I don't know if this is the correct solution but it's worked for me. In your tomcat webapps folder (eg. usr/share/tomcat7/webapps) replace the contents of the "ROOT" folder with the contents from the "Thingworx" folder. I first copied the contents of the "ROOT" folder to a new folder called "tomcat" so I could still access the tomcat manager app at <serverIP>/tomcat. Let me know if you have any questions
You have a couple of options:
Remove the out-of-the-box ROOT/ directory from tomcat and rename your war file to ROOT.war before deploying it.
Deploy your war as (from your example) war_name.war and configure the context root in conf/server.xml to use your war file :
<Context path="" docBase="war_name" debug="0" reloadable="true"></Context>
The first one is easier, but a little more kludgy. The second one is probably the more elegant way to do it.