Attenuation factor and attenuation coefficient as functions of distance from epicenter
1. Introduction
Tsunamis are ocean waves generated by the displacement of a large volume of water due to earthquakes, volcanic eruptions, landslides or other causes above or below the ocean floor (e.g., Karling, 2005; Parker, 2012). The great Indian Ocean tsunami of December 2004 will be remembered for its ferocity, devastation and unprecedented loss of life for a long time (Stewart, 2005; The Indian Ocean Tsunami, 2011). It is also the same tsunami which has galvanized the international community to set up warning systems and undertake preventive measures against the onslaught of future tsunamis in the vulnerable regions around the globe. A surge of scientific studies on all aspects of the tsunami is in evidence in the literature. And a volume entitled “
There are three distinct stages of a tsunami event: (1) Generation; (2) Propagation; and (3) Inundation/landfall (cf. Cecioni & Belloti,2011). The generation stage is the most complex and most difficult to analyze, since each tsunami is different and no single mechanism can account for all tsunamis. The inundation stage is also different for different areas affected, and again, no single scenario can describe all affected areas. The propagation stage covers the most extensive area, and is the only one that can be attacked by simple theory and analysis,even though detailed numerical models are found in the literature (see, for example, Imteaz, et al., 2011, and the references therein). These models consist of solving hydrodynamic equations with suitable boundary conditions that necessarily involve tedious numerical integrations. Such models, unfortunately, fall within the realm of the specialists, and are, by and large, outside the reach of the broader audience.
This chapter takes an alternative approach to the study of tsunami propagation in the open ocean. It commences with the theory of water wave propagation in general and applies them to tsunami propagation in particular, using analytical models. It is based upon first principles of physics and avoids numerical analysis, which is thus accessible to the broader scientific community. It only requires the knowledge of general science and basic calculus-based physics. The derivations of the relevant equations are relegated to the appendices for quick reference so that the need to search for them outside this article is kept to a minimum.
2. Theory of water waves and tsunamis
The theory of water waves is well-documented in the literature (e.g., Coulson, 1955; Sharman, 1963; Towne, 1967; Elmore & Heald, 1969). The wave velocity of waves on water of density
Here
When the gravity and surface tension terms are equal,
For water waves,
For waves of
Thus
and the group velocity
The group velocity is faster than the wave velocity! The individual wave-crests fall behind the group while new crests build up at the forward edge of the group (cf. Towne, 1967). Such waves are called
For longer wavelengths (
and
Such waves are called
In the first case,
The group velocity is, from Eq. (8):
Owing to the equality of the wave velocity and the group velocity, such waves are non-dispersive.
In the second case,
and from Eq. (8):
In such waves, the individual waves travel faster than the group and rapidly diminish in amplitude as they cross the group (Towne, 1967).
Tsunami waves in the deep ocean typically have wavelengths of about 200 km and velocity of 700 kilometers per hour. They maintain their form in dispersion-less propagation for long distances with little dissipation of energy. They are thus identified as ‘long waves in shallow water’ even as they travel above the deepest parts of the ocean, the ‘long’ and ‘shallow’ adjectives referring to the relative magnitudes of the wavelength and depth of water. Many of the properties of the tsunamis, such as velocity and dispersion-less propagation follow from this description (e.g., Margaritondo, 2005; Helene & Yamashita, 2006; Tan & Lyatskaya, 2009). However, the linearized theory of water waves is only an approximation to the true situation and a rigorous theory calls for the inclusion of the non-linear term in Bernoulli’s equation. For example, the form of these waves having a narrow crest and very wide trough can only be accounted for by a complete non-linear theory (cf. Elmore & Heald, 1969).
Tsunamis also exhibit the characteristics of ‘canal waves’, first observed by Scott Russell in Scotland in 1834 (Russell, 1844). These disturbances travel like a single wave over long distances and maintain their shapes with little loss of energy. The mathematical theory of such waves, now-a-days called ‘solitary waves’ or ‘solitons’, was developed by Boussinesq (1871), Korteweg & de Vries (1895) and Lord Rayleigh (1914). Their analyses show that in a dispersive medium, the non-linear effect can exactly cancel out the dispersive effect to preserve the form of the wave(cf. Stoker, 1957; Lamb, 1993).
3. Tsunami propagation models
One-dimensional propagation models are the easiest to construct and analyze. Even though they may not represent the real situation for tsunami propagation in the open ocean, valuable results can come out of these models. The ‘long waves in shallow water’ model noted above represents a one-dimensional model (
In the ‘long waves in shallow water’ model, the group velocity is equal to the wave velocity according to Eqs. (9) and (10). This helps to explain the dispersion-less propagation of the tsunamis. An alternative derivation of these equations is provided by Margaritondo (2005). By assuming that the vertical displacement of water is proportional to distance from the bottom, Margaritondo (2005) arrives at the same result by applying the energy conservation law. As stated earlier, however, a rigorous derivation would call for the solitary wave solution of non-linear differential equation obtained from Navier-Stokes equation (cf. Stoker, 1957).
Eq. (9) has profound consequences in the inundation phase as the tsunami makes landfall. The wave energy density
where
A 1 m high wave at a depth of 1000 m would become 5.62 m at a depth of 1 m! That goes to illustrate the devastating effects of tsunamis as they make landfall. Helene & Yamashita (2006) have shown that the Eq. (14) holds true when the depth of the ocean floor varies gradually instead of having abrupt steps, which helps the tsunamis to maintain their characteristics as they approach land.
Helene & Yamashita (2006) have further shown how a tsunami will bend dramatically around an obstacle and strike land in the shadow regions. Since
Useful as they are, the one-dimensional models of are not appropriate when the tsunami propagates in the open ocean from a well-defined epicenter since the waves will spread out in concentric circles, which calls for two-dimensional models. In Model B (Tan & Lyatskaya, 2009), waves propagate outwards on a flat two-dimensional ocean from the epicenter. When the energy conservation principle is imposed, the energy density of the wave falls off inversely as the distance from the epicenter
The wave amplitude therefore falls off inversely as the square-root of the radial distance
The formal derivation of the results (15) and (16) are given in Appendix B where it is mentioned that strictly speaking, they apply at distances away from the epicenter.
For long distance propagation, the curvature of the Earth must be taken into consideration. This is incorporated in
where
The amplitude variation depends solely on the polar angle. Since the epicenter is considered to be located at the north pole (
The comparison of wave amplitudes in Model C (true) and Model B (approximate) is given in Appendix C. We have:
Table C.1 shows that the difference in the two solutions is slight for small values of
When the tsunami is caused by an oceanic plate sliding under a continental plate, the subduction zone can be described by a finite line source instead of a localized point source. In that case,
The wave amplitude in Model D is finite at the origin as opposed to being infinite in Models B and C. However, away from the source, the solutions for Models B and D rapidly converge, as the wave-fronts become more circular. For
Tsunamis are vast and highly complex geophysical phenomena, each having a character of its own. It is impossible to construct one model for any tsunami even with a high-speed numerical code. Further, each stage of the tsunami – generation, propagation and inundation, has to be modeled and studied separately. Nonetheless, simplified models based on first principles are able to explain individual aspects of this very complex geophysical phenomenon without detailed numerical computations.
4. Model applications
Viewed from space, the Earth is a watery planet with the oceans covering a full 71% of the surface area. The world ocean consists of three inter-connected oceans of the Pacific, Atlantic and Indian oceans, which comprise 51.5%, 25.6% and 22.9% of the water surface, respectively (we disregard ‘Southern Ocean’ as a separate entity). The Pacific ocean is thus larger than the two other oceans put together. It alone covers 34% or just over a full one-third of the Earth’s surface and is comfortably larger than all the landmasses (at 29% of the Earth’s surface) put together. The Pacific ocean is thought to be the remnant of ‘Panthalassa’, the world ocean, when all the landmasses were joined together as ‘Pangaea’.
The Pacific ocean provides an ideal venue for tsunami propagation studies for several reasons. First, as stated above, it is the largest body of water, covering a full one-third of the globe. Second, it is bounded by active tectonic plate junctions, studded with volcanoes called the ‘Ring of Fire’. Tsunamis produced at these hotspots can traverse the length and breadth of the ocean with relative ease. Third, there are no landmasses or large islands to block or interfere with the propagation of tsunamis formed in the ocean. Fourth, the ocean itself is dotted with small islands which pose little interference with tsunami propagation, but provide valuable platforms for recording tsunami wave amplitudes. Many of these islands are volcanic in origin and are sources of tsunamis themselves. Fifth, the longest stretch of ocean water is found between the Japan archipelago in the north-west and southern Chile on the south-east covering a distance of over 17,000 km or 85% of the distance between the North pole and the South Pole. At both the ends of this diameter lie some of the most active plate tectonic regions and tsunamis from either ends have traversed this favorite racetrack.Last but not least, an astonishing 80% of all tsunamis are recorded in the Pacific ocean.
With this geographical backdrop, we now proceed to study representative tsunami event to illustrate the validity of our propagation models. Model A, even though uni-dimensional, is a valuable tool for all tsunami events, as it correctly furnishes the velocity given the depth of the ocean, or vice-versa. It further predicts the travel times, which are vital for warning purposes. These results are independent of the direction, given the isotropy of space. Model A fails when the amplitude of the wave is to be studied, in which case Model B, C or D is called into consideration. In the following, we provide examples where one of the latter models, in conjuction with Model A, is used to analyze historic tsunami events. The data are taken from the National Oceanic and Atmospheric Administration website at www.ngdc.noaa.gov/hazard/tsu_travel_time_events.shtml.
4.1. Hawaii tsunami of 1975
On 29 November 1975, a magnitude 7.2 earthquake occurred on the southern coast of the island of Hawaii with the epicenter at 19.3oN and 155.0oW at a focal depth of 8 km (cf. Pararas-Carayannis, 1976). The earthquake, the largest local one since 1868, generated a locally damaging submarine landslide tsunami which was recorded at 76 tide gauge stations in Alaska, California, Hawaii, Japan, Galapagos Islands, Peru and Chile. The tsunami caused $1.5 million damage in Hawaii, 2 deaths and 19 injuries (Dudley & Lee, 1988). From the travel times registered, the tsunami reached Guadalupe Island, Mexico, 3864 km away in 5 h 9 m at an average speed of 750 kph, while it took 6 h 8 m to reach Tofino Island, Canada, 4210 km away, at the average speed of 686 kph. The slower speed in the first case is likely to be due to the fact that the tsunami had to bend considerably before heading towards its destination (cf. Helene & Yamashita, 2006).
Fig. 1 is a scatter plot of the wave amplitude versus the propagation distance. Since the maximum distance was under 8000 km, the flat space approximation holds (vide Appendix C) and Model B is applicable. The variation of the wave amplitude in this model is given by Eq. (16):
where
The current tsunami data yield:
where
and
By summing Eqs. (24) and (25) over the
and
By eliminating
The actual wave amplitude falls off slightly faster than that predicted by Model B, which is based on conservation of energy. The difference may be assumed to represent the loss of energy due to as yet unidentified causes. The prime candidate appears to be the generation of atmospheric internal gravity waves by tsunamis, which can transport energy and momentum vertically through the atmosphere and produce travelling ionospheric disturbances (cf. Hickey, 2011). Assuming an exponential attenuation factor, we can write (vide Appendix B)
The attenuation factor then follows as
The attenuation factor and the attenuation coefficient are calculated for this event as functions of the distance from the epicenter and entered in Table 1. It shows that the amplitude loss is most rapid near the epicenter with almost 50% of it dissipated in the first 400 km. The dissipation slows down dramatically thereafter and at 10,000 km, it is increases only to 65%. Consequently, the attenuation coefficient is not a constant but a function of the distance from the epicenter. By all accounts, the attenuation is small, suggesting the validity of models A and B.
4.2. The Great Japan tsunami of 2011
The Japan archipelago is one of the most earthquake-prone regions of the world. On 11 March 2011, a 9.0 magnitude earthquake struck on the east coast of Honshu. The epicenter was at 38.322oN latitude and 142.369oE longitude, 72 km east of Oshika peninsula with the hypocenter at a depth of 32 km below sea level (cf. http://www.tsunamiresearchcenter. com/news/earthquake-and-tsunami-strikes-japan; http://itic.ioc-unesco.org/index.php). It was the greatest earthquake to strike Japan and one of the greatest in recorded history. It was comparable to the 2004 Indian Ocean earthquake. There were an estimated 16,000 deaths, 27,000 injured and 3,000 missing (http://www.npa.go.jp/archive/keibi/ higaijokya_e.pdf) with total property damage of $235 billion (according to World Bank reports), making it the costliest natural disaster of all time.
Fig. 2 provides a geometrical perspective of the 2011 Japan tsunami. The geodesic lines from the epicenter shown in the figure are great circles with a longitudinal separation of 90o, which define a ‘lune’ that covers one quarter of the Earth’s surface area. Intersecting the great circles are ‘circles of latitude’ at angular distances of
The 2011 Japan tsunami was felt throughout the Pacific Ocean. Wave amplitudes were recorded at over 293 stations scattered in and around the Pacific. Fig. 3is a scatter plot of the wave amplitude versus the distance from the epicenter. The highest amplitudes were recorded near the epicenter. Amidst a considerable scatter, a well-defined trend in the wave amplitudes emerges from the figure. The wave amplitude diminished rapidly as a function of the distance from the source, becoming nearly constant around the 10,000 km mark, and showing a discernible rise thereafter in accord with Model C. But for the intervention of the South American landmass, the waves would have converged at the anti-podal point in south Atlantic Ocean, and barring losses, the original wave amplitude restored.
The variation of the wave amplitude according to Model C can be written as [from Eq. (18)]:
where
where
The constant
giving
The regression lines according to Models B and C are superimposed on the data points in Fig. 3. As is shown in Appendix C, the difference between the two wave amplitudes is slight up to about the equator mark (10,000 km), after which the two curves begin to diverge. The predicted amplitude in Model B continues to fall according to Eq. (21), whereas that in Model C begins to rise according to Eqs. (32) and (33). The actual data clearly supports Model C and validates the convergence effect of the Earth’s curvature on the wave amplitude. The same effect was earlier observed in the 8.3 magnitude Kuril islands earthquake of 2006 (Tan & Lyatskaya, 2009).
Finally, Eq. (9) of Model A furnishes a means to determine the average depth of the ocean along the travel path given the distance and the travel time to the destination:
From the observed travel times and the geodesic distances from the epicenter to 13 destinations along the coast of Chile, the average travel speeds were calculated from which the average depth of the ocean along the travel path determined (Table 2). The mean average speed of 739 kph yielded a mean average depth of 4303 m for the Pacific Ocean along these paths, which compares favorably with various estimates found in the literature: e.g., 4282 m (Herring & Clarke, 1971), 4190 m (Smith & Demopoulos, 2003), 4267 m (http://oceanservice.noaa.gov), and 4080 m (britannica online encyclopedia).
The average depth of the Pacific Ocean (4300 m) is considerably grater than those of the Atlantic Ocean (3600 m) and Indian Ocean (3500 m) (cf. Herring & Clarke, 1971). Further, the smaller north-western half of the Pacific Ocean is substantially deeper than the larger south-eastern remainder, even though separate depth figures are hard to find in the literature. In order to estimate the average depth of north-western Pacific Ocean, we consider the travel times of the tsunami to reach various destinations on the coasts of the Hawaiian islands, which lie entirely in that region (Table 3). Travel time data from the epicenter to 8 destinations in the Hawaiian islands yield a mean average speed of 798 kph for a mean average depth of 5016 m for north-western Pacific Ocean along these paths. This confirms the fact that north-western Pacific Ocean is considerably deeper than the south-eastern remainder. These travel time studies further re-affirm the validity of the ‘long wave in shallow water’ approximation for tsunami propagation.
4.3. The Great Chilean tsunami of 1960
On Sunday, May 22, 1960, at 19:11 GMT (15.11 LT), a super-massive earthquake occurred off the coast of south central Chile, with epicenter at 39.5oS latitude, 74.5oW longitude and focal depth of 33 km (cf. http://earthquake.usgs.gov/earthquakes/world/events/ 1960_05_22_tsunami.php). It happened when a piece of the Nazca Plate of the Pacific Ocean subducted beneath the South American Plate. The magnitude of the earthquake of 9.5 makes it the most powerful earthquake in recorded history (Kanamori, 2010). The tsunami generated by the earthquake, along with coastal subsidence and flooding, caused tremendous damage along the Chilean coast where an estimated 2,000 people lost their lives (http://neic.usgs.gov/neis/eq_depot/world/1960_05_22articles.html). The resulting tsunami raced across the Pacific Ocean causing the death of 61 people in Hawaii and 200 others in Japan and elsewhere (USGS reports). The estimated damage costs were near half a billion dollars.
The Great Chilean tsunami of 1960 is similar to the Great Japan tsunami of 2011 coming from the opposite direction, only having greater amplitude. Both of these tsunamis, as well as many other analogous ones, traversed the longest stretch of continuous water covering over 85% of the distance between the poles. Thus the amplitude variations with distance of both of these tsunamis were similar. The strength of the Great Chilean tsunami was such that reflected waves from the Asian coasts were detectable (http://www.soest.hawaii.edu/GG/ASK/chile-tsunami.html).
Fig. 4 provides the geometrical perspective of the 1960 Chilean tsunami. As in the earlier example, the geodesic lines from the epicenter shown in the figure are great circles with a longitudinal separation of 90o, which define a ‘lune’ that covers one quarter of the Earth’s surface area. Intersecting the great circles are ‘circles of latitude’ at angular distances of
The Great Chilean tsunami of 1960 was felt throughout the Pacific Ocean. Over 1,000 measurements of wave amplitudes were recorded at 815 stations scattered in and around the Pacific. Fig. 5is a scatter plot of the wave amplitude versus the distance from the epicenter. The highest amplitudes were recorded near the Chilean coast and at the diametrically opposite end, mostly on the Japanese coasts, where hundreds of data points were clustered. There is a second cluster past the 10,000 km mark at the Hawaiian Islands, where numerous measurements were taken. The over-all trend of the data points closely agrees with that predicted by the ‘spherical ocean’ Model C.
The least-squares regression lines of the data points as obtained from Eqs. (32) - (35) are shown in Fig. 5, with the amplitude constant
From the observed travel times and distances from the epicenter to destinations on the eastern coast of Japan, the average travel speeds were calculated and the average depth of the ocean determined along these paths (table 4). The mean average depth of 4,231 m compares favorably with the value of 4,303 m obtained from the 2011 Japan tsunami (Table 2). Also calculated were the travel speeds from the epicenter to locations on the Hawaiian Islands and the average depth of the ocean (Table 5). The mean average depth of 3,972 m reaffirms the fact that the north-western Pacific Ocean, at 5,016 m (Table 3) is substantially deeper than its south-eastern compliment.
4.4. The Great Indian Ocean tsunami of 2004
The great 2004 Indian Ocean earthquake occurred off the west coast of Sumatra, Indonesia, on Boxing Day, December 26, 2004. Its revised magnitude of 9.2 makes it the second largest earthquake in recorded history, after the 9.5 magnitude Chilean earthquake of 1960 (cf. http://walrus.usgs.gov/tsunami/sumatraEQ/; Lay, et al., 2005). In terms of human casualties, however, it was the greatest natural disaster in recorded history, by far. The earthquake generated a super-massive tsunami that took the lives of an estimated 230,000 people in Indonesia, Sri Lanka, India, Thailand and elsewhere (Mörner, 2010). More than 1,000,000 people were displaced in the aftermath following the tsunami, which was eventually registered at every coast of the world ocean.
This great earthquake was caused by the subduction of the Indo-Australian plate under the Eurasian plate near the Andaman and Nicobar Islands chain and its extension southwards under the Bay of Bengal (Fig. 6). Its hypocenter is listed at 3.295oN altitude and 95.982oW longitude. However, its fault-line was 1000 km long, which roughly consisted of two linear segments (cf. Kowalik, et al., 2005): (1) a 700 km long section off the west coasts of Andaman and Nicobar Islands; and (2) a 300 km section west of Aceh province of Sumatra, Indonesia (Fig. 6). The alignments of both the segments were generally north-south, with the southern segment titled slightly towards the south-easterly direction. The shorter southern segment had the more intense earthquake and generated the greater tsunami. Consequently, the epicenter lied on this segment of the fault-line. The earthquake is also variously referred to as the Sumatra-Andaman earthquake or the Boxing Day earthquake.
It is evident from Fig. 6 that the eastward tsunami from the southern segment of the fault-line (henceforth referred to as the Sumatra fault-line) had a direct impact on the Aceh province in northern Sumatra, where the highest waves of over 50 m were registered. More than half of all casualties were reported there. The westward tsunami from this fault-line, on the other hand, passed harmlessly over the open ocean, reaching the east coast of South Africa, and entering the Atlantic Ocean. The northern segment of the fault-line (henceforth called the Andaman-Nicobar fault-line) produced tsunami which affected greater areas of landmass. Phuket lied near the perpendicular bisector of this fault and took a direct hit from the tsunami as did other coastal locations of Thailand and Myanmar. To the west, the east coasts of Sri Lanka and southern India were greatly affected. The tsunami rolled over the Maldive Islands and reached the eastern coast of Africa (Somalia, in particular), causing damage there. In consolation, Bangladesh, a densely populated area to the north, was spared the devastation.
Assuming that the tsunami from the Sumatra fault-line was largely intercepted by the island (vide Fig. 6), we proceed to analyze the tsunami propagation from the Andaman-Nicobar fault-line based on Models B and D. For this, we have to consider the mid-point of the fault-line (approximately 9oN latitude and 92.5oE longitude) as the origin of the tsunami instead of the epicenter which lay on the Sumatra fault-line. Distances are now reckoned from this new center. If the coordinates (latitude, longitude) of the source and destination points be (
The linear distance between the two points on the surface of the Earth is then
Fig. 7 is a scatter plot of the wave amplitudes as functions of the distance from the center of the Andaman-Nicobar fault-line. The data betray distinct clumps for Thailand, Sri Lanka, India and Maldives. At the far end are data for the Somalia coast. Also shown are the isolated data points for Andaman and Seychelles islands. Superimposed on the data points are the model amplitudes according to Models B and D. The model amplitudes are virtually indistinguishable for distances upwards of 1000 km (cf. Appendix D). In comparison with the Hawaii and Japan earthquakes (Figs. 1 and 3), the wave amplitudes are significantly higher, which is indicative of the magnitude of the earthquake. Unlike Hawaii and Japan earthquakes, however, there is a dearth of data points with high amplitudes near the origin. This is because the earthquake occurred beneath the ocean and there was no land close to it. Nevertheless, the lack of high wave amplitudes close to the fault-line appears to support the validity of Model D.
As before, the wave amplitudes according to Model B is given by (vide Eq. D.12):
where
The data yield a value of
Here
From the travel times and the travel distances from the epicenter on the Sumatra fault-line to five destinations on the South African coast and one on the Antarctic coast, the average speed of the tsunami was calculated, and the average depth of the Indian Ocean determined (Table 6). The average speed of 700 kph translates to an average depth of 3,840 m which is quite consistent with the reference figure of 3,963 m found in the literature (Herring & Clarke, 1971). Once again, the tsunami speed and depth of the ocean predicted by the ‘long waves in shallow water’ model turns out to be reliable.
5. Conclusion
Tsunamis are complex geophysical phenomena not easily amenable to theoretical rendering. The formal theory based on solitary wave model is not easily accessible to the great majority of the scientific readers. This chapter has demonstrated that the ‘long waves in shallow water’ approximation of the tsunami explains many facets of tsunami propagation in the open ocean. The one-dimensional model provides accurate assessments of the general properties such as dispersion-less propagation, speed of propagation, bending of tsunamis around obstacles and depth of the ocean, among others. Two-dimensional models on flat and spherical ocean substantially account for the wave amplitudes for far-reaching tsunami propagation. Finally, the finite line-source model satisfactorily predicts the wave amplitudes near the source when the tsunami is caused by a long subduction zone.
Appendix
A. Propagation of water waves in one dimension
A wave is a transfer of energy from one part of a medium to another, the medium itself not being transported in which process (e.g., Coulson, 1955). The individual particles of the medium execute simple harmonic motions in one or two dimensions, depending upon the nature of the wave. For sound waves in air, the oscillations are parallel with the direction of propagation. Such waves are called
In an earthquake, as many as four kinds of waves are produced, of which two are
Waves on the surface of water are similar to the Rayleigh waves. They are influenced by three physical factors: (1)
The theory of gravity wave propagation in one dimension is well documented in the literature. A particularly elegant treatment is found in Elmore & Heald (1969). Conventionally,
where
According to the potential theory,
where
If
and
At the bottom, the vertical component of velocity must vanish:
At the surface of water, the pressure is atmospheric pressure
where
Our task now is to find the velocity potential which satisfies Laplace’s equation (A.4) and the two boundary conditions (A.7) and (A.9). We assume a trial solution
and apply the method of separation of variables. From Eq. (C.4), we get
where the separation constant
and
The general solutions to Eqs. (A.12) and (A.13) are, respectively
and
For a wave travelling in the forward direction,
The boundary condition at the bottom (A.7) dictates that
and
where
and
Eqs. (A.19) and (A.20) yield:
Hence, the velocity potential has a simple harmonic time-dependence with angular frequency
If we choose
The wave velocity of the gravity wave is thus
B. Wave propagation on two-dimensional flat surface
The equation of a wave emanating from a point source in a two-dimensional plane is conveniently expressed in plane polar coordinates (
where
To apply the method of separation of variables, let
Then Eq. (B.2) becomes
Dividing both sides by
The
whose solution is
The
or
This is Bessel’s equation of the zeroth order. A novel technique to solve this equation is found in Irving & Mullineaux (1959). Let
Then Eq. (B.9) assumes the form
For large
giving the solution
Hence, from Eq. (B.10):
and
Since the wave is propagating outwards, we retain the + sign only, giving
In terms of the wave number k,
The amplitude of the wave falls off inversely as the square-root of the distance from the source:
The intensity of the wave (i.e., the energy density) is thus inversely proportional to the distance from the source:
There is a simple alternative procedure to obtain Eq. (B.18) from Eq. (B.19) (Tan & Lyatskaya, 2009). Assuming conservation of energy, the energy spreads out in concentric circles of radius
When
In Eq. (B.20),
C. Wave propagation on two-dimensional spherical surface
For long-distance propagation of waves on a spherical surface, the curvature of the surface must be taken into account (Tan & Lyatskaya, 2009). Use spherical coordinates (
Hence the amplitude of the wave varies inversely as the square-root of sin
The amplitude and intensity of the wave decrease from the north pole (
In order to compare the spherical surface solution with that of the flat surface solution (cf. Bhatnagar, et al., 2006), we notice that the corresponding circular wave-fronts on the flat surface will have radii of
Hence, the ratio of the amplitudes of the waves is
Table C.1 shows the ratios of the wave intensities and amplitudes in the spherical and flat surface solutions. The values are independent of the radius of the sphere and are therefore applicable to all spherical surfaces. The departures of the spherical surface solutions from those of the flat surface solutions are slight for small values of
D. Wave propagation from finite line source in two dimensions
Consider a finite line source of strength
where
The potential due to a line source constitutes a well-known problem in electrostatics (cf. Abraham & Becker, 1950).The equi-potential lines on which
The amplitude of the wave can be obtained by a practical approach similar to that of Tan & Lyatskaya (2009). An approximate expression for the perimeter of an ellipse with semi-major axis
From the property of the ellipse,
As the elliptical wave-front propagates outwards, its intensity diminishes inversely as
Hence, the wave amplitude varies as
Along the
One can compare the wave amplitude in Model D with that of a point source of equivalent strength (Model B). Remembering Eq. (B.18), we can re-write Eq. (D.7) as:
Thus
Table D.1 shows the comparative values of the wave amplitudes in the line source (Model D) and point source (Model B) models. They differ greatly near the origin (ρb = 0), but the solutions begin to converge away from the origin. At ρb =.2c, the Model D value is only half that of Model B. For ρb> c, the difference is below 10%.For ρb = 10c, the two solutions are virtually indistinguishable.
Acknowledgement
This study was partially supported by a grant from the National Science Foundation HBCU-UP HRD 0928904.
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