179 



WAVES AND TIDES. 



WAVES AXD TIDES. 



'.an 



utter U in a itato of disturbance, by the new position* of the eight 

 article.) commuting the angular points of the former parallelepiped. 

 But, M the water is supposed to be in a rectangular canal, the 

 ixeut of the parallelepiped in the direction of the breadth of the canal 

 U supposed to be constant; and therefore it U sufficient to assume 

 the equality of the pamllelograini which form a aide of each in the 

 direction of the length of the canal 



The caual being of uniform depth, the " equation of continuity " ia 

 xpreMedby 



/*<Jx 

 T / j (between and ]/) 



where x and y are respectively the horizontal and vertical coordinates 

 of a particle of fluid, and where z and T are respectively the horizontal 

 and vertical displacement* of the particle by the action of the dis- 

 turbing forces : the equation expresses a relation between those 

 coordinates and the disturbances or displacement*. 



An equation of the pressure experienced by any particle from the 

 force* which act upon it is next found in the following manner : 

 Let p represent the pressure in every direction on the lower part of a 

 disturbed molecule of water in consequence of the height or weight ot 

 the nlament of particles above it : then, the verticil coordinate of the 

 particle being y' or y + T, suppose in the element dt of time the vertical 

 coordinate to become y 1 + Sy ' (the vertical height of the filament above 

 the molecule in that position being increased by the general rising of 

 the wave), the pressure on the upper part of the molecule will bo 



dp 

 greater than before, and may be represented by p + Jjy ty' i conse- 



quently the molecule may be supposed to be pressed downwards by a 

 * 



force represented by -j-, Sy'. 



Now, in order to render the expression 



for the hydrostatical pressure homologous to that which is employed 

 for the force of gravity, it must be considered as accelerative, or as a 

 motive-power divided by the mass; and therefore the accelerative 



dp 

 pressure downwards becomes g-y , which being added to rj, representing 



dp 

 the force of gravity and supposed to be constant, there arises ^-7 + g 



for the whole acceleration of the molecule downwards : hence there is 

 obtained the equation 



This equation being integrated between the limits for the bottom oi 

 the molecule and the top of the wave, gives the bydrostatical force by 

 which a vertical filament of water descends, or that by which it is 

 carried forward horizontally. 



Let the slender column of water above the molecule have a hori 

 zontal breadth equal to A in the direction of .T ; then the horizontal 

 pressure in front, by which the column is forced backwards, wil 

 exceed the pressure by which it is carried forwards by a force repre 



dp dp 



sented by ^ dh, or by an acceleration represented by -T- ; therefore 



dp 

 the horizontal acceleration forwards is j- : if extraneous forces, ai 



the attraction of the sun or moon on the molecule, and the effects o. 

 friction, be together represented by F, when estimated in the direction 



dp 

 of *, there arises the expression F -7- for the whole acceleration 



forwards; then the "equation of motion" becomes 

 rf|x rfg 



which gives relation* between the terms x, Y, ,r, ?/, and t. This 

 " equation of equal pressure" and the " equation of continuity" con 

 stitute the theory of the motion of fluids in canals of uniform breadth 



The general equation representing the disturbance or displacement 

 of particle of water is the same as that which expresses the dis 

 turbance of a particle of light in the iindul.it ory theory; and, in 

 order to indicate oscillatory motion, both the horizontal and vertica 

 displacement* are represented by terms containing the sines or cosines 

 of angles depending on the time I. 



If it be assumed that 



X = BCOS(( nu) -1-8 eln (nt-mx), 



B and s being functions of y, the above equations of continuity ani 

 of equal presture give, on tho supposition that gravity is constant, tha 

 no extraneous force* act, and retaining for the present only the fire 



power of -j- , or of the horizontal displacement 



rfp + rfP * 



From tlioio two equations are obtained the values of x and T in 

 terras of A cos (nt - m.r) and B sin (HI- i. 



These values will not be altered if .* is increased or diminished by 

 ie, two, three. &c. whole circumferences, that U, if ,r is increased or 



diminished by , , &c., while < remains the same ; therefore 



s tho value of the increments of x which correspond to points \ 

 the particles of water are in the same condition with respect to di* 



urbanoe, that is, is the length of a wave. Again, the values will 



not bo altered if n< is increased or diminished by whole circumferences, 



2 4" 

 that U, if t is increased or diminished by , , &o., while x remain* 



lie same ; therefore is tho increment of time which corresponds to 

 lie particles of water being successively in the like state of disturb- 

 ance, that isj is the period of a wave, or the time between two 

 successive formations of a wave-Bummit at the same place. Therefore 

 - is the velocity of the wave ; and, from the value found for it by the 



.heory, it follows that the velocity depends on in and on tho depth of 

 ,he water : the latter being constant, the velocity depends ou thn 

 ength of the wave, or it depends on the time in which a pan 

 water makes a complete vibration. If the length of a wave or the 

 ;ime of its vibration U given, the velocity will vary with the depth of 

 the water. 



From a table of the computed velocities of waves of different lengths, 

 and with different depths of water, it is found that when the length of 

 the wave is not greater than the depth of the water, the velocity of tin- 

 wave is proportional to the square root of its length : also when the 

 length is not less than one thousand times the depth of water, tho 

 velocity is proportional to the square root of the depth, and is the same 

 as that which a body would acquire in falling from rest through a 

 height equal to half that depth. The greatest horizontal and vertical 

 displacement of a particle being computed for different values of tho 

 length of the wave and the depth of the water, it appears that when 

 the latter is great, compared with the former, as in the open sea, the 

 motion of the water far below the surface is very small compared with 

 the motion at the surface, and at a depth equal to the length of wave 

 it is only about ,J S of the motion at the surface. On the same suppo- 

 sition, the greatest horizontal motion is equal to the greatest vertical 

 motion. When the length of the wave is great compared with the 

 depth of the water, as in tide-waves, the horizontal motion of tho 

 particles is nearly the same from the surface to the bottom, and the 

 vertical motion varies with the distance from the bottom. " On the same 

 supposition, the vertical motion of the superior particles is much less 

 thau their horizontal motion. 



The movement of a particle of water near the surface may be deter- 

 mined from the values given by the theory to x and Y : if the waves 

 are small, so that A may be considered as equal to B, we have 

 (x* + Y*) I = c, a constant ; which, being the equation of a circle, it 

 follows that the particles move in the circumference of a circle whose 

 radius is A ; but if the length of the wave is great compared with tho 

 depth of water, the equation is that of an ellipse. These last deductions 

 from the theory are conformable to what has been observed in exj-i-ri- 

 mental waves, as above mentioned. It follows that, in a long tide- wave 

 flowing up a channel, the horizontal velocity in the direction of the 

 wave's motion is the greatest at the summit of the wave that is, at 

 high-water : at the place of greatest depression that is, at low 

 the motion is most rapid downwards ; and at tho mean level the 

 water is for a time stationary. 



In investigating theoretically the phenomena of waves by whatever 

 cause produced, if the lengths of the waves are very great con 

 with the depth of the canal in which they move, it becomes necessary 



to retain the second and even higher powers of , or of thehorizont.il 



dx 

 displacement, in the equations of continuity and of equal pressure ; 



but the vertical oscillations being then small, the value of _JJ may be 



of 



neglected.- Then, if the perturbating actions of the sun and moon are 

 not considered, the integration of the differential equation of equal 

 pressure gives a value of the vertical displacement at the surface, or 

 the height of the wave above the mean elevation, in terms which 

 contain k sin (nt-mx) andlcx sin (2a 2mx), k being the depth of 

 water in the canal. Tracing an undulating line whose ordinates are 

 the values of that vertical height, corresponding to different values of 

 x, the horizontal distance from the mouth of the canal, which is sup- 

 posed to open to the sea; it is found that, near the opening, the front 

 and rear slopes of the waves are of equal lengths and of similar forms ; 

 but as the distance from the sea become* greater, the front slope i* 

 shorter and steeper, and the rear slope longer and more gentle. At a 

 great distance the latter becomes nearly horizontal in tho middle, 

 and at length it divides into two parts, so that the wave becomes 

 double. Near the sea, also, the time occupied by the rise of the wave 

 i equal to the time occupied by its descent : at a certain distance the 

 rise takes place in less time t lian the descent ; and at a fctill greater 

 distance the descent, after having been rapid, ia checked, or changed 



