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  • Primer
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Wave analysis tools

Nature Reviews Methods Primers volume 5, Article number: 21 (2025) Cite this article

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Abstract

This Primer provides an overview of a fundamental set of analysis methods for studying waves, vibrations and related oscillatory phenomena — including instabilities, turbulence and shocks — across diverse scientific fields. These phenomena are ubiquitous, from astrophysics to complex systems in terrestrial environments, and understanding them requires careful selection of techniques. Misapplication of analysis tools can introduce misleading results. In this Primer, the fundamental principles of various wave analysis methods are first reviewed, along with adaptations to address complexities such as nonlinear, non-stationary and transient signal behaviour. These techniques are applied to identical synthetic datasets to provide a quantitative comparison of their strengths and limitations. Details are provided to help select the most appropriate analysis tools based on specific data characteristics and scientific goals, promoting reliable interpretations and ensuring reproducibility. Additionally, the Primer highlights best ethical practices for data deposition and the importance of open-code sharing. Finally, the broad applications of these techniques are explored in various research fields, current challenges in wave analysis are discussed, and an outlook on future directions is provided, with an emphasis on potential transformative discoveries that could be made by optimizing and developing cutting-edge analysis methods.

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Fig. 1: Unveiling wave phenomena with a visual guide to wave analysis.
Fig. 2: Synthetic datasets.
Fig. 3: Performance of diverse analysis methods on the intricate synthetic 1D time series.
Fig. 4: Dominant frequency maps and mean power spectra.
Fig. 5: Comparison of k − ω filtering and POD analysis.
Fig. 6: Cross-correlation analysis of two synthetic 1D time series using FFT and wavelet techniques.

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Code availability

The synthetic datasets and codes used for the wave analyses presented in this Primer, including the generation of all displayed figures, are publicly available via the WaLSAtools repository on GitHub (https://github.com/WaLSAteam/WaLSAtools), archived at Zenodo (https://doi.org/10.5281/zenodo.14978610). Further information and documentation are available online at https://WaLSA.tools.

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Acknowledgements

The authors acknowledge the scientific discussions with the Waves in the Lower Solar Atmosphere (WaLSA) team, which has been supported by the Research Council of Norway (project number 262622), The Royal Society (award number Hooke18b/SCTM)284, and the International Space Science Institute (ISSI Team 502). The authors also thank H. Schunker for her insightful comments and suggestions. S.J. gratefully acknowledges support from the Niels Bohr International Academy, at the Niels Bohr Institute, University of Copenhagen, Denmark, and the Rosseland Centre for Solar Physics, University of Oslo, Norway. D.B.J. and T.J.D. acknowledge support from the Leverhulme Trust via the Research Project Grant RPG-2019-371. D.B.J., S.J. and S.D.T.G. thank the UK Science and Technology Facilities Council (STFC) for the consolidated grants ST/T00021X/1 and ST/X000923/1. D.B.J. and S.D.T.G. also acknowledge funding from the UK Space Agency via the National Space Technology Programme (grant SSc-009). S.B. acknowledges the support from the STFC Grant ST/X000915/1. V.F., G. Verth and S.S.A.S. acknowledge the support from STFC (ST/V000977/1 and ST/Y001532/1) and the Institute for Space-Earth Environmental Research (ISEE, International Joint Research Program, Nagoya University, Japan). V.F. and G. Verth thank the Royal Society International Exchanges Scheme, in collaboration with Monash University, Australia (IES/R3/213012), Instituto de Astrofisica de Canarias, Spain (IES/R2/212183), National Observatory of Athens, Greece (IES/R1/221095), and the Indian Institute of Astrophysics, India (IES/R1/211123), for the support provided. R.G. acknowledges the support from Fundação para a Ciência e a Tecnologia through the research grants UIDB/04434/2020 and UIDP/04434/2020. E.K. acknowledges the support from the European Research Council through the Consolidator Grant ERC-2017-CoG-771310-PI2FA and from the Agencia Estatal de Investigación del Ministerio de Ciencia e Innovación through grant PID2021-127487NB-I00. R.J.M. acknowledges the support from the UK Research and Innovation Future Leader Fellowship (RiPSAW-MR/T019891/1). L.A.C.A.S. acknowledges the support from the STFC Grant ST/X001008/1. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 101097844 — project WINSUN). N.Y. acknowledges the support from the DST INSPIRE Faculty Grant (IF21-PH 268) and the Science and Engineering Research Board – Mathematical Research Impact Centric Support grant (MTR/2023/001332).

Author information

Authors and Affiliations

  1. Max Planck Institute for Solar System Research, Göttingen, Germany

    Shahin Jafarzadeh, Daniele Calchetti & Sami K. Solanki

  2. Astrophysics Research Centre, Queen’s University Belfast, Belfast, UK

    Shahin Jafarzadeh, David B. Jess, Samuel D. T. Grant, Peter H. Keys & Timothy J. Duckenfield

  3. Department of Physics and Astronomy, California State University Northridge, Los Angeles, CA, USA

    David B. Jess

  4. Italian Space Agency (ASI), Rome, Italy

    Marco Stangalini

  5. Department of Geography and Planning, School of Environmental Sciences, University of Liverpool, Liverpool, UK

    Jonathan E. Higham

  6. Niels Bohr International Academy, Niels Bohr Institute, Copenhagen, Denmark

    Martin E. Pessah

  7. Centre for Fusion, Space and Astrophysics, University of Warwick, Coventry, UK

    Sergey Belov

  8. Plasma Dynamics Group, School of Electrical and Electronic Engineering, The University of Sheffield, Sheffield, UK

    Viktor Fedun & Suzana S. A. Silva

  9. Operations Department, ESA Directorate of Science, Greenbelt, MD, USA

    Bernhard Fleck

  10. Geophysical and Astronomical Observatory, Faculty of Science and Technology, University of Coimbra, Coimbra, Portugal

    Ricardo Gafeira

  11. Instituto de Astrofísica e Ciências do Espaço, University of Coimbra, Coimbra, Portugal

    Ricardo Gafeira

  12. Department of Physics, University of Coimbra, Coimbra, Portugal

    Ricardo Gafeira

  13. Department of Physics and Astronomy, Georgia State University, Atlanta, GA, USA

    Stuart M. Jefferies

  14. Institute for Astronomy, University of Hawaii, Honolulu, HI, USA

    Stuart M. Jefferies

  15. Instituto de Astrofísica de Canarias, La Laguna, Spain

    Elena Khomenko

  16. Departamento de Astrofísica, Universidad de La Laguna, Tenerife, Spain

    Elena Khomenko

  17. Department of Mathematics, Physics and Electrical Engineering, Northumbria University, Newcastle upon Tyne, UK

    Richard J. Morton, Luiz A. C. A. Schiavo & Rahul Sharma

  18. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA, USA

    Aimee A. Norton

  19. Indian Institute of Astrophysics, Bangalore, India

    S. P. Rajaguru

  20. Institute for Solar Physics (KIS), Freiburg, Germany

    Oskar Steiner & Gangadharan Vigeesh

  21. Istituto ricerche solari Aldo e Cele Daccò (IRSOL), Locarno, Switzerland

    Oskar Steiner

  22. Plasma Dynamics Group, School of Mathematical and Physical Sciences, The University of Sheffield, Sheffield, UK

    Gary Verth

  23. Department of Physics, Indian Institute of Science Education and Research Thiruvananthapuram, Vithura, India

    Nitin Yadav

Authors
  1. Shahin Jafarzadeh
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  2. David B. Jess
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  3. Marco Stangalini
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  4. Samuel D. T. Grant
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  5. Jonathan E. Higham
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  6. Martin E. Pessah
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  7. Peter H. Keys
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  8. Sergey Belov
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  9. Daniele Calchetti
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  10. Timothy J. Duckenfield
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  11. Viktor Fedun
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  12. Bernhard Fleck
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  13. Ricardo Gafeira
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  14. Stuart M. Jefferies
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  15. Elena Khomenko
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  16. Richard J. Morton
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  17. Aimee A. Norton
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  18. S. P. Rajaguru
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  19. Luiz A. C. A. Schiavo
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  20. Rahul Sharma
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  21. Suzana S. A. Silva
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  22. Sami K. Solanki
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  23. Oskar Steiner
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  24. Gary Verth
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  25. Gangadharan Vigeesh
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  26. Nitin Yadav
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Contributions

Introduction (S.J., D.B.J., M.S., S.D.T.G., M.E.P., P.H.K., R.J.M., S.K.S. and O.S.); Experimentation (S.J., D.B.J., M.S., S.D.T.G., J.E.H., M.E.P., P.H.K., S.B., V.F., B.F., R.G., S.M.J., E.K., R.J.M., A.A.N., S.P.R., L.A.C.A.S., R.S., S.S.A.S., S.K.S., O.S., G. Verth, G. Vigeesh and N.Y.); Results (S.J., D.B.J., M.S., S.D.T.G., J.E.H., P.H.K., D.C., T.J.D., V.F., B.F., R.G., S.M.J., R.J.M., A.A.N., S.P.R., L.A.C.A.S., R.S., S.S.A.S., S.K.S., O.S., G. Vigeesh and N.Y.); Applications (S.J., D.B.J., M.S., S.D.T.G., M.E.P., V.F. and R.J.M.); Reproducibility and data deposition (S.J., D.B.J., M.S., S.D.T.G., D.C., T.J.D., R.G., G. Vigeesh and N.Y.); Limitations and optimizations (S.J., D.B.J., M.S., S.D.T.G., P.H.K., S.K.S. and O.S.); Outlook (S.J., D.B.J., M.S., S.D.T.G., S.K.S. and O.S.); overview of the Primer (all authors).

Corresponding author

Correspondence to Shahin Jafarzadeh.

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Competing interests

The majority of the authors of this Primer are members of the Waves in the Lower Solar Atmosphere (WaLSA) team, which developed and maintains the WaLSAtools repository, extensively used for all the analyses in the Primer.

Peer review

Peer review information

Nature Reviews Methods Primers thanks Maciej Trusiak, Vincenzo Pierro and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Related links

WaLSA team: https://WaLSA.team

WaLSAtools documentation: https://WaLSA.tools

WaLSAtools repository: https://github.com/WaLSAteam/WaLSAtools

Supplementary information

Supplementary Information (download PDF )

43586_2025_392_MOESM2_ESM.mp4 (download MP4 )

Supplementary Video 1 Synthetic spatio-temporal datacube analysed in the Primer. The datacube features 50 concentric circular regions, each containing ten sinusoidal waves with distinct frequencies, amplitudes, and phases. To enhance complexity, additional features are superimposed, including a transient polynomial signal, transverse motion, a fluting-like instability, and a quasi-periodic signal. These diverse features, along with added noise, create a rich dataset for analysing wave parameters and their spatio-temporal distributions.

43586_2025_392_MOESM3_ESM.mp4 (download MP4 )

Supplementary Video 2 Frequency-filtered spatial POD modes reconstructed for the ten dominant frequencies. The spatial patterns (modes) associated with each frequency were first obtained through POD analysis of the synthetic spatio-temporal datacube. Then, the ten identified dominant frequencies were fitted to the temporal coefficients of the first 22 modes (capturing 99% of the total variance), using a non-linear least-squares fit to imposed sinusoids. The modes are shown for each time step of the series, with each image covering 130 × 130 pixels.

43586_2025_392_MOESM4_ESM.mp4 (download MP4 )

Supplementary Video 3 Combined Welch power spectrum with various numbers of POD modes included. The vertical dotted lines mark the ten base frequencies used to construct the synthetic data; the red dashed line identifies the 95% confidence level (estimated from 1000 bootstrap resamples). The ten strongest peaks of this combined spectrum align with the base frequencies used to construct the synthetic spatio-temporal datacube, while additional weaker peaks (lying below the 95% confidence level) are likely due to spurious signals and noise. Importantly, these ten strongest peaks remain consistent as the number of modes included in the combined power spectrum changes, demonstrating the robustness of POD in identifying dominant frequencies.

43586_2025_392_MOESM5_ESM.mp4 (download MP4 )

Supplementary Video 4 Spatial modes (130 × 130 pixels each), temporal coefficients, and Welch power spectra of the 200 SPOD modes. The SPOD analysis was performed on the synthetic spatio-temporal datacube with a Gaussian filter kernel. This illustrates the frequency pairing phenomenon, where each frequency is associated with two distinct spatial modes with corresponding temporal coefficients. The first 20 SPOD modes represent the ten base frequencies used to construct the synthetic signal, effectively capturing the primary features of the data. Their power spectra indicate significant energy content at these frequencies. Beyond the 20th mode, the power drops significantly, and the modes exhibit other frequencies, likely due to noise and spurious signals.

Glossary

Dispersion

The frequency-dependent separation of signals into wave components owing to differing propagation speeds, such as white light refracting in a prism.

Eigenvalue

A measure of the variance or energy associated with a specific eigenvector (spatial pattern of oscillation).

Eigenvector

In wave analysis, an eigenvector represents a spatial pattern of oscillation that retains its shape whereas its amplitude might change over time.

Frequency

The number of oscillatory cycles in a given time, typically the amount per second measured in Hertz (Hz).

Frequency resolution

The smallest difference in frequency that can be detected. It is primarily determined by the length of the time series.

Harmonic

Characteristic frequencies that an oscillation will prefer to exhibit. These are integer multiples of the natural, fundamental frequency of the oscillation.

Magnetohydrodynamic

(MHD). Study of the interactions between a magnetic field and a fluid (gas or plasma).

Nonlinear signals

Signals that deviate from linear behaviour, often exhibiting distortions, harmonics or abrupt changes, such as shocks.

Non-stationary signal

A time series whose statistical properties, such as the mean or variance, evolve over time.

Nyquist frequency

The highest frequency that can be accurately represented in a discrete signal, equal to half the sampling frequency.

Optically thick

When a medium readily absorbs light, only the surface layer is typically observable.

Oscillations

Periodic variations of a measurement, typically around an equilibrium point, which may be fixed or dynamic.

Phase

A time-dependent variable that describes the position of a point within a wave cycle at a given time. Typically expressed as an angle, representing a fraction of the wave cycle.

Phase lag

The angular difference between corresponding points on two waves of the same frequency, representing the delay of one wave relative to the other.

Quasi-periodic

An oscillation with a changing amplitude and/or frequency over time. They are often owing to an inconsistent driver and, thus, can appear and vanish with time.

Sampling frequency

The number of samples taken per unit of time from a continuous signal to create a discrete representation.

Shocks

Waves propagating above the local sound speed, causing abrupt changes in pressure, density and temperature within the medium, such as a sonic boom.

Spectral analysis

The study of frequency characteristics within data, including power, phase and coherence across a range of frequencies.

Spectral leakage

Artefact in spectral power caused by finite data length, non-integer wave cycles, or windowing effects, spreading power into adjacent frequencies.

Spectral power

The distribution of detected wave power in data as a function of frequency.

Transverse motion

Displacement perpendicular to a reference direction. In waves, it refers to oscillations that are perpendicular to the direction of wave propagation.

Wavelength

The distance covered by one full cycle of a wave.

Wavenumber

The number of wavelengths per unit of distance. The angular variant describes the number of radians per unit of distance.

Wave power

A measure of the energy flow associated with waves. It is proportional to the square of the wave amplitude.

Waves

Coherent collections of oscillations that propagate through space over time.

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Jafarzadeh, S., Jess, D.B., Stangalini, M. et al. Wave analysis tools. Nat Rev Methods Primers 5, 21 (2025). https://doi.org/10.1038/s43586-025-00392-0

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