Nonlinear Resonance of Cavities Filled with Bubbly Liquids: A Numerical Study with Application to the Enhancement of the Frequency Mixing Effect

<div>Maximum difference-frequency pressure amplitude in the cavity <svg height="10.2124pt" id="M155" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 20.3131 10.2124" width="20.3131pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.132)"><path d="M517 51L485 54C448 58 441 63 441 115V712C404 700 337 684 285 678V653C357 648 362 645 362 580V437C339 446 309 449 295 449C159 449 38 340 38 201C38 61 143 -12 223 -12C234 -12 261 -6 301 17L362 53V-12C420 9 495 22 517 26V51ZM362 85C338 67 301 51 266 51C201 51 128 109 128 228C128 373 212 411 259 411C296 411 338 395 362 360V85Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,12.189,3.132)"><path d="M797 0V26C739 32 732 36 732 103V296C732 394 682 449 605 449C576 449 550 437 529 423C504 407 475 389 446 366C425 418 382 449 334 449C303 449 279 437 253 421C222 403 201 385 180 371V452C135 432 85 419 41 411V388C99 379 102 374 102 310V103C102 38 93 32 27 26V0H238V26C189 32 180 38 180 103V338C210 363 250 390 289 390C351 390 377 348 377 275V103C377 37 368 32 306 26V0H520V26C465 32 456 38 456 101V296C456 314 455 326 453 338C491 369 529 390 565 390C628 390 653 345 653 274V107C653 36 642 32 583 26V0H797Z"></path></g></svg> vs. frequency (around <span class="nowrap"><svg height="12.7178pt" id="M156" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 17.0809 12.7178" width="17.0809pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.721,3.132)"><path d="M517 51L485 54C448 58 441 63 441 115V712C404 700 337 684 285 678V653C357 648 362 645 362 580V437C339 446 309 449 295 449C159 449 38 340 38 201C38 61 143 -12 223 -12C234 -12 261 -6 301 17L362 53V-12C420 9 495 22 517 26V51ZM362 85C338 67 301 51 266 51C201 51 128 109 128 228C128 373 212 411 259 411C296 411 338 395 362 360V85Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.526,3.132)"><path d="M495 163C480 117 462 85 444 65C421 39 387 34 332 34C290 34 256 36 236 47C218 57 213 77 213 131V526C213 612 222 616 301 622V650H40V622C122 616 128 611 128 526V126C128 41 120 34 36 28V0H489C498 31 519 126 525 157L495 163Z"></path></g></svg>)</span> in Case 2 (constant <span class="nowrap"><svg height="12.7178pt" id="M157" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 11.7923 12.7178" width="11.7923pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.721,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg>,</span> varied <span class="nowrap"><svg height="12.7178pt" id="M158" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 11.7923 12.7178" width="11.7923pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M619 670C619 686 593 712 555 712S459 686 410 634S335 504 320 430H250L219 400L222 388H312L258 73C223 -133 201 -166 187 -180C175 -191 158 -199 140 -199C123 -199 88 -188 74 -172C68 -166 63 -164 54 -171C38 -185 23 -201 23 -215C23 -236 60 -261 93 -261C122 -261 161 -247 207 -200C268 -138 300 -71 337 94C365 220 376 277 394 387L501 399L521 430H401C432 623 464 665 501 665C524 665 544 651 567 627C577 617 583 618 592 625C601 631 619 651 619 670Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.721,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g></svg>)</span> for (a) <svg height="12.6656pt" id="M159" style="vertical-align:-3.42945pt" version="1.1" viewbox="-0.0498162 -9.23615 71.1713 12.6656" width="71.1713pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 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570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z"></path></g><g transform="matrix(.013,0,0,-0.013,37.914,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,40.878,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g><g transform="matrix(.013,0,0,-0.013,51.476,0)"><path d="M513 0V26C475 28 448 43 413 81C379 116 277 234 258 264C301 314 319 334 344 359C383 396 404 407 473 418V443L269 434V408C317 403 320 395 299 364C280 337 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190 39 159 39 110C39 47 88 -12 159 -12C189 -12 237 25 277 52C282 35 288 21 301 8C312 -3 333 -12 348 -12L433 39ZM275 84C256 65 221 48 195 48C164 48 124 73 124 124C124 161 146 180 185 198C206 208 254 229 275 240V84Z"></path></g></svg> and three values of <svg height="10.2124pt" id="M160" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 16.9061 10.2124" width="16.9061pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.816,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg> and (b) <svg height="12.6656pt" id="M161" style="vertical-align:-3.42945pt" version="1.1" viewbox="-0.0498162 -9.23615 71.1713 12.6656" width="71.1713pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,7.384,3.132)"><path d="M245 635C92 635 37 457 37 312C37 149 91 -12 244 -12C395 -12 449 166 449 312C449 469 395 635 245 635ZM243 598C332 598 358 454 358 312C358 173 334 26 245 26C158 26 128 174 128 313S152 598 243 598Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.816,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.411,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,31.674,0)"><path d="M137 343C167 482 260 545 321 574C357 591 397 603 429 609L423 641C382 634 335 622 295 608C189 570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z"></path></g><g transform="matrix(.013,0,0,-0.013,37.914,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,40.878,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g><g transform="matrix(.013,0,0,-0.013,51.476,0)"><path d="M513 0V26C475 28 448 43 413 81C379 116 277 234 258 264C301 314 319 334 344 359C383 396 404 407 473 418V443L269 434V408C317 403 320 395 299 364C280 337 250 298 228 276C213 260 195 247 165 241V712C126 699 69 682 21 677V653C82 648 87 642 87 576V104C87 40 81 32 14 26V0H237V26C173 32 165 38 165 104V215C171 217 176 218 181 218C189 218 198 212 212 196C267 129 330 49 376 -2L513 0Z"></path></g><g transform="matrix(.013,0,0,-0.013,58.146,0)"><path d="M46 650V622C120 617 128 613 128 525V125C128 42 120 34 40 28V0H311V28C221 34 212 39 212 124V281L286 262C297 261 316 261 331 263C429 275 526 338 526 468C526 533 501 579 462 609C422 638 364 650 293 650H46ZM212 559C212 588 215 600 223 606C230 613 251 618 279 618C361 618 430 572 430 464C430 337 350 302 285 302C252 302 225 309 212 314V559Z"></path></g><g transform="matrix(.013,0,0,-0.013,65.178,0)"><path d="M433 39L423 65C413 59 399 54 387 54C370 54 352 69 352 114V299C352 352 342 392 307 422C285 440 255 449 225 449C168 437 102 399 75 379C56 365 44 353 44 339C44 315 69 296 87 296C101 296 111 303 116 319C124 349 133 371 145 385C156 397 171 404 190 404C241 404 275 364 275 291V274C253 256 180 229 120 209C65 190 39 159 39 110C39 47 88 -12 159 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Shock and Vibration

fig4

Figure 4

Figure 4: Nonlinear Resonance of Cavities Filled with Bubbly Liquids: A Numerical Study with Application to the Enhancement of the Frequency Mixing Effect 

Figure 4 | Nonlinear Resonance of Cavities Filled with Bubbly Liquids: A Numerical Study with Application to the Enhancement of the Frequency Mixing Effect 