{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:53:42Z","timestamp":1760241222991,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,12,23]],"date-time":"2019-12-23T00:00:00Z","timestamp":1577059200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Agency for Research (Agencia Estatal de Investigaci\u00f3n, AEI), Ministry of Science, Innovation, and Universities of Spain (Ministerio de Ciencia, Innovaci\u00f3n y Universidades), and the European Regional Development Fund (FEDER)","award":["DPI2017-84758-P"],"award-info":[{"award-number":["DPI2017-84758-P"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Techniques based on ultrasound in nondestructive testing and medical imaging analyze the response of the source frequencies (linear theory) or the second-order frequencies such as higher harmonics, difference and sum frequencies (nonlinear theory). The low attenuation and high directivity of the difference-frequency component generated nonlinearly by parametric arrays are useful. Higher harmonics created directly from a single-frequency source and the sum-frequency component generated nonlinearly by parametric arrays are attractive because of their high spatial resolution and accuracy. The nonlinear response of bubbly liquids can be strong even at relatively low acoustic pressure amplitudes. Thus, these nonlinear frequencies can be generated easily in these media. Since the experimental study of such nonlinear waves in stable bubbly liquids is a very difficult task, in this work we use a numerical model developed previously to describe the nonlinear propagation of ultrasound interacting with nonlinearly oscillating bubbles in a liquid. This numerical model solves a differential system coupling a Rayleigh\u2013Plesset equation and the wave equation. This paper performs an analysis of the generation of the sum-frequency component by nonlinear mixing of two signals of lower frequencies. It shows that the amplitude of this component can be maximized by taking into account the nonlinear resonance of the system. This effect is due to the softening of the medium when pressure amplitudes rise.<\/jats:p>","DOI":"10.3390\/s20010113","type":"journal-article","created":{"date-parts":[[2019,12,24]],"date-time":"2019-12-24T05:56:15Z","timestamp":1577166975000},"page":"113","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Nonlinear Maximization of the Sum-Frequency Component from Two Ultrasonic Signals in a Bubbly Liquid"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1935-9681","authenticated-orcid":false,"given":"Mar\u00eda Teresa","family":"Tejedor Sastre","sequence":"first","affiliation":[{"name":"NANLA, Departamento de Matem\u00e1tica Aplicada, Ciencia e Ingenier\u00eda de los Materiales y Tecnolog\u00eda Electr\u00f3nica, Universidad Rey Juan Carlos, Tulip\u00e1n s\/n, 28933 M\u00f3stoles, Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1578-2602","authenticated-orcid":false,"given":"Christian","family":"Vanhille","sequence":"additional","affiliation":[{"name":"NANLA, Departamento de Matem\u00e1tica Aplicada, Ciencia e Ingenier\u00eda de los Materiales y Tecnolog\u00eda Electr\u00f3nica, Universidad Rey Juan Carlos, Tulip\u00e1n s\/n, 28933 M\u00f3stoles, Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,12,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/S0041-624X(97)00041-3","article-title":"Ultrasonics of non-linear contacts: Propagation, reflection and NDE applications","volume":"36","author":"Solodov","year":"1998","journal-title":"Ultrasonics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1126\/science.280.5360.82","article-title":"Ultrasound-Stimulated Vibro-Acoustic Spectrography","volume":"280","author":"Fatemi","year":"1998","journal-title":"Science"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3946","DOI":"10.1121\/1.2126917","article-title":"Nonlinear resonant ultrasound spectroscopy (NRUS) applied to damage assessment in bone","volume":"118","author":"Muller","year":"2005","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.ultras.2019.02.003","article-title":"Fundamental wave amplitude difference imaging for detection and characterization of embedded cracks","volume":"96","author":"Haupert","year":"2019","journal-title":"Ultrasonics"},{"unstructured":"Shung, K.K. (2005). Diagnostic Ultrasound: Imaging and Blood Flow Measurements, CRC Press.","key":"ref_5"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"R67","DOI":"10.1088\/0031-9155\/46\/5\/201","article-title":"Three-dimensional ultrasound imaging","volume":"46","author":"Fenster","year":"2001","journal-title":"Phys. Med. Biol"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1016\/S0301-5629(99)00060-5","article-title":"Clinical use of ultrasound tissue harmonic imaging","volume":"25","author":"Tranquart","year":"1999","journal-title":"Ultrasound Med. Biol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0887-2171(01)90014-9","article-title":"Tissue harmonic imaging tecniques: Physical principles and clinical aplications","volume":"22","author":"Desser","year":"2001","journal-title":"Eur. Semin. Ultrasound CT MR"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/S0899-7071(01)00335-7","article-title":"Tissue harmonic imaging sonography of breast lesions: Improved margin analysis, conspicuity, and image quality compared to conventional ultrasound","volume":"25","author":"Rosen","year":"2001","journal-title":"Clin. Imaging"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"114911","DOI":"10.1063\/1.2821174","article-title":"Simultaneous sum-frequency and vibroacoustography imaging for nondestructive evaluation and testing aplications","volume":"102","author":"Mitri","year":"2007","journal-title":"J. Appl. Phys."},{"doi-asserted-by":"crossref","unstructured":"Zhang, W., Liu, Y., Ratilal, P., Cho, B., and Makris, N.C. (2017). Active nonlinear acoustic sensing of an object with sum or difference frequency Fields. Remote Sens., 9.","key":"ref_11","DOI":"10.3390\/rs9090954"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1121\/1.1918525","article-title":"Parametric acoustic array","volume":"354","author":"Westervelt","year":"1963","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_13","first-page":"117","article-title":"Broadband unidirectional ultrasound propagation using sonic crystal and nonlinear medium","volume":"2","author":"Sinha","year":"2013","journal-title":"Emerg. Mater. Res."},{"key":"ref_14","first-page":"396","article-title":"Emission of harmonic and combination-frequency waves by bubbles","volume":"18","author":"Zabolotskaya","year":"1973","journal-title":"Acoust. Phys."},{"unstructured":"Hamilton, M.F., and Blackstock, D.T. (1998). Nonlinear Acoustics, Academic Press.","key":"ref_15"},{"unstructured":"Naugolnykh, K., and Ostrovsky, L. (1998). Nonlinear Wave Processes in Acoustics, Cambridge University Press.","key":"ref_16"},{"unstructured":"Grieser, F., Choi, P.K., Enomoto, N., Harada, H., Okitsu, K., and Yasui, K. (2015). Sonochemistry and the Acoustic Bubble, Elsevier.","key":"ref_17"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.euje.2004.02.001","article-title":"Microbubbles and ultrasound: From diagnosis to therapy","volume":"5","author":"Dijkmans","year":"2004","journal-title":"Eur. J. Echocardiogr."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3570","DOI":"10.1121\/1.417222","article-title":"Low-frequency acoustic wave generation in a resonant bubble layer","volume":"100","author":"Druzhinin","year":"1996","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2051","DOI":"10.1016\/j.ultras.2014.07.004","article-title":"Nonlinear frequency mixing in a resonant cavity: Numerical simulations in a bubbly liquid","volume":"54","author":"Vanhille","year":"2014","journal-title":"Ultrasonics"},{"key":"ref_21","first-page":"881","article-title":"A numerical model for the study of the difference frequency generated from nonlinear mixing of standing ultrasonic waves in bubbly liquids","volume":"34","author":"Vanhille","year":"2017","journal-title":"Ultrasonics"},{"key":"ref_22","first-page":"597","article-title":"Numerical models for the study of the nonlinear frequency mixing in two and three-dimensional resonant cavities filled with a bubbly liquid","volume":"39","author":"Vanhille","year":"2017","journal-title":"Ultrasonics"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"726","DOI":"10.1016\/j.rinp.2018.10.007","article-title":"Numerical study of frequency mixing in a focused ultrasonic field in bubbly liquids from a dual-frequency spherical source","volume":"11","author":"Vanhille","year":"2018","journal-title":"Results Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1018","DOI":"10.1121\/1.395376","article-title":"Oscillations of a cloud of bubbles of small and not so small amplitude","volume":"82","author":"Omta","year":"1987","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1002\/fld.833","article-title":"Behaviour of a bubble cluster in an ultrasound field","volume":"47","author":"Matsumoto","year":"2005","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1703","DOI":"10.1121\/1.4945452","article-title":"Nonlinear acoustic propagation in bubbly liquids: Multiple scattering, softening and hardening phenomena","volume":"139","author":"Doc","year":"2016","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_27","first-page":"1570508","article-title":"Nonlinear resonance of cavities filled with bubbly liquids: A numerical study with application to the enhancement of the frequency mixing effect","volume":"2018","author":"Vanhille","year":"2018","journal-title":"Shock Vib."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/S0887-2171(01)90015-0","article-title":"Detection of liver lesions with contrast harmonic imaging and \u2018burst\u2019-mode","volume":"22","author":"Leen","year":"2001","journal-title":"Semin. Ultrasound CT MR"},{"key":"ref_29","first-page":"1531","article-title":"Ultrasound contrast harmonic imaging of abdominal organs","volume":"215","author":"Wilson","year":"2000","journal-title":"Radiology"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.ndteint.2003.09.003","article-title":"Finite element simulation of non-destructive damage detection with higher harmonics","volume":"37","author":"Hurlebaus","year":"2004","journal-title":"NDT E Int."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4095","DOI":"10.1016\/j.matdes.2009.05.009","article-title":"Non-linear ultrasonic evaluation of damaged concrete based on higher order harmonic generation","volume":"30","author":"Shah","year":"2009","journal-title":"Mater. Des."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/1\/113\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:45:03Z","timestamp":1760190303000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/1\/113"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,23]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["s20010113"],"URL":"https:\/\/doi.org\/10.3390\/s20010113","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2019,12,23]]}}}