Tidal Range and Spring-Neap Cycle Calculator

Find the spring, neap and mean tidal range of a coast from its M2 and S2 amplitudes and see the 14.77 day spring-neap cycle on a 30 day tide chart.

🌕 Tidal Range and Spring-Neap Cycle Calculator
M2 amplitude (principal lunar semidiurnal)
m
06
S2 amplitude (principal solar semidiurnal)
m
03
Mean water level above datum (Z0)
m
-210
K1 amplitude (optional, for tide type)
m
O1 amplitude (optional, for tide type)
m
Spring range 2(M2 + S2)
—
Neap range 2|M2 − S2|
—
Mean range 2 M2
—
Spring / neap ratio
—
Spring high water
—
Neap high water
—
Spring low water
—
Neap low water
—
M2 period
—
S2 period
—
Spring-neap beat period
—
Form factor F
—
Tide type
—

🌕 What is the Tidal Range and Spring-Neap Cycle?

Tidal range is the vertical difference between consecutive high water and low water. It is the number that decides how far a harbor floods and drains, how strong the tidal currents become, and how much power a tidal barrage could generate. Tides are produced by the gravitational pull of the moon and sun, and tidal science breaks the water level into a sum of cosine waves called harmonic constituents. The two most important are M2, the principal lunar semidiurnal wave with a period of 12.4206 hours, and S2, the principal solar semidiurnal wave with a period of exactly 12 hours.

Because the two periods differ by about 25 minutes, the waves drift in and out of step. When they are in phase, at new and full moon, the crests add and the range is largest. These are the spring tides, and they have nothing to do with the season. When they are out of phase, near the first and last quarter moon, the S2 trough cancels part of the M2 crest and the range is smallest. These are the neap tides. The pattern repeats every 14.77 days, so a coast sees two springs and two neaps in each lunar month.

This calculator uses the two-constituent model. Give it the M2 and S2 amplitudes for a site and it returns the spring, neap and mean range, the high and low water heights above your datum, the spring-neap ratio and the beat period. If you also enter the diurnal amplitudes K1 and O1, it computes the tidal form factor and tells you whether the tide is semidiurnal, mixed or diurnal. The 30 day chart shows 1441 simulated water levels so you can see the fast twice-daily oscillation under the slow fortnightly envelope.

Typical uses include comparing sites for tidal energy, planning intertidal fieldwork around spring lows, estimating how much extra clearance a quay needs, and teaching how the moon and sun combine. A common misconception is that spring tides occur in spring or that neaps are weak because the moon is far away. In fact springs and neaps depend only on the alignment of the sun, earth and moon. For the timing of that alignment, see the synodic period calculator.

📐 Formula

h(t)  =  Z0 + M2 cos(2πt / 12.4206 h) + S2 cos(2πt / 12 h)
h(t) = water level above datum (m) at time t in hours; Z0 = mean water level above datum (m)
M2, S2 = amplitudes (half ranges) of the lunar and solar semidiurnal constituents (m)
Spring range: Rspring = 2 (M2 + S2)
Neap range: Rneap = 2 |M2 − S2|, and mean range: Rmean = 2 M2
Spring / neap ratio: (M2 + S2) / |M2 − S2|
High water: HWspring = Z0 + M2 + S2, HWneap = Z0 + |M2 − S2|
Beat period: Tsn = 1 / (1/TS2 − 1/TM2) = 1 / (1/12 − 1/12.4206) h = 354.37 h = 14.7653 days
Form factor: F = (K1 + O1) / (M2 + S2): below 0.25 semidiurnal, 0.25 to 3 mixed, above 3 diurnal
Example: M2 = 1.5 m, S2 = 0.5 m gives spring range 4.00 m, neap range 2.00 m, mean range 3.00 m.

Adding two cosines of nearly equal frequency gives a wave at the average frequency whose amplitude swings between M2 + S2 and |M2 − S2|. That slowly changing amplitude is the spring-neap envelope drawn on the chart. The tide type classification follows the Courtier form factor used in standard oceanography texts.

📖 How to Use This Calculator

Steps

1
Enter the M2 and S2 amplitudes. Type the M2 (principal lunar) and S2 (principal solar) amplitudes in meters, taken from a tide table or harmonic constants for your site.
2
Set the mean water level. Enter the mean water level above your chart datum, or leave it at 0 to read heights relative to mean sea level.
3
Add K1 and O1 if you want the tide type. Optionally enter the K1 and O1 diurnal amplitudes to compute the form factor and classify the tide as semidiurnal, mixed or diurnal.
4
Read the ranges and the chart. Compare the spring, neap and mean ranges, the high and low water heights, and the 30 day chart showing the spring-neap envelopes.

💡 Example Calculations

The site amplitudes below are rounded, illustrative values of the right order of magnitude for each region, not official harmonic constants. Use your own tide gauge constants for real work.

Example 1 — Bay of Fundy (Huge Range)

M2 = 5.3 m, S2 = 1.2 m, Z0 = 7 m, K1 = 0.15 m, O1 = 0.1 m

1
Spring range = 2 (5.3 + 1.2) = 13.00 m, neap range = 2 (5.3 − 1.2) = 8.20 m, mean range = 2 × 5.3 = 10.60 m
2
Spring high water = 7 + 6.5 = 13.50 m and spring low water = 7 − 6.5 = 0.50 m; neap high water = 11.10 m and neap low water = 2.90 m
3
Spring / neap ratio = 6.5 / 4.1 = 1.59, and the form factor F = 0.25 / 6.5 = 0.038, a semidiurnal tide
Spring range 13.00 m, neap range 8.20 m, F = 0.038 (semidiurnal)
Try this example →

Example 2 — Gulf of Mexico (Small, Diurnal)

M2 = 0.06 m, S2 = 0.02 m, Z0 = 0 m, K1 = 0.15 m, O1 = 0.14 m

1
Spring range = 2 (0.06 + 0.02) = 0.16 m, neap range = 2 (0.06 − 0.02) = 0.08 m, mean range = 0.12 m
2
Spring / neap ratio = 0.08 / 0.04 = 2.00; spring high water = 0.08 m and spring low water = -0.08 m
3
Form factor F = (0.15 + 0.14) / 0.08 = 3.625, above 3, so the tide is classed as diurnal
Spring range 0.16 m, F = 3.625 (diurnal)
Try this example →

Example 3 — English Channel (Semidiurnal)

M2 = 2.2 m, S2 = 0.7 m, Z0 = 3.7 m, K1 = 0.07 m, O1 = 0.07 m

1
Spring range = 2 (2.2 + 0.7) = 5.80 m, neap range = 2 (2.2 − 0.7) = 3.00 m, mean range = 4.40 m
2
Spring high water = 3.7 + 2.9 = 6.60 m, spring low water = 0.80 m, neap high water = 5.20 m, neap low water = 2.20 m
3
Spring / neap ratio = 2.9 / 1.5 = 1.93, and F = 0.14 / 2.9 = 0.048, strongly semidiurnal
Spring range 5.80 m, neap range 3.00 m, ratio 1.93
Try this example →

Example 4 — Mid-Ocean Island (Mixed)

M2 = 0.17 m, S2 = 0.05 m, Z0 = 0 m, K1 = 0.1 m, O1 = 0.06 m

1
Spring range = 2 (0.17 + 0.05) = 0.44 m, neap range = 2 (0.17 − 0.05) = 0.24 m, mean range = 0.34 m
2
Spring / neap ratio = 0.22 / 0.12 = 1.83; spring high water = 0.22 m, neap high water = 0.12 m
3
Form factor F = 0.16 / 0.22 = 0.727, between 0.25 and 3, so the tide is mixed (mainly semidiurnal)
Spring range 0.44 m, neap range 0.24 m, F = 0.727 (mixed)
Try this example →

❓ Frequently Asked Questions

How do you calculate tidal range from M2 and S2 amplitudes?+
Spring range is twice the sum of the amplitudes, 2 (M2 + S2), and neap range is twice their difference, 2 |M2 - S2|. The mean range is 2 M2. With M2 = 1.5 m and S2 = 0.5 m the spring range is 4.00 m, the neap range 2.00 m and the mean range 3.00 m.
What is the spring-neap cycle?+
The spring-neap cycle is the fortnightly rise and fall of the tidal range. When the moon and sun pull in line (new and full moon) the M2 and S2 waves add and the range is largest. At the quarter moons they partly cancel and the range is smallest. The cycle repeats every 14.77 days.
Why is the spring-neap period 14.77 days?+
It is the beat period of the M2 tide (12.4206 h) and the S2 tide (12 h). The beat period is 1 / (1/12 - 1/12.4206) hours, which is 354.37 hours or 14.7653 days. It is half the 29.53 day synodic month.
What is the difference between M2 and S2?+
M2 is the principal lunar semidiurnal constituent, with a period of 12.4206 hours, driven by the moon. S2 is the principal solar semidiurnal constituent, with a period of exactly 12 hours, driven by the sun. M2 is usually larger, often by a factor of about 2 to 4.
What is the tidal form factor?+
The form factor F = (K1 + O1) / (M2 + S2) compares the diurnal and semidiurnal constituents. It is below 0.25 for semidiurnal tides, 0.25 to 3 for mixed tides and above 3 for diurnal tides. Mixed tides are often split at 1.5 into mainly semidiurnal and mainly diurnal.
Which place has the largest tidal range in the world?+
The Bay of Fundy in Canada has the largest tidal range, with a mean spring range of about 14.5 m at Burntcoat Head in Minas Basin (the Guinness World Records figure) and extreme ranges of about 16 m reported there. The huge range is caused by near resonance of the bay with the M2 tide and by the funnel shape of the basin.
Why are the tides in the Gulf of Mexico so small?+
The Gulf is nearly enclosed and has little response at semidiurnal periods, so M2 and S2 are only a few centimeters to a few decimeters. Diurnal constituents K1 and O1 dominate, giving a form factor above 3 and mainly one tide a day with a small range.
What is the mean tidal range?+
Mean range is the average difference between high and low water over many tides. In the two-constituent model it equals 2 M2, because the S2 contribution averages out over the spring-neap cycle. Official mean range is measured from tide gauge records over a 19 year epoch.
What does the mean water level or datum offset mean?+
It is the height of the mean water level above your chart datum, such as lowest astronomical tide. Adding it to the harmonic waves shifts every high and low water. With a level of 3.7 m and amplitudes of 2.2 m and 0.7 m, spring high water is 6.60 m above the datum.
What happens if M2 equals S2?+
The neap range becomes zero because the lunar and solar waves cancel completely at neaps, and the spring-neap ratio is infinite. Real tides never reach this extreme because the S2 amplitude is smaller than M2 almost everywhere and other constituents remain.
How accurate is a two-constituent tide model?+
It reproduces the fortnightly cycle and the correct order of magnitude of the range, but real tidal predictions use 30 to 100 constituents. The elliptical orbit of the moon (N2), daily inequality (K1, O1) and shallow-water terms (M4, MS4) can shift heights and times by tens of percent.
Why do spring tides not happen at exactly new and full moon?+
In real oceans there is an age of the tide: springs lag the moon phase by one to three days because of friction and the response time of the ocean basin. The model here starts both waves in phase, so it places springs at days 0, 14.77 and 29.53.
Tags: Tidal Range Calculator Spring Neap Cycle Calculator M2 S2 Tidal Constituents Spring Tide Range Neap Tide Range Tidal Form Factor Tide Height Calculator Calculator Formula Free

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How do you calculate tidal range from M2 and S2 amplitudes?

Spring range is twice the sum of the amplitudes, 2 (M2 + S2), and neap range is twice their difference, 2 |M2 - S2|. The mean range is 2 M2. With M2 = 1.5 m and S2 = 0.5 m the spring range is 4.00 m, the neap range 2.00 m and the mean range 3.00 m.

What is the spring-neap cycle?

The spring-neap cycle is the fortnightly rise and fall of the tidal range. When the moon and sun pull in line (new and full moon) the M2 and S2 waves add and the range is largest. At the quarter moons they partly cancel and the range is smallest. The cycle repeats every 14.77 days.

Why is the spring-neap period 14.77 days?

It is the beat period of the M2 tide (12.4206 h) and the S2 tide (12 h). The beat period is 1 / (1/12 - 1/12.4206) hours, which is 354.37 hours or 14.7653 days. It is half the 29.53 day synodic month.

What is the difference between M2 and S2?

M2 is the principal lunar semidiurnal constituent, with a period of 12.4206 hours, driven by the moon. S2 is the principal solar semidiurnal constituent, with a period of exactly 12 hours, driven by the sun. M2 is usually larger, often by a factor of about 2 to 4.

What is the tidal form factor?

The form factor F = (K1 + O1) / (M2 + S2) compares the diurnal and semidiurnal constituents. It is below 0.25 for semidiurnal tides, 0.25 to 3 for mixed tides and above 3 for diurnal tides. Mixed tides are often split at 1.5 into mainly semidiurnal and mainly diurnal.

Which place has the largest tidal range in the world?

The Bay of Fundy in Canada has the largest tidal range, with a mean spring range of about 14.5 m at Burntcoat Head in Minas Basin (the Guinness World Records figure) and extreme ranges of about 16 m reported there. The huge range is caused by near resonance of the bay with the M2 tide and by the funnel shape of the basin.

Why are the tides in the Gulf of Mexico so small?

The Gulf is nearly enclosed and has little response at semidiurnal periods, so M2 and S2 are only a few centimeters to a few decimeters. Diurnal constituents K1 and O1 dominate, giving a form factor above 3 and mainly one tide a day with a small range.

What is the mean tidal range?

Mean range is the average difference between high and low water over many tides. In the two-constituent model it equals 2 M2, because the S2 contribution averages out over the spring-neap cycle. Official mean range is measured from tide gauge records over a 19 year epoch.

What does the mean water level or datum offset mean?

It is the height of the mean water level above your chart datum, such as lowest astronomical tide. Adding it to the harmonic waves shifts every high and low water. With a level of 3.7 m and amplitudes of 2.2 m and 0.7 m, spring high water is 6.60 m above the datum.

What happens if M2 equals S2?

The neap range becomes zero because the lunar and solar waves cancel completely at neaps, and the spring-neap ratio is infinite. Real tides never reach this extreme because the S2 amplitude is smaller than M2 almost everywhere and other constituents remain.

How accurate is a two-constituent tide model?

It reproduces the fortnightly cycle and the correct order of magnitude of the range, but real tidal predictions use 30 to 100 constituents. The elliptical orbit of the moon (N2), daily inequality (K1, O1) and shallow-water terms (M4, MS4) can shift heights and times by tens of percent.

Why do spring tides not happen at exactly new and full moon?

In real oceans there is an age of the tide: springs lag the moon phase by one to three days because of friction and the response time of the ocean basin. The model here starts both waves in phase, so it places springs at days 0, 14.77 and 29.53.

📌 Quick Tips

💡Harmonic amplitudes are half ranges. A tide with M2 = 1.5 m and S2 = 0.5 m has a spring range of 4 m, not 2 m, because the range runs from the crest to the trough.
💡The spring-neap beat period is 14.77 days, not the 29.53 day synodic month. Springs happen twice a month, at new and full moon, because the S2 and M2 waves come into phase twice per lunar cycle.
💡If S2 is larger than M2 the neap range uses the absolute difference |M2 - S2|. In the two-constituent model the phase of the beat flips, but the envelope size is the same.
💡Add K1 and O1 amplitudes for your site to classify the tide. A form factor above 3 means one high water per day, as in much of the Gulf of Mexico, so the semidiurnal numbers describe only a small part of the signal.
💡Real ranges also depend on the lunar perigee (N2) and on shallow-water effects (M4). Treat this two-constituent result as a clean first estimate and use official tide tables for navigation.