Mathematician here, but I can also speak for the physicists in this regard. When we say “1d”, “2d” or “3d”, we refer to “space”, i.e. space dimensions. Those diagrams you refer to are called “space-time” diagrams, and reflect the situation in space (y-axis) at a given time (t-axis).
A first remark can be made here: the wave equation is not symmetric in space and time; as a consequence, space and time are fundamentally different (this is even more clear on the heat equation, where not only do we have space-time asymmetry, but also irreversibility). A second remark is that the wave equation is hyperbolic, meaning it has underlying geometrical objects called “characteristic curves”.
This “characteristics” are quite special, as are trajectories in space-time where the solution of the wave equation (the wave profile) looks constant. In the simplest of scenarios, this characteristics are straight lines in space-time (i.e. x-ct=const), and have the remarkable property of separating space-time.
For the wave equation, there are two characteristic curves: x-ct=const and x+ct=const. The first one represents a wave traveling forward in space and the second one represents a wave traveling backwards. Together, they form a “light-cone”, and break space-time into two: “space like” space-time (up and down regions of the cone) and “time like” space-time (left and right region of the cone). For every event in space-time, there is a light cone, and nothing “space like” can communicate with something “time like” without sending waves traveling faster than the wave equation’s propagating speed (commonly called “c”), but any wave traveling faster than “c” would break uniqueness of solutions (information paradoxes). Geometrically, this means that no two places, A and B, in space-time where t_A > t_B can communicate without sending signals traveling faster that the speed of propagation “c” and thus breaking uniqueness, hence no point in the future can speak with a point in the past, and this implies causality.
Adding more space brings even more structure: 3d wave equation “averages out” information, and 2d wave equation solutions lead to saturation of information.
It is remarkable that such a “basic” equation (linear, 1d, second order) can have so many properties, and that such properties go along very well with our experienced reality.
A first remark can be made here: the wave equation is not symmetric in space and time; as a consequence, space and time are fundamentally different (this is even more clear on the heat equation, where not only do we have space-time asymmetry, but also irreversibility). A second remark is that the wave equation is hyperbolic, meaning it has underlying geometrical objects called “characteristic curves”.
This “characteristics” are quite special, as are trajectories in space-time where the solution of the wave equation (the wave profile) looks constant. In the simplest of scenarios, this characteristics are straight lines in space-time (i.e. x-ct=const), and have the remarkable property of separating space-time.
For the wave equation, there are two characteristic curves: x-ct=const and x+ct=const. The first one represents a wave traveling forward in space and the second one represents a wave traveling backwards. Together, they form a “light-cone”, and break space-time into two: “space like” space-time (up and down regions of the cone) and “time like” space-time (left and right region of the cone). For every event in space-time, there is a light cone, and nothing “space like” can communicate with something “time like” without sending waves traveling faster than the wave equation’s propagating speed (commonly called “c”), but any wave traveling faster than “c” would break uniqueness of solutions (information paradoxes). Geometrically, this means that no two places, A and B, in space-time where t_A > t_B can communicate without sending signals traveling faster that the speed of propagation “c” and thus breaking uniqueness, hence no point in the future can speak with a point in the past, and this implies causality.
Adding more space brings even more structure: 3d wave equation “averages out” information, and 2d wave equation solutions lead to saturation of information.
It is remarkable that such a “basic” equation (linear, 1d, second order) can have so many properties, and that such properties go along very well with our experienced reality.