Stationary vs Progressive Waves

Identifying Phase Relationships: In a progressive wave, the phase difference between two points depends on their distance $x$ and is given by $\Delta \phi = \frac{2\pi x}{\lambda}$. In a stationary wave, all points between two adjacent nodes are in phase ($0$ rad), while points on opposite sides of a node are in anti-phase ($\pi$ rad).

Calculating Wavelength: The distance between two adjacent nodes (or two adjacent antinodes) in a stationary wave is exactly half a wavelength ($\frac{\lambda}{2}$). This relationship is used to determine the wavelength of the original progressive waves.

Determining Amplitude: To find the amplitude of a stationary wave at a specific point, one must consider the superposition of the two constituent waves. At an antinode, the amplitude is $2A$ (where $A$ is the amplitude of the progressive wave), while at a node, it is zero.

Visual Identification: When presented with a diagram, check if the wave profile is shifting or if specific points are permanently at zero displacement. If points are fixed at zero, it is a stationary wave.

Phase Calculations: Always remember that phase in stationary waves is binary (either $0$ or $\pi$). Do not use the progressive wave phase formula for points on a stationary wave unless you are calculating the phase of the constituent waves.

Wavelength Pitfall: A common error is assuming the distance between a node and the next antinode is $\frac{\lambda}{2}$. It is actually $\frac{\lambda}{4}$. Always verify the distance between two identical features (node to node) to find $\frac{\lambda}{2}$.

Boundary Conditions: Pay attention to the ends of the medium. A fixed end (like a tied string) must be a node, while a free end (like the open end of a pipe) is typically an antinode.

The 'No Motion' Myth: Students often think particles at nodes don't move because the wave is 'stationary'. While the nodes are indeed stationary, the particles at antinodes are moving with the highest velocity in the system.

Energy Transfer: It is a misconception that stationary waves have no energy. They contain significant energy; it is simply 'trapped' between the nodes and does not propagate down the medium.

Amplitude Uniformity: In progressive waves, every particle eventually reaches the same maximum displacement. In stationary waves, a particle near a node will never reach the same maximum displacement as a particle at an antinode.