The speed of waves in channels.

Observation of waves in a channel shows that a seemingly unlimited variety of waves can occur. The wind blowing on still water in a channel may produce small ripples on one day and organised well-defined waves moving along the channel on the next depending on the speed and direction of the wind. But flowing water often has stationary criss-cross wave patterns that are generated by the flow and, at other points, a stationary pattern of stippled waves. These waves may be of intellectual interest to the observer but the interest to the mechanical engineer is in the wave that moves along the channel as the result of a disturbance.

 

We have to have a way of creating a wave to study. Let us start by imagining a long horizontal channel of rectangular section that is partly filled with water that is at rest. Now suppose that a block that fills the channel is at rest on the bed. The arrangement is shown diagrammatically in the upper diagram in figure 10-17. Suppose now that the movable block is suddenly made to move steadily along the channel at a velocity . It is a matter of observation that this causes a wave to be transmitted quite quickly along the surface and that in front of the wave the water is at rest and behind the wave moving at the speed of the moving block at a new depth. This wave will appear to roll along the surface.

 

It is possible to find a useful expression for the speed of this wave using the ideas of one-dimensional flow, continuity and momentum and, if we find this expression for a wave moving in stationary water, we can also regard it as being the speed of a wave relative to the water.

 

If we start at the instant at which the block starts to move at some velocity  we can say that in some subsequent time interval  the wave will have moved a distance  where  is the speed of the wave. During this time a force caused by the difference in levels before and after the wave will have given a mass of water occupying the space between the block and the wave a new velocity .

 

If we make the usual simplifications we can say that the net force equals :-

                                  .

This force will have acted on a mass of water equal to  to give it velocity c. Therefore, from the statement that the impulse of a force equals the change in momentum that it produces, we can write:-

                               from which :-

                                                      

We can also relate these quantities by saying that the stationary mass, , is equal to the  moving  mass, . From this .

 

This now requires some manipulation to produce an expression that relates  and the two depths in the channel . If we expand the square term in the impulse equation and divide through by  we get:-  . But  or , and from this :- 

                                                         

This is an interesting result as it says that the speed of the wave depends on the square root of the arithmetic mean of the depths before and after the wave and that we can deduce that a wave of a given height would travel more quickly in a deep channel than in a shallow one and, that in a given channel, a large wave would travel more quickly than a small one.

 

This has certain implications. Suppose that we consider our moving block to have taken a short but finite time to reach its speed. During that time the depth in front of it will have been increasing and we can consider that a series of waves of different heights have been transmitted. As the speed of the wave increases as the depth increases the later waves will catch up with the earlier ones and the wave will first become steeper and then roll. Once it has started the rolling will continue. As there is nothing in the derivation of the expression for the speed of the wave that would preclude its use for the block moving in the opposite direction we can consider the “negative” wave. The falling level will produce a series of waves each slower moving than the one before it. The negative wave will become longer and longer.

 

Had the block been made to move in the opposite direction it would not have led to a similar wave producing a drop in level and water keeping up with the moving block. Instead, the surface takes on a slope with the leading edge of the slope advancing quickly and the unbalanced pressure force acts to accelerate the water slowly. Clearly either of these two effects could be superimposed on water that is moving in a channel. Sometimes this occurs naturally especially in tidal rivers where a rapid rise in the tide will produce a bore. More often positive and negative waves are produced by rapid adjustment of sluice gates. These effects can be demonstrated in laboratory channels.

 

A further point of interest is that a very small wave will move at a speed given by  where d is the depth and then the speed of the wave will be equal to the critical speed, that is, Fr for the wave will be 1.

 

These waves have links to shock waves in gases and the intensifying of the positive-going waves and the attenuation of the negative-going waves has some part to play in producing wakes behind ships and aeroplanes and buulets.

 

Quite large waves can be produced in channels and around 1850 a service of high speed, horse drawn, passenger boats ran on some canals “surfing” along on such waves. The technique was to make the horses pull vigorously at the start so that a wave would form in front of the boat. Then, with an extra effort the boat could be lifted over the wave to get in front of it. From there on the boat would be carried forward by the wave and the effort required from the horses much reduced although the boat was moving much more quickly than it would otherwise do with the same effort. The boat had produced an isolated wave moving along the canal and, in modern parlance, surfed along on the wave. The boats covered 4 miles in about 20 minutes and ran to a timetable.[1]

I spotted the picture in 10-17 in a calendar and could think of no reason for the wave that is almost square across the canal in line with the stem of the boat. It is clearly not a part of the bow wave. When I was writing this chapter and thinking about the waves in a channel I recalled meeting a river “steamer” on the River Thames at Kew when the water level was low and I was in an eight-oared racing boat. Ahead of the steamer the water drained away from us and flowed under the steamer leaving us in danger of going aground. The propellers just pulled the water away from under our boat. The propeller of the narrow boat in 10-18 must be doing the same and the result is this hydraulic “drop” ahead of the boat. My text also required that there be a wave across the canal behind the boat in which the level rises. I had the picture in 10-19 that had been cut from a newspaper and the drop is evident again on the right of the picture and the “jump” as well. It is very intriguing. It pays to look for these odd examples of physics at work and make a scrap-book of them.

 

Even prolonged study of a particular wave pattern in the sea seems not to lead to any real understanding of what is going on. However there are two quite different forms of wave motion one of which occurs in deep water and one in shallow water.

 

In deep water the motion of particles of water as a consequence of wave motion on the surface varies with depth. The particles behave as if they form elastic strings with one end fixed to some deep point that is stationary and the other to a point in the surface that moves in a circle about another fixed point which is vertically above the first. The radius vector for each circle is progressively out of phase with its neighbour and the resultant motion is one that approximates to the observed motion in waves.

 

The vertical component of the motion at depth transfers no energy horizontally but the net effect is one of continuous transfer of momentum forwards and near to the surface a series of particles that are equally spaced horizontally in the direction of motion of the waves will have equal phase differences. Then for particles spaced out by half a wave length one particle will be at the top of its circle and travelling forwards when the other is at the bottom of its circle and travelling backwards. The combination of such motions is such that the particles when at the top carry energy forwards but underneath the particles simply fill the space under the crests and do not transmit energy.

 

As waves come ashore on a shelving beach they roll as is all too evident. The tsunami that hit South East Asia was almost undetectable as it travelled in the open sea but, when it came ashore the wavelength was very long and the backwards flow deep down under the wave first pulled the sea away from the beaches and then the tsunami came roaring inland as a great rolling wave.

 

Waves are worth watching.