The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves).

A plane wave in two or three dimensions is like a sine wave in one dimension except that crests and troughs aren’t points, but form lines (2-D) or planes (3-D) perpendicular to the direction of wave propagation. Figure \(\PageIndex{1}\) shows a plane sine wave in two dimensions.

Our goal is to mathematically model the vibrations of the membrane surface. We let deflection of membrane from equilibrium at. u(x, y, t) = position (x, y) and time t. For a fixed t, the surface z = u(x, y, t) gives the shape of the membrane at time t. for 0 < x < a, 0 < y < b.

So far, we’ve looked at waves in one dimension, traveling along a string or sound waves going down a narrow tube. But waves in higher dimensions than one are very familiar — water waves on the surface of a pond, or sound waves moving out from a source in three dimensions.

The wave equation is a second-order linear partial differential equation describing the behaviour of mechanical waves; its two (spatial) dimensional form can be used to describe waves on a surface of water: $$ \frac{\partial^2 u}{\partial t^2} - c^2\left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) = 0 $$

Jan 11, 2023 · The wave equation we introduced in the previous section describes one-dimensional waves. However, in most cases waves propagate in either two-dimensions (such as the surface of water) or in three-dimensions (such as sound or light).

This applet is a simulation that demonstrates scalar waves (such as sound waves) in two dimensions. It demonstrates the wave principles behind slit diffraction, zone plates, and holograms. To get started with the applet, just go through the items in …

Dive into the fascinating world of 2D waves! Understand reflection, refraction, and diffraction through interactive demonstrations. Gain valuable insights applicable to physics and engineering.

Waves can exist in two or three dimensions, however. One example is a plane wave where the wave front or crest of the wave makes a line (in two dimensions) or a plane (in three dimensions). Circular waves (in two dimensions) and spherical waves (in three dimensions) also exist.

Lecture Video: 2D and 3D waves, Snell’s Law. The main focus of the lecture is the mathematical description of 2D and 3D waves. Prof. Lee also shows that the wave description of light leads to reflection law and Snell’s laws in geometrical optics.