9.7: Deep-Water Waves

Individual water molecules in waves in deep water (depth >L/2) move in vertical circular orbits oriented in the direction of wave travel (Fig. 9-10). The forward motion at the top of the orbit is very slightly greater than the backward motion at the bottom of the orbit, hence there is a very small net forward movement of water in the wave, although this forward movement of water is so small that it is negligible (Fig. 9-10b). Water motion within a wave is not restricted to water depths between crest and trough. Scuba divers who have dived where long-period swell waves are present know that the surging wave motion can be felt 10 m or more below the surface, even when the surface swell waves are less than a meter high.

Figure 9-10. Orbital motion of water particles beneath waves. (a) The orbits in which individual particles move within a wave decrease in diameter exponentially with increasing depth. The wave motion becomes essentially zero at a depth equal to one-half of the wavelength. (b) The particle orbits are not exactly closed. Instead, each particle moves forward a very small distance (exaggerated in this figure) during each orbit. Therefore, there is a very small net mass transport of water in the direction of the wave.

Water immediately below the surface moves in an orbit whose diameter is equal to the wave height, whereas water farther below the surface moves in orbits whose diameter decreases as depth increases. The decrease in orbital diameter with increasing depth depends on the wavelength. The diameter is reduced to one-half of the wave height when the depth is equal to one-ninth of the wavelength, and it is almost zero at a depth of one-half of the wavelength (Fig. 9-10).

The speed of a wave (the speed at which the wave form of the wave moves forward) in deep water also depends on wavelength and is approximated by the following equation:

C = √gL/2π

where g is the acceleration due to gravity. Because g (9.8 m•s^–2) and π (3.142) are constants, we can simplify this equation to: 

C = 1.25√L

C is measured in meters per second (m•s^–1), and L in meters (m).

Thus, the speed of deep-water waves (depth >L/2) increases with increasing wavelength. Because wave speed is also equal to L/T, we can calculate the wavelength of a deep-water wave if we know its period:

C = L/T = 1.25

L = 1.252 × T² = 1.5625T²

Table 9-1 lists wavelengths and celerities for deepwater waves of different periods. The table can be useful for scuba divers in planning a dive. If the divers count the number of waves that pass their boat per minute, they can calculate the average wave period. From Table 9-1, they can then determine the L/2 and L/9 depths for this wave period. If the L/9 depth exceeds the depth of the planned dive, the divers can expect to feel a strong wave surge, though it will be less than that at the surface, throughout the dive. At depths greater than L/2, there will be no wave surge.

Table 9-1a. Characteristics of Deep-Water Waves (metric)

Wavelength (m)	Period (s)	Frequency (waves per minute)	Depth of No Motion (L/2) (m)	Depth at Which Motion Reduced by 50% (L/9) (m)	Deep Water Celerity (m•s^–1)
1.6	1	60	0.8	0.2	1.6
6.2	2	30	3.1	0.7	3.1
14	3	20	7.1	1.6	4.7
25	4	15	13	2.8	6.2
39	5	12	20	4.3	7.8
56	6	10	28	6.2	9.4
77	7	8.6	38	8.5	11
100	8	7.5	50	11	12
126	9	6.7	63	14	14
156	10	6.0	78	17	16
225	12	5.0	113	25	19
306	14	4.3	153	34	22
400	16	3.7	200	44	25
506	18	3.3	253	56	28
624	20	3.0	312	69	31
975	25	2.4	488	108	39
1405	30	2.0	703	156	47

Table 9-1b. Characteristics of Deep-Water Waves (imperial)

Wavelength (ft.)	Period (s)	Frequency (waves per min)	Depth of No Motion (L/2) (ft.)	Depth at Which Motion Reduced by 50% (L/9) (ft.)	Deep Water Celerity (mph)
5	1	60	0.5	0.6	3.5
20	2	30	10	2.2	7.0
45	3	20	23	5.0	10
80	4	15	40	8.9	14
126	5	12	63	14	17
181	6	10	90	20	21
246	7	8.6	123	27	24
322	8	7.5	161	36	28
407	9	6.7	203	45	31
502	10	6.0	251	56	35
724	12	5.0	362	80	42
985	14	4.3	493	109	49
1287	16	3.7	644	143	56
1628	18	3.3	814	181	63
2008	20	3.0	1004	223	70
3138	25	2.4	1569	349	87
4521	30	2.0	2261	502	105

Wave Trains

Waves usually travel in groups called “wave trains.” As the leading wave in a wave train travels, some of its energy is transferred to the molecules of undisturbed water in front of it. This energy is used to initiate orbital motion of the previously undisturbed water molecules. However, only half of the energy of the individual wave is transferred forward; the remaining energy is transferred back to the second wave in the train. By examining the pressure gradient under the wave, we can see how this happens (Fig. 9-4). The second wave in the train also transfers energy forward to the first and back to the third wave. As the wave train progresses, energy is lost from the front wave of the wave train. The front wave progressively loses height and eventually disappears. The energy transferred backward from the first wave is transferred progressively rearward from one wave to another, until it passes the last wave and builds a new wave at the rear of the wave train.

In deep water, a wave train and the energy associated with it move at one-half the speed of individual waves within the wave train. The time it takes for waves of a particular wavelength to cross the ocean must be calculated from this group speed, which is one-half of the individual wave speed specified in Table 9-1.

As a result of the rearward transfer of energy in the wave train, the front wave in the train continuously decays, and a new wave is continuously created at the rear of the train (Fig. 9-11). Individual waves form at the back of the train progress forward as their predecessors decay and, eventually, reach the front of the train, where they decay themselves. We can see this process by carefully watching ripples caused by a stone thrown into a lake or the waves of a ship’s wake (Fig. 9-12).

Figure 9-11. Waves travel in groups called wave trains. As the leading wave of the group travels forward, it transfers half of its energy forward to initiate motion in the undisturbed surface ahead and transfers the other half to the wave behind it to maintain the wave motion. The leading wave is constantly decaying (losing energy) as it transfers energy forward to displace the undisturbed water surface ahead of it. As a result, the leading wave in the wave train is constantly disappearing, while a new wave is constantly being formed at the back of the train. To visualize this process, follow wave number 5 in this diagram. Notice that the train travels at half the speed of an individual wave.

Figure 9-12. A boat’s bow and stern waves travel out from the ship as wave trains. If you look carefully at the bow wave of the leading boat, you can see that the leading wave disappears as it moves away from where the boat has passed and is replaced by another wave behind it. The second wave in turn disappears to be replaced by another one behind it, and so on. The wave energy is also slowly spread as the wave crest lengthens by dispersion, and the wave train eventually disappears. Note how the bow and stern waves of the two boats combine to form complex wave interference patterns.

Wave Interference

Only very rarely are waves in a specific part of the ocean regularly spaced and all of the same height. The reason for the irregularity in height and spacing is that waves of several different periods and moving in different directions pass a given location simultaneously. If we plot the sea surface height as a function of time, we get the type of complicated wave pattern shown in Figure 9-13a. Surprisingly, the pattern in Figure 9-13a is simply the sum of the five simple waves shown in Figure 9-13b.

Figure 9-13. The waveforms observed at any point in the ocean are often complex. The complex waveform in part (a) is simply the result of interference (addition) of the five simple sine waves in part (b).

When wave trains of similar wave heights but slightly different periods arrive at the same time, the sea surface will appear to alternate between periods of low and high waves. During relatively calm periods (the segments where the waveform depicted in Figure 9-14b is almost flat), the crest of one wave arrives at almost the same time as the trough of another wave, and the elevations of the two waves cancel each other out (Fig. 9-14). During the intervals when the waves are high, the crests of the two waves arrive at the same time and combine to form a larger wave (Fig. 9-14). When waves of several different, but similar, periods arrive simultaneously, the situation is more complex. However, the process of wave addition and subtraction, called “wave interference,” is what causes waves to vary in height in irregular patterns.

Figure 9-14. When two sets of waves of very similar wavelengths arrive at the same location at the same time, their addition can produce large, long-period variations of the wave height in the resulting waveform. (a) These two simple waves have slightly different wavelengths. (b) When they are combined, they produce a very different waveform. This is why we may see periodic sets of very much higher (and lower) waves than normal arriving at the shore.

The irregular pattern of waves caused by wave addition is important to surfers, who await sets of big waves. Contrary to some popular beliefs, the periodicity in wave heights does not always follow a consistent pattern. Sets of waves that are especially high may appear every few minutes or only every few hours at any given location on any day. People who visit coastlines when the sea is rough should be aware of this phenomenon. It is not safe to assume that a dry position on the rocks or beach is beyond the reach of the waves. More than a few unfortunate individuals are washed off the rocks or swept off beaches and drown every year because they have made that assumption. It is wise always to anticipate that waves much larger than those that have arrived in the preceding hours might suddenly appear.

Wave Dispersion

Storms create waves with a wide range of periods. The speed of deep-water waves increases as period (and wavelength) increase (Table 9-1). Hence, waves with longer periods and longer wavelengths travel faster than shorter-period waves, and longer-period waves move away from the area where they were generated more rapidly than shorter-period waves do. Wave trains of different periods become separated as they move away from the storm center where they were created (Fig. 9-15). This phenomenon is called “wave dispersion” and causes waves to be sorted by period (and wavelength). Because waves are sorted by period as they move away from their origin, the longer-period waves from a storm arrive at locations far from the storm before the shorter-period waves do. If the storm that created the waves occurred about 100 km away, waves with a 5-s period will arrive about 3 h after waves with a 10-s period. If the storm occurred about 1000 km away, the 5-s waves will arrive about 36 h after the 10-s waves. The waves from the more distant storm will have traveled farther and been dispersed by wavelength to a greater extent (Fig. 9-15).

Figure 9-15. Within the fetch of a storm, the waves are irregular because they are the sum of many separate waves of different wavelengths, as shown in Figure 9-12. Long-wavelength waves move faster than shorter-wavelength waves, such that, as waves move away from the storm, the long-wavelength waves move ahead of the shorter-wavelength waves. This sorting of the waves by wavelengths is called wave dispersion. Because they are separated by wavelength, the waves appear to become smoother as they travel away from the storm fetch. The greater the distance from the fetch, the greater the difference in the arrival times of waves of two different wavelengths.Normally, waves of more than one storm arrive simultaneously, so the shorter-period waves from any given storm are usually obscured by longer-period waves from another storm. The shorter-period waves can be observed if detailed wave records are analyzed mathematically. Oceanographers are able to calculate the locations of storms many thousands of kilometers away by analyzing waves arriving at two or more locations. The wave dispersion at each location reveals the distance of the storm from that location, and the distances of the storm from two widely separated locations will reveal the storm location. For example, storms near Antarctica can be located by wave measurements made in locations such as the United Kingdom and Brazil. Although this method is reliable, satellite observations of cloud patterns are now a much easier way to locate storms.

As waves disperse by wavelength in their direction of travel, wave steepness decreases because the waves also spread laterally, and their energy is distributed over a wider area. The reduction in steepness is accompanied by a change in the wave shape, making it closer to the ideal smooth sine wave shape (Fig. 9-2a). Thus, at some distance from the area where they are generated, the waves are sorted by wavelength and form a swell—smooth undulations without sharp or breaking crests (Fig. 9-15). A swell originates from a sea and is caused by the combination of wave (wavelength) dispersion and lateral dispersion of wave energy (spreading of the wave front like the spreading of a ripple as it moves away from the point of impact of a stone) as the waves travel from their origin.