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Wave energy balance for protected basins
510
I . A . STEPANOV
Wave energy balance for protected b a s i n s I. A. STEPANOV Institute for Water Transport, Leningrad
Okeanologiya 1961.1, (5) : 851-855. BY means of the equation for the balance of energy, first published by MAKKAVEEVin 1937, one may calculate the transfer of energy in the direction of a wave. KARAUSHEV(1960) completed this equation by a term which represents the transfer of the wave energy in a transverse direction, i.e. along the wave crest. For standing waves (swell) in a deep basin the balance becomes simplified and it may be represented by cylindrical co-ordinates : 3x
'
,"
~k
~/
=0
(1)
where ~ is the wave energy per unit area, U is the grouped rate, r and l are the cylindrical co-ordinates, and m a coefficient of proportionality. The energy balance (1) has been applied for the practical calculation of the wave motion in large reservoirs (behind a promontory) and in a harbour. Expressing equation (1) as terminal differences, KARAUSHEVuses the grid method. In order to obtain a solution when the problem is stated in this way, it is necessary to make two or three successive approximations. The solution is clearly rather complex and time-consuming, which deters many designers from making such calculations. In addition, there is an unknown variation in the coefficient in the last term of equation (1), because the value of rn has been assumed to be constant. It is necessary to recognize the possibility of a variation in the coefficient m with respect to the longitudinal co-ordinates, in order to compare the solution of the balanced equation obtained for different water-masses with the results of the study of large-scale spatial models and with calculations by other methods (RYBCHEVSKY, LOGINOV, OFIISEROV, JOHNSON). The same conclusion may be drawn from a consideration of the radial character of energy distribution and of changes in the dispersion effect along its course. Furthermore, an indirect confirmation of the variation of m may be found in the examples of calculations given by KARAUSHEV,in which a successful agreement with the actual data is only obtained by means of unequal steps in the grid (KARAUSHEV, 1960). The size o f the grid was calculated by the use of two different variants, and in both variants the size of the steps on the grid depends on the intervals of the coefficient m. If the equation is solved for m, i.e. the problem is reversed and a solution obtained for equal intervals on the grid, it can be established that, in both cases, this coefficient is proportional to the longitudinal interval, m can only be constant with an unrestricted wave flow. This will be restricted to natural water-masses protected from wave-action, or to protected harbours. The longitudinal distance will be measured in the direction of the wave-motion from the beginning of a co-ordinate placed on a promontory or else on the structure which protects the watermass. In the general case, in view of the variability of m, it is convenient to divide it into two p a r t s - the variable p and the constant a 2 m
--
p
a 2
(2)
A study of large-scale models of marine and inland harbours, which was carried out in the Hydrotechnical Laboratory of the Institute for Water Transport in Leningrad (LIVT), showed that the constancy of a 2 bears a somewhat restricted character; a is constant only for a particular configuration of the harbour, because, in its own turn, it depends upon the ratio between the wave-length ~ and the width of the harbour entrance B :
In addition to making the alteration (2), equation (1) is expressed in terms of the rectangular co-ordinates x and y; also for the same depth of water-mass the grouped rate is constant :
Wave energy balance for protected basins
511
33 = P a z b23 3x ~yZ '
introducing the intermediate variable s : 3z9
b3 bs
3s _ P a 2 _ . bx by z '
fulfilling the condition --=p
or
3,
s =
dx
pdx,
f
(4)
the equation of balance is then obtained in a homogeneous form which is convenient from the operational point of view : ~ = a~ __35 3. 3s ~ y2
(5)
It can be shown that the heterogeneous balance equation, which includes the term R expressing the progressive loss of energy, reduces to a form similar to (5) (STEPANOV, 1961a). This is particularly important for shallow waters. The scope of the variable s may be changed in relation to the form of the function p, which is equivalent to an extension of the co-ordinates. The following will hold, for example, if the coefficient should be constant p = 1, s = X , and if we accept the above-mentioned linear dependence of the coefficient upon the longitudinal co-ordinates, then according to the conditions in (4) : p ~
x,
s ~
xz
and we may write the balance equation for a protected water-mass as follows : 33 -- a s . ~---~. 3 (x ~) ~ yZ
(6)
From the generally accepted relationships : ~ =
~'hZ ;
~-
k
h
-h-o-
%~o' 3
where h and 3 are the height of the waves and the energy at the calculated point under consideration, h0 and 30 are the wave height and energy at the start of the calculated section, for example, at the harbour entrance, k is the relative height of the wave at the calculated point. The general solution of the balance equation is then expressed in the form : i=n
= I I k~
(7)
I=1
where k~ is a particular solution, n the number of factors (or particular solutions) entering into the calculations. The particular solutions kt depend upon the actual conditions of the problem and may be shown in ' p u r e ' theoretical schemes. As the method for obtaining such solutions will not be discussed in detail here, we will restrict ourselves to a brief reference to certain particular solutions of k, previously obtained as a first approximation. The most important influence which determines the general pattern of the wave-motion penetrating into a water-mass, must be regarded as a diffraction. When the wave-motion is of a turbulent pulsating character the influence of diffraction is calculated as the particular solution k~. For a water-mass protected by a promontory or a breakwater (ST~ANOV, 1960a) this solution has the form :
512
I . A . ST~ANOV
(8) where ~ = erG represents the ' e r o s i o n ' function (JANKE and EMo~, 1949) y tgO = - .
X
For the case of the rotation of the axis of the breakwater close to the direction of the wave-motion, it is necessary to take into account the constriction of the diffraction sector :
kl'=
-
+
2-
F o r large t~---the angle of rotation of the breakwater in relation to the rays of the initial wavem o t i o n - - t h e second term in the root reduces to zero, and formula (8a) reverts to the form of (8). Solution (8) illustrates the basic diffraction pattern and only in the immediate vicinity of the breakwater is there any inaccuracy, due to the neglect of the shielding effect of the breakwater and the collapse of the waves directly against the shield. However, this reduction in the wave-motion relates to a zone which is deliberately protected against wave-action, and practically speaking it is insignificant, because the local wave energy here comprises only 1-4 per cent of the initial wave energy. However, this error is eliminated by the introduction of a correction for the distribution of wave-motion from the first Frenel zone. In this case a good agreement is obtained not only for the main area of the water-mass, but also close" to the shield or breakwater. F o r the calculation of the wave-motion penetrating into a water-mass through a strait or harbour entrance, the solutions (8) will be combined. This method provides a solution for displaced breakwaters; the theoretical equation for the main ray of energy (8) is obtained incidentally. The linear loss of energy may be calculated from the solutions of kz and k3, which have the form of a geometric l:rogression, which may be approximately expressed as an exponential equation. Thus, for viscous losses, which are the sum of turbulent viscosity and dispersion in air, equation (9) was obtained : D
kz = (1 -- p~)~-a
(9)
where D is the distance travelled by the wave (course of loss), and/~2 ~ 0.00015. F o r the calculation of the losses in shallow water-masses (in which the depth exceeds a critical value) with the resulting friction against the bottom and partial transformation of the waves there is the expression :
or the approximation
where ;~¢ is a coefficient of roughness, similar to the one used by BROVIKOV and OFH'SEROV (1958); HIS the depth of water at the calculated point ; f h (H/A) is a hyperbolic function relative to the depth. Strictly speaking, formula (10) is applicable to constant depths. However, one may agree with the ' Rele ' C R m t ~ G n ) assumption that the results of the calculations also extend to cases of variable depth. As the investigations by R. Wn~OEL (1951) showed, for ' waves spreading over an inclined bottom, the same relationships hold for any point as if the depth of water had been the same as at this point, but constant.' The limits of applicability of this thesis are debatable, but for approximate calculations it is subject to a deviation of at least 1--4 per cent. Even the strictest critics admit a deviation o f 1 per cent, while with a milder approach, based on laboratory investigations, a deviation of up to I0 per cent is admitted.
Wave energy balance for protected basins
513
Some uncertainty as to the limiting conditions has arisen as a result of the undoubted losses of energy against the walls along which the wave front is moving. Usually these losses have been ignored or else their distribution at a distance from the walls has not been taken into account. Many examples could be given in which such inaccuracy in the limiting conditions has sometimes had a substantial influence upon the results of investigations. In the best cases such losses have been determined as an average for the whole water-mass. According to the data obtained under natural conditions in a marine harbour, the losses of energy against a vertical concrete wall were as high as 15 per cent, while in another harbour with long pile-types defences, the losses were as high as 15-30 per cent. The size of such losses and their distribution at a distance from the walls may be calculated on the basis of the particular solution k~, which was announced at the First Interdepartmental Conference on Problems of Modelling in the Atmosphere and Hydrosphere, held in Leningrad in 1960 (STEvANOV, 1960b), and previously discussed in a seminar held in the V. E. T ~ o N o v Hydrotechnical Laboratory : k4 = /
1
(11) D
'
where r = ~5~bd~ d~; a table of values of • has been compiled; ~bis the function given in formula (8); "q = y ] 2 a x = tgO]2a is the relative co-ordinate; ~4 is the energy extinction fraction, which depends on the type of material of which the wall is constructed. All the examples given of particular solutions (8)-(11) satisfy the balance equation and have been verified for limiting conditions. In accordance with the symbolic representation (7) the relative height of the wave-motion caused by the penetration of a wave-crest into a protected water-body will be as follows : k = kl k2 k3 k4. (12) Refraction, the reflection of waves from the surroundings, the concentration of wave,energy when the water suddenly shoals, and other important circumstances which have been or am being investigated by others, may be calculated by such methods, and certainly should be when the conditions are appropriate. Although the relative values of the various particular solutions differ (for instance, the influence of k~ is high, that of kz negligible), the problem as to the participation of the corresponding k~ in the calculations should be resolved on the basis of the actual conditions of the object under investigation. In general, if one of the k is not required to participate in the calculations, it will automatically receive the value of unity, this follows from the structure of the formulae (8)-(11). The subsequent calculation of all the active factors enables a satisfactory agreement to be obtained between the calculated wave-motion and the experimental data obtained from large-scale models of 1 : 50, 1 : 60, 1 : 75 and under natural conditions. The inaccuracy of such calculations usually lies within the limits of error of the natural measurements, which are about 15-20 per cent. The suggested method has been used to calculate the wave-motion in a harbour on a large reservoir (Leningrad Hydroelectric and Water-transport) in an outer harbour on another reservoir and in a marine harbour (by ALISOVAand TROITSKY);the agreement with the natural data was satisfactory. CONCLUSIONS The method described is intended for the approximate calculation of the height of wave-motion in protected water-masses. The basis of the calculations is the equation for the balance of waveenergy. The combination of a hydraulic statement of the problem together with the principles established in hydrodynamics has proved beneficialin bringing the solution to a form in whichit is acceptable for applied calculations. Onlyfundamental problems have been examinedin this paper : a general solution is presented, and essential referenceshave been made to particular solutions and to the possibility of their practical application. It is noted that the calculations are in satisfactory agreement with experimental data, corresponding to the error in the measurement of waves, which is 15-20 per cent under natural conditions. The original data, particulars as to the developmentof the formulae, useful graphs and nomogrammes, and also definiteexamples of the calculation of particular cases and a fuller bibliography may be found in the papers listed below.
514
I . A . Sm, ANOV REFERENCES
JANKE, E. and EMDE, F. (1949) Tables of Functions with Formulae and Curves. State Technical Publishing House. KARAUSI-mV,A. V. (1960) Problems of the Dynamics of Natural Water Currents. Hydrometeorol. Publishing House. MAKZAV~EV,V. M. (1937) On the processes of growth and extinction of waves of short length in relation to their separation in the direction of the wind. Trudy Gos. gidrol, insta (5). OFrrSEROV, A. S. (1958) Problems of the Methods of Laboratory Investigation of Waves and of the Linear Energy Losses by Wave-motion. State Publishing House. S ~ S E L , V. K. and S~PANOV, l. A. (1960) On the accuracy of investigation of waves on space models. Report on the First Interdepartmental Conference on Problems of Modelling in the Atmosphere and Hydrosphere. U.S.S.R. Academy of Sciences, Marine Hydrophysics Institute. STm'ANOV,I. A. (1960a) The extinction of waves by a single breakwater. Trudy Leningr. insta vodn. transp. (VIII). STEPANOV,I. A. (1960b) The influence of a longitudinal wall on wave-motion. Report on the First Interdepartmental Conference on Problems of Modelling in the Atmosphere and Hydrosphere. U.S.S.R. Academy of Sciences, Marine Hydrophysics Institute. STEPANOV, I. A. (1961a) The use of a balance of wave-energy for restricted water-masses. Thesis, XVth Sci.-Techn. Conf. Leningr. Inst. Water-Transport. STEPANOV,I. A. (1961b) The calculation of the wave-regime within a harbour. Rechnoi Transport (River Transport) (2). SreVANOV,I. A. (I 961c) Calculation of the extinction of wave-motion in harbours and outer harbours. Trudv Leningr. Vodn. Transp. (LIVT) (XIII). WEmEL, R. (1951) The experimental study of wave height. Coll. papers on Bases of Forecasting of Wind Waves, Swell and Breakers. Foreign Literature Publishing House. Note---All references to articles from Russian journals etc. are in Russian.
I . A . STEPANOV
Wave energy balance for protected b a s i n s I. A. STEPANOV Institute for Water Transport, Leningrad
Okeanologiya 1961.1, (5) : 851-855. BY means of the equation for the balance of energy, first published by MAKKAVEEVin 1937, one may calculate the transfer of energy in the direction of a wave. KARAUSHEV(1960) completed this equation by a term which represents the transfer of the wave energy in a transverse direction, i.e. along the wave crest. For standing waves (swell) in a deep basin the balance becomes simplified and it may be represented by cylindrical co-ordinates : 3x
'
,"
~k
~/
=0
(1)
where ~ is the wave energy per unit area, U is the grouped rate, r and l are the cylindrical co-ordinates, and m a coefficient of proportionality. The energy balance (1) has been applied for the practical calculation of the wave motion in large reservoirs (behind a promontory) and in a harbour. Expressing equation (1) as terminal differences, KARAUSHEVuses the grid method. In order to obtain a solution when the problem is stated in this way, it is necessary to make two or three successive approximations. The solution is clearly rather complex and time-consuming, which deters many designers from making such calculations. In addition, there is an unknown variation in the coefficient in the last term of equation (1), because the value of rn has been assumed to be constant. It is necessary to recognize the possibility of a variation in the coefficient m with respect to the longitudinal co-ordinates, in order to compare the solution of the balanced equation obtained for different water-masses with the results of the study of large-scale spatial models and with calculations by other methods (RYBCHEVSKY, LOGINOV, OFIISEROV, JOHNSON). The same conclusion may be drawn from a consideration of the radial character of energy distribution and of changes in the dispersion effect along its course. Furthermore, an indirect confirmation of the variation of m may be found in the examples of calculations given by KARAUSHEV,in which a successful agreement with the actual data is only obtained by means of unequal steps in the grid (KARAUSHEV, 1960). The size o f the grid was calculated by the use of two different variants, and in both variants the size of the steps on the grid depends on the intervals of the coefficient m. If the equation is solved for m, i.e. the problem is reversed and a solution obtained for equal intervals on the grid, it can be established that, in both cases, this coefficient is proportional to the longitudinal interval, m can only be constant with an unrestricted wave flow. This will be restricted to natural water-masses protected from wave-action, or to protected harbours. The longitudinal distance will be measured in the direction of the wave-motion from the beginning of a co-ordinate placed on a promontory or else on the structure which protects the watermass. In the general case, in view of the variability of m, it is convenient to divide it into two p a r t s - the variable p and the constant a 2 m
--
p
a 2
(2)
A study of large-scale models of marine and inland harbours, which was carried out in the Hydrotechnical Laboratory of the Institute for Water Transport in Leningrad (LIVT), showed that the constancy of a 2 bears a somewhat restricted character; a is constant only for a particular configuration of the harbour, because, in its own turn, it depends upon the ratio between the wave-length ~ and the width of the harbour entrance B :
In addition to making the alteration (2), equation (1) is expressed in terms of the rectangular co-ordinates x and y; also for the same depth of water-mass the grouped rate is constant :
Wave energy balance for protected basins
511
33 = P a z b23 3x ~yZ '
introducing the intermediate variable s : 3z9
b3 bs
3s _ P a 2 _ . bx by z '
fulfilling the condition --=p
or
3,
s =
dx
pdx,
f
(4)
the equation of balance is then obtained in a homogeneous form which is convenient from the operational point of view : ~ = a~ __35 3. 3s ~ y2
(5)
It can be shown that the heterogeneous balance equation, which includes the term R expressing the progressive loss of energy, reduces to a form similar to (5) (STEPANOV, 1961a). This is particularly important for shallow waters. The scope of the variable s may be changed in relation to the form of the function p, which is equivalent to an extension of the co-ordinates. The following will hold, for example, if the coefficient should be constant p = 1, s = X , and if we accept the above-mentioned linear dependence of the coefficient upon the longitudinal co-ordinates, then according to the conditions in (4) : p ~
x,
s ~
xz
and we may write the balance equation for a protected water-mass as follows : 33 -- a s . ~---~. 3 (x ~) ~ yZ
(6)
From the generally accepted relationships : ~ =
~'hZ ;
~-
k
h
-h-o-
%~o' 3
where h and 3 are the height of the waves and the energy at the calculated point under consideration, h0 and 30 are the wave height and energy at the start of the calculated section, for example, at the harbour entrance, k is the relative height of the wave at the calculated point. The general solution of the balance equation is then expressed in the form : i=n
= I I k~
(7)
I=1
where k~ is a particular solution, n the number of factors (or particular solutions) entering into the calculations. The particular solutions kt depend upon the actual conditions of the problem and may be shown in ' p u r e ' theoretical schemes. As the method for obtaining such solutions will not be discussed in detail here, we will restrict ourselves to a brief reference to certain particular solutions of k, previously obtained as a first approximation. The most important influence which determines the general pattern of the wave-motion penetrating into a water-mass, must be regarded as a diffraction. When the wave-motion is of a turbulent pulsating character the influence of diffraction is calculated as the particular solution k~. For a water-mass protected by a promontory or a breakwater (ST~ANOV, 1960a) this solution has the form :
512
I . A . ST~ANOV
(8) where ~ = erG represents the ' e r o s i o n ' function (JANKE and EMo~, 1949) y tgO = - .
X
For the case of the rotation of the axis of the breakwater close to the direction of the wave-motion, it is necessary to take into account the constriction of the diffraction sector :
kl'=
-
+
2-
F o r large t~---the angle of rotation of the breakwater in relation to the rays of the initial wavem o t i o n - - t h e second term in the root reduces to zero, and formula (8a) reverts to the form of (8). Solution (8) illustrates the basic diffraction pattern and only in the immediate vicinity of the breakwater is there any inaccuracy, due to the neglect of the shielding effect of the breakwater and the collapse of the waves directly against the shield. However, this reduction in the wave-motion relates to a zone which is deliberately protected against wave-action, and practically speaking it is insignificant, because the local wave energy here comprises only 1-4 per cent of the initial wave energy. However, this error is eliminated by the introduction of a correction for the distribution of wave-motion from the first Frenel zone. In this case a good agreement is obtained not only for the main area of the water-mass, but also close" to the shield or breakwater. F o r the calculation of the wave-motion penetrating into a water-mass through a strait or harbour entrance, the solutions (8) will be combined. This method provides a solution for displaced breakwaters; the theoretical equation for the main ray of energy (8) is obtained incidentally. The linear loss of energy may be calculated from the solutions of kz and k3, which have the form of a geometric l:rogression, which may be approximately expressed as an exponential equation. Thus, for viscous losses, which are the sum of turbulent viscosity and dispersion in air, equation (9) was obtained : D
kz = (1 -- p~)~-a
(9)
where D is the distance travelled by the wave (course of loss), and/~2 ~ 0.00015. F o r the calculation of the losses in shallow water-masses (in which the depth exceeds a critical value) with the resulting friction against the bottom and partial transformation of the waves there is the expression :
or the approximation
where ;~¢ is a coefficient of roughness, similar to the one used by BROVIKOV and OFH'SEROV (1958); HIS the depth of water at the calculated point ; f h (H/A) is a hyperbolic function relative to the depth. Strictly speaking, formula (10) is applicable to constant depths. However, one may agree with the ' Rele ' C R m t ~ G n ) assumption that the results of the calculations also extend to cases of variable depth. As the investigations by R. Wn~OEL (1951) showed, for ' waves spreading over an inclined bottom, the same relationships hold for any point as if the depth of water had been the same as at this point, but constant.' The limits of applicability of this thesis are debatable, but for approximate calculations it is subject to a deviation of at least 1--4 per cent. Even the strictest critics admit a deviation o f 1 per cent, while with a milder approach, based on laboratory investigations, a deviation of up to I0 per cent is admitted.
Wave energy balance for protected basins
513
Some uncertainty as to the limiting conditions has arisen as a result of the undoubted losses of energy against the walls along which the wave front is moving. Usually these losses have been ignored or else their distribution at a distance from the walls has not been taken into account. Many examples could be given in which such inaccuracy in the limiting conditions has sometimes had a substantial influence upon the results of investigations. In the best cases such losses have been determined as an average for the whole water-mass. According to the data obtained under natural conditions in a marine harbour, the losses of energy against a vertical concrete wall were as high as 15 per cent, while in another harbour with long pile-types defences, the losses were as high as 15-30 per cent. The size of such losses and their distribution at a distance from the walls may be calculated on the basis of the particular solution k~, which was announced at the First Interdepartmental Conference on Problems of Modelling in the Atmosphere and Hydrosphere, held in Leningrad in 1960 (STEvANOV, 1960b), and previously discussed in a seminar held in the V. E. T ~ o N o v Hydrotechnical Laboratory : k4 = /
1
(11) D
'
where r = ~5~bd~ d~; a table of values of • has been compiled; ~bis the function given in formula (8); "q = y ] 2 a x = tgO]2a is the relative co-ordinate; ~4 is the energy extinction fraction, which depends on the type of material of which the wall is constructed. All the examples given of particular solutions (8)-(11) satisfy the balance equation and have been verified for limiting conditions. In accordance with the symbolic representation (7) the relative height of the wave-motion caused by the penetration of a wave-crest into a protected water-body will be as follows : k = kl k2 k3 k4. (12) Refraction, the reflection of waves from the surroundings, the concentration of wave,energy when the water suddenly shoals, and other important circumstances which have been or am being investigated by others, may be calculated by such methods, and certainly should be when the conditions are appropriate. Although the relative values of the various particular solutions differ (for instance, the influence of k~ is high, that of kz negligible), the problem as to the participation of the corresponding k~ in the calculations should be resolved on the basis of the actual conditions of the object under investigation. In general, if one of the k is not required to participate in the calculations, it will automatically receive the value of unity, this follows from the structure of the formulae (8)-(11). The subsequent calculation of all the active factors enables a satisfactory agreement to be obtained between the calculated wave-motion and the experimental data obtained from large-scale models of 1 : 50, 1 : 60, 1 : 75 and under natural conditions. The inaccuracy of such calculations usually lies within the limits of error of the natural measurements, which are about 15-20 per cent. The suggested method has been used to calculate the wave-motion in a harbour on a large reservoir (Leningrad Hydroelectric and Water-transport) in an outer harbour on another reservoir and in a marine harbour (by ALISOVAand TROITSKY);the agreement with the natural data was satisfactory. CONCLUSIONS The method described is intended for the approximate calculation of the height of wave-motion in protected water-masses. The basis of the calculations is the equation for the balance of waveenergy. The combination of a hydraulic statement of the problem together with the principles established in hydrodynamics has proved beneficialin bringing the solution to a form in whichit is acceptable for applied calculations. Onlyfundamental problems have been examinedin this paper : a general solution is presented, and essential referenceshave been made to particular solutions and to the possibility of their practical application. It is noted that the calculations are in satisfactory agreement with experimental data, corresponding to the error in the measurement of waves, which is 15-20 per cent under natural conditions. The original data, particulars as to the developmentof the formulae, useful graphs and nomogrammes, and also definiteexamples of the calculation of particular cases and a fuller bibliography may be found in the papers listed below.
514
I . A . Sm, ANOV REFERENCES
JANKE, E. and EMDE, F. (1949) Tables of Functions with Formulae and Curves. State Technical Publishing House. KARAUSI-mV,A. V. (1960) Problems of the Dynamics of Natural Water Currents. Hydrometeorol. Publishing House. MAKZAV~EV,V. M. (1937) On the processes of growth and extinction of waves of short length in relation to their separation in the direction of the wind. Trudy Gos. gidrol, insta (5). OFrrSEROV, A. S. (1958) Problems of the Methods of Laboratory Investigation of Waves and of the Linear Energy Losses by Wave-motion. State Publishing House. S ~ S E L , V. K. and S~PANOV, l. A. (1960) On the accuracy of investigation of waves on space models. Report on the First Interdepartmental Conference on Problems of Modelling in the Atmosphere and Hydrosphere. U.S.S.R. Academy of Sciences, Marine Hydrophysics Institute. STm'ANOV,I. A. (1960a) The extinction of waves by a single breakwater. Trudy Leningr. insta vodn. transp. (VIII). STEPANOV,I. A. (1960b) The influence of a longitudinal wall on wave-motion. Report on the First Interdepartmental Conference on Problems of Modelling in the Atmosphere and Hydrosphere. U.S.S.R. Academy of Sciences, Marine Hydrophysics Institute. STEPANOV, I. A. (1961a) The use of a balance of wave-energy for restricted water-masses. Thesis, XVth Sci.-Techn. Conf. Leningr. Inst. Water-Transport. STEPANOV,I. A. (1961b) The calculation of the wave-regime within a harbour. Rechnoi Transport (River Transport) (2). SreVANOV,I. A. (I 961c) Calculation of the extinction of wave-motion in harbours and outer harbours. Trudv Leningr. Vodn. Transp. (LIVT) (XIII). WEmEL, R. (1951) The experimental study of wave height. Coll. papers on Bases of Forecasting of Wind Waves, Swell and Breakers. Foreign Literature Publishing House. Note---All references to articles from Russian journals etc. are in Russian.