density decreasing as you go upwards. If you disturb the fluid(s) from hydrostatic equilibrium you can get oscillations. The restoring force on a piece of fluid comes from the difference between the buoyancy force on it and its weight and is vertical. These oscillations are often called gravity waves.
2. “Waves. Berkeley Physics Course - Volume 3.”, by Frank S. Crawford (McGraw-Hill Book Company, New York, 1968) gives a pretty complete derivation for gentle water waves of small amplitude in section 7.3, except that it ignores viscosity.
3. Some books say that water waves are transverse. They are wrong. In a water wave, a piece of fluid moves in an ellipse. The ellipse lies in the plane that contains both the vertical and the direction in which the wave is moving.
4. In case you have trouble picking out the correct definitions, a solid is homogeneous if it is the same everywhere. This of course can’t be true at the atomic scale but it can be true on larger scales. If a solid is isotropic, any direction in the solid acts the same as any other direction in the solid.
5. “A treatise on the mathematical theory of elasticity”, by A.E.H. Love, 4th edition (Dover Publications, New York, 1944) which is a reprint of a 1927 edition by Cambridge Press. Article 214 discusses Rayleigh waves.
6. “Some problems of geodynamics; being an essay to which the Adams prize in the University of Cambridge was adjudged in 1911”, by A.E.H. Love (Dover Publications, New York, 1967), which is a reprint of the 1911 edition by Cambridge Press, chapter XI, especially sections 176 through 181.
7. In most places the earth is neither homogeneous nor isotropic, nor yet is it perfectly elastic, but it is close enough for solutions derived for homogeneous isotropic solids to be useful.
8. Ground waves in radio are sometimes called Zenneck waves.

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