(II) The displacement of a transverse wave traveling on a...

(II) The displacement of a transverse wave traveling on a string is represented by $D_{1}=4.2 \sin (0.84 x-47 t+2.1)$ where $D_{1}$ and $x$ are in $\mathrm{cm}$ and $t$ in $s$ . $(a)$ Find an equation that represents a wave which, when traveling in the opposite direction, will produce a standing wave when added to this one. (b) What is the equation describing the standing wave?

Key Concepts

Wave Propagation Direction

The direction of propagation of a wave is determined by the sign and structure of the phase term in its wave equation. Reversing the direction of travel of a wave can be achieved by changing the sign of the spatial term in the phase, which is essential in problems involving waves moving in opposite directions.

Superposition Principle

The superposition principle states that when two or more waves travel through the same medium, the resultant displacement at any point is the algebraic sum of the displacements produced by each wave independently. This principle underlies the formation of more complex wave phenomena, including the creation of standing waves when waves of equal amplitude and frequency travel in opposite directions.

Transverse Waves

A transverse wave is a disturbance in which the particles of the medium move perpendicular to the direction of wave propagation. This type of motion is typical for waves on a string, where the displacement of the string occurs vertically while the wave travels horizontally.

Traveling Wave Equation

A traveling wave can be mathematically expressed using a sinusoidal function that incorporates amplitude, wavenumber, angular frequency, and phase shift. This function describes how the displacement of the medium varies with both space and time, encapsulating essential information about the wave's spatial profile and temporal evolution.

Standing Waves

Standing waves are formed when two traveling waves of identical frequency and amplitude move in opposite directions and interfere. Unlike traveling waves, standing waves exhibit fixed nodes and antinodes, resulting from the cancellation and reinforcement of the wave displacements at specific positions. The mathematical representation of a standing wave typically factors into a product of a spatial function and a temporal function.

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