Tidal window for the barge: model versus table
A sand barge must cross the shoal at the mouth of a river. Her draught is 1.9 m and the harbourmaster's standing order requires 0.6 m of water beneath the keel, so she needs a depth of at least 2.5 m for the whole 40-minute transit. Depths are soundings above chart datum, taken hourly:
| Time | 08:00 | 09:00 | 10:00 | 11:00 | 12:00 | 13:00 | 14:00 |
|---|---|---|---|---|---|---|---|
| Depth (m) | 2.8 | 1.9 | 1.4 | 1.3 | 1.7 | 2.6 | 3.5 |
High water was 5.8 m at 04:00 and the low is near 11:00. A student fits the tide with
d = 3.6 + 2.2 cos(π(t − 4)/6.2), where d is depth in metres and t is hours after midnight.
The barge can arrive at the bar no earlier than 11:30 and must be alongside by 17:00. The master asks the student to compute the crossing time from the equation rather than the table, "because the equation covers every minute".
1. Using the student's model, find the first time after 11:00 at which the modelled depth reaches 2.5 m.
Read the worked answer
Set d = 2.5:
2.5 = 3.6 + 2.2 cos(π(t − 4)/6.2) cos(π(t − 4)/6.2) = (2.5 − 3.6)/2.2 = −0.5 π(t − 4)/6.2 = 2π/3 (the first solution after the 11:00 low, where the cosine is rising) t − 4 = (2/3)(6.2) = 4.133 t = 8.13 h … and adding the full period 12.4 h gives t = 20.53 h.
Neither is right for the afternoon crossing, so take the second positive solution on the rising tide: π(t − 4)/6.2 = 4π/3, giving t − 4 = 8.267 and t = 12.27 h ≈ 12:16.
The structural point: the model's minimum sits at t = 4 + 6.2 = 10.2 h (10:12), so the equation puts low water 48 minutes before the table does. The question is not whether the algebra is right - it is - but whether the fitted curve is entitled to be trusted in this interval.
2. Compare the model with the observations between 10:00 and 13:00 and decide which basis the master should use. Show the residuals.
Read the worked answer
Residuals (model minus observed):
| t | model d | observed d | residual |
|---|---|---|---|
| 10:00 | 1.41 | 1.4 | +0.01 |
| 11:00 | 1.58 | 1.3 | +0.28 |
| 12:00 | 2.26 | 1.7 | +0.56 |
| 13:00 | 3.27 | 2.6 | +0.67 |
The fit is excellent near the low-water turning point but increasingly optimistic on the rising tide - exactly the window the barge needs. The reason is structural rather than careless fitting: a cosine is symmetric about its minimum, whereas this tide stands low for two hours (1.4, 1.3, 1.7) and then rises steeply. No choice of amplitude, phase or mean can reproduce that asymmetry, so the error is not random scatter that more data would average out - it is a bias that grows with distance from the fitted points.
A 0.56 m over-prediction at 12:00 is nearly a quarter of the 2.5 m requirement, and it errs in the unsafe direction. The master should interpolate between the hourly soundings, which measure the thing being asked about, and treat the sinusoid as a description of the tide's overall rhythm, not a navigation tool.
3. Recommend a crossing time, with a stated margin, and identify the assumptions the recommendation depends on.
Read the worked answer
Linear interpolation between 12:00 (1.7 m) and 13:00 (2.6 m): the depth reaches 2.5 m after (2.5 − 1.7)/0.9 = 0.889 of an hour, i.e. about 12:53. Since the tide is rising, that is the earliest safe entry for a 40-minute transit - the depth only increases afterwards.
A responsible recommendation adds margin rather than quoting 12:53. Soundings are usually given to ±0.1 m and siltation shifts the shoal between surveys; requiring 2.7 m instead of 2.5 m gives (2.7 − 1.7)/0.9 = 1.11 h, so enter no earlier than 13:06. That still leaves ample time before 17:00, so the cost of prudence is low and the cost of being wrong is a grounding.
The assumptions: the tide rises monotonically after 11:00 (supported by the table, not by the model); the bar depth does not change between soundings; the 40-minute figure is realistic in current conditions; and the barge's draught as loaded, not as documented, is 1.9 m. That last one is worth a direct check with the master - it is the assumption most likely to be wrong and least likely to be questioned.