Sea surface waves are important for marine safety and coastal engineering, but mapping the wave properties at complex shorelines, such as coastal archipelagos, is challenging. The wave spectrum,

Since the 1950s the wave spectrum has been the central way to define the properties of random sea-surface wind waves

In the coastal region, waves are important for coastal engineering, erosion, small-vessel safety, and biological processes. Coastal waves deviate from deep-water open-sea waves, but their exact properties depend on the shoreline structure. On sloping beaches the limitation by the water depth is a major factor shaping the wave properties through bottom friction, depth-induced wave breaking, and shallow-water non-linear wave interactions

One of the most complex nearshore conditions can be found in coastal archipelagos where islands, the irregular shoreline, the slanting fetch, and the decreasing depth affect the attenuation and local growth of the waves. Collections of large islands, in the scale of kilometres, can be found in, for example, the Gulf of Mexico, outside of Louisiana, or between Vancouver and Seattle (the San Juan Islands). In Europe an example is the Aegean Sea, which separates parts of Turkey and Greece from the Mediterranean Sea. Denser archipelagos, where the island sizes are of the order of hundreds of metres, are even more complex. An archipelago made up of a large number of small islands has a strong effect on the wave field by attenuating the waves and diffracting the remaining wave energy behind them. At the same time groups of islands practically define new fetches for local wave systems to grow from, thus giving birth to unique wave conditions. Examples of such archipelagos are the Thousand Islands at the US–Canadian border or the coastline of Maine. In Europe dense coastal archipelagos can be found especially in the Baltic Sea, with examples being the Stockholm archipelago and the Archipelago Sea. Also the coastline near the Finnish capital, Helsinki, has a characteristic archipelago with heavy commercial and recreational marine traffic.

Although coastal archipelagos are usual – almost typical – in the Baltic Sea, there is a limited amount of data available on their effect on waves.

The shape of the wave spectrum is of pure theoretical interest. Still, it also has direct and indirect consequences for practical applications, such as for estimating the expected height of single waves. An atypical spectral shape also affects the applicability and reliability of engineering formulas using integrated wave parameters and alters the sampling variability in standard wave buoy measurements. The spectral shape can be atypical at any location because of swell or decaying and turning winds. Nonetheless, in the archipelago an atypical spectral shape forms even under steady wave conditions, thus giving the wave field inside the archipelago its prevailing properties. These properties need to be identified and quantified in order to fundamentally understand waves in archipelagos.

This paper aims to fill the knowledge gap regarding the properties of the wave field inside dense coastal archipelagos. The study relied on spatially extensive wave buoy measurements; all data were collected in the Helsinki archipelago, which is located in the Gulf of Finland, the Baltic Sea. The data and methods are introduced in Sect.

We conducted wave measurements at 14 locations off Helsinki in the Gulf of Finland (GoF). An overview of the sites can be found in Fig.

The nearshore measurements with the G4 buoys – conducted as part of a commissioned work by the city of Helsinki – were made for about a month at each location between 2012 and 2014 (Table

Both the G4 and the Mk-III use a sampling frequency of 1.28

For the G4 buoys, the low-frequency artefacts, which have later been identified to be caused by the filter response to a missing GPS signal, were corrected following

The bathymetry and the measurement locations. GoF (green) is the open-sea wave buoy. The stations are divided into groups: outer archipelago (O1–O3), transition zone (T1–T3), inner archipelago (I1–I3), and sheltered archipelago (S1–S4). The plus (+) marks the Harmaja weather station. The Kalbådagrund weather station is outside of the map. The black box marks the area of Fig.

A more detailed map of the measurement locations close to the Helsinki shoreline. This area is marked with a black box in Fig.

We used wind measurements from two of the Finnish Meteorological Institute's operational automatic weather stations. Harmaja (measuring height 17.5

The weather stations provided the wind speed and direction averaged over 10 min. We calculated the 30 min averages from these data to coincide with the time series used to compute a single wave spectrum. The mean wind speed at Harmaja and Kalbådagrund for the years 2016–2018 were 6.1 and 7.8

The measurement time and depth at the different sites (see Fig.

The wave buoys calculated the wave spectrum

All integrals were evaluated up to 0.58

A spectral version of the zero down-crossing period was defined as

while the mean frequency was given by

We defined the peak frequency as

i.e. the frequency where the wave spectrum has its maximum value. Because of the discrete frequency intervals and statistical variations in the spectrum, several methods for calculating the peak frequency have been proposed. In this paper we calculated

We also determined wave parameters directly from the 30 min vertical displacement time series,

The significant wave height,

The traditional definition of the significant wave height is the mean height of the highest one-third of the individual waves in the time series. To distinguish it from the significant wave height calculated using the variance, we will denote this parameter

The zero down-crossing period,

Assuming that

Several parameters to quantify the spectral width have been proposed. The width parameter

The Goda peakedness parameter was needed in the definition of the Benjamin–Feir index

Data were available from more locations than the 14 presented in this paper

As a loose definition of well defined wave conditions we used the 80th percentile of the significant wave height as a cut-off for each location. The 80th percentiles for the 13 coastal locations were determined using all available data. For the GoF, only data from the years 2016–2018 were used to keep the measurement period comparable with the Suomenlinna (T2) observations. In addition we used a cut-off of

The choice of the 80th percentile was a compromise between (i) removing the smallest wave heights, e.g.

Because the short waves are generated by the shortest fetch, they are least affected by the varying spectral shape inside the archipelago. The chosen spectra were therefore normalised using the values at the high frequencies (

The frequencies were then normalised with respect to the mean frequency and the spectra,

The 13 measurement stations in the archipelago were divided into four groups based on a visual estimation of the geographical conditions. The attenuation coefficients for the significant wave height compared to the GoF wave buoy were used as a crude check to ensure that the visual determination of the amount of sheltering was reasonable. The attenuation coefficients,

The common denominator throughout the archipelago is that the waves travel slower than the wind. Thus, the longer waves propagating from the open sea are not swell. In this paper we will show that the sheltering effect of the archipelago is a continuum and several reasonable classifications could therefore be made. The main purpose of the classification was to make the results more presentable.

The main result of this section is that the wave spectrum transitioned from a, more traditional, single-peaked spectrum to a flat spectrum inside the archipelago. The transition was continuous, as readily seen in Fig.

When moving closer towards the coast the spectral shape started to flatten out in the transition zone (blue). Länsikari (T1) and Suomenlinna (T2) are located very close to each other (Fig.

In the inner archipelago (red) the spectral shape had collapsed around the peak and exhibited a constant energy in a broad frequency interval (

In the sheltered archipelago (magenta) even these attenuated low-frequency peaks were no longer visible. The mean spectrum at Koivusaari (S1) was still flat (in a similar fashion to sites I1–I3), but for the other sheltered locations the spectra almost transitioned back to a single-peaked shape – the local fetch was starting to dominate over the very attenuated longer waves. The tail of the spectrum was not determined reliably, since these short waves were often not captured by the wave buoy.

The mean wave spectra divided into the open sea and four archipelago areas: outer archipelago (O1–O3), transition zone (T1–T3), inner archipelago (I1–I3), and sheltered archipelago (S1–S4). An overview of the locations can be found in Fig.

We quantified the change in width (or more exactly, narrowness) of the spectrum using the

The spectral width in the transition zone was in between those of the outer and inner archipelago (

Quantifying the spectral width is no trivial matter, but the good agreement between the

Although the mean spectral profiles were shown to change when moving through the archipelago towards the shore, the spectral shape also varied with the wind direction because of the anisotropic fetch conditions. We used the wind direction because the instability of the spectral peak at Suomenlinna (T2) made it hard to define the dominant wave direction. The wave direction at the GoF buoy, again, collapses to be aligned with the gulf, thus causing a misalignment of up to 50

The most peaked spectra at Suomenlinna (T2) were generated by southerly winds (Fig.

The mean wave spectra at Suomenlinna (T2) as a function of the wind direction.

This section presents some implications of the results of Sect.

When appropriate, the results make use of all available data. Nonetheless, especially Sect.

The confidence limits of observed wave spectra follow a

The final degrees of freedom (d.o.f.) of the integral of a measured spectrum depend on the shape of the spectrum in the following way

It immediately follows that d.o.f.

In Sect.

The increase in the d.o.f. in the archipelago had a direct implication for the confidence limits of the significant wave height: the confidence limits at the GoF wave buoy were 50 % larger than at the inner archipelago points (Table

Because the spectral shape depended strongly on the wind direction at Suomenlinna (T2), the confidence limits for easterly winds were close to those of the inner archipelago, while south-westerly – and especially southerly – winds resulted in confidence limits in line with the open sea (Fig.

The correlation between the d.o.f. and the spectral width parameter

The degrees of freedom of the Suomenlinna (T2) wave variance as a function of the wind direction

Mean values of the spectral narrowness parameter (

The significant wave height is the most central and widely used wave parameter. Still, it can be defined in a couple of different ways. The connection between the definition using the mean height of the highest one-third of the single waves and the definition based on the variance of the vertical displacement was made based on the assumption of a narrowband spectrum, deep water, and that the height of single waves are Rayleigh distributed with the parameter

Studies have shown, however, that the assumption of a Rayleigh distribution (with a parameter

where

We determined the fit between

Even for the southerly waves at Suomenlinna (T2)

Comparison of

The different wave height and crest height parameters at the Gulf of Finland (GoF) and Suomenlinna (T2). The values have been determined using a linear fit through the origin of the P80 data set. The ratio

The highest expected single wave is often of considerable interest, and usually this single wave is given relative to the significant wave height. The estimate is made based on the assumption that the height of the single waves are either Rayleigh or Weibull distributed. The estimated highest single wave thus depends on the assumed distribution and the number of waves encountered during the measurement period (

We determined the highest single wave from the vertical displacement time series of the P80 data sets. For the GoF, the connection between the single wave height and the significant wave height was determined to be

The linear regression to the Suomenlinna (T2) data was

The maximum crest height,

None of the aforementioned dimensionless wave or crest heights had any correlation with the dimensionless depth,

The maximum single wave measured at Suomenlinna (T2) was

The wave height of the single highest wave (

As for the significant wave height, the zero-crossing period,

We compared these two definitions of the zero-crossing period using a linear fit to the P80 data sets. For the GoF data, the two definitions agreed well, with a linear fit giving a proportionality coefficient of 1.02. For the Suomenlinna (T2) data, the linear fit was

The ratio

Often the full spectrum is not available, and the characteristics of the wave field are described using a limited set of integrated parameters. If directionality is ignored, the choice is usually a measure for the height and a measure for the length, or equally well the frequency. A unimodal spectrum, for example, is quite well described by the significant wave height and the peak frequency. Nevertheless, the flat spectral shape in the archipelago leads to a low stability of the peak frequency. The mean frequency, again, is stable but biased compared to the peak frequency for the more unimodal spectra in the outer archipelago.

We determined this error function for each location separately using the P80 data set. The minimum was achieved between

In addition to a best estimate of

In the GoF data roughly 65 % of the energy was below the mean frequency regardless to the wind direction (

The characteristic wave frequency

For Suomenlinna (T2),

Choosing

The mean values and the scatter (standard deviation) of the characteristic wave frequency

The spectral shape affected the relation between the different definitions of the significant wave height (

The reduction of the single highest wave in a wider spectrum has been explained by the de-correlation of the following crests and troughs: a deep trough is less likely to be followed by a high wave crest, even if the maximum and minimum water levels are not affected

The d.o.f. of the wave variance (

Section

The overall results of this study showed that general archipelago conditions need to be quantified using at least three parameters. If the total energy is known, a frequency will give – in some sense – the location where the energy of the wave spectrum is concentrated. The spectral width, again, quantifies how narrowly the energy is distributed around this frequency. Traditionally the variation in spectral width has been relatively small, but in an archipelago setting it is dominant. Clearly, advancing our knowledge of waves in archipelagos hinges on the development of an analytical parameterisation for the archipelago type spectrum. Both

This study was done using the most extensive wave data set that is available from dense archipelago areas. Still, since the material was not primarily collected for fundamental research purposes, it has a few limitations. The first limitation is the sheltered archipelago locations (S1–S4), where the standard 40

The second limitation is the short duration of the measurements in 12 of the 14 locations (Table

An extensive field measurement campaign consisting of wave buoy measurements from 14 locations was performed in the Helsinki archipelago during 2012–2018. Multi-year time series were available from two operational wave buoys in the middle of the Gulf of Finland (GoF) and in the middle of the archipelago (Suomenlinna, T2). Measurements from the other sites in the archipelago lasted for about a month. These measurements were used to study the shape of the wave spectrum in the archipelago and the consequences that the variations in the spectrum have for derived wave parameters.

The mean spectral shape in the middle of the GoF was unimodal with a distinct peak. No peak was identifiable close to the shoreline, where the spectrum was best described by a wide energy-carrying range with almost constant variance density. At Suomenlinna (T2), located in between these two extremes, the spectral shape varied strongly with the wind direction because of the anisotropic fetch conditions. For south-westerly, and especially southerly, winds, the spectral shape was peaked. For easterly winds, the spectral shape was wide, being close to that of the sites near the shore. The wide spectral shape in the archipelago was not created by swell, since even the longer waves travelled slower than the wind. Rather, the spectra reflected complex wind sea conditions where waves grow from different fetches and are attenuated by the islands.

The mean shape of the spectra was well quantified by the spectral narrowness parameter (

The spectral shape affected the ratio

The highest single wave

The traditional peak frequency,

The data are available through the following DOI:

JVB initiated the study based on previous conceptualisations of KK and HP. KK and JVB took part in designing the field measurements. JVB produced the methodology and performed the analysis. JVB did the visualisation. JVB wrote the article with contributions from HP and KK. HP supervised the study.

The authors declare that they have no conflict of interest.

We want to acknowledge the work done by the technical staff at FMI, namely Tuomo Roine, Heini Jalli, and Riikka Hietala. The efforts of Hannu Jokinen in processing the raw wave buoy data are also gratefully acknowledged. Most of the wave buoy observations in this study have been collected through work commissioned by the city of Helsinki. The data is used in this paper with their kind permission. We thank the anonymous reviewers for their constructive critique and comments. They helped us improve our article.

This research has been supported by the University of Helsinki and the Svenska Kulturfonden (grant no. 17/103386).

This paper was edited by Judith Wolf and reviewed by two anonymous referees.