Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 362184 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.1313 1211552 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.1314 1211552 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.2008 4164600 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.2022 4192360 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.2030 4210072 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.2030 4227080 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.2038 4234816 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.2038 4234816 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2564 4891608 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2588 4976840 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2594 4997768 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2700 5114232 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2701 5114488 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Deep Water Gravity Waves - Infinite Depth

The equations satisfied by waves are

for water of depth (1)

at (2)

at (3)

(1) can be solved by separation of variables technique. Assume(there will also be an arbitrary factor ofwhich we deal with later). (1) becomessince the left hand side is a function ofonly and the right hand side is a function ofonly, so both sides are equal to the same constant

Ifthe solution will be exponential, tending toasand the same problem occurs ifsoto givewhere

This equation has solutions

The corresponding equation foriswhich has solutionshence

atsoand

In the same way we can solve (2) by the separation of variables method to findAssume (ignoring the factor)to giveas before, and as beforeelseasso putto give

Now writeandand sincewe haveso by picking a suitable point on the wave surface and a suitable time we have