Standing waves are produced by the superposition of two...

Standing waves are produced by the superposition of two waves $y_{1}=0.05 \sin (3 \pi t-2 x)$ and $y_{2}=0.05 \sin (3 \pi t+2 x)$ where $\mathrm{x}$ and $\mathrm{y}$ are in meters and $\mathrm{t}$ is in seconds. The speed (in $\mathrm{ms}-1$ ) of each wave is $\ldots \ldots$ (A) $1.5$ (B) $3.0$ (C) $3 \pi / 2$ (D) $3 \pi$

Standing waves are produced by the superposition of two waves $y_{1}=0.05 \sin (3 \pi t-2 x)$ and $y_{2}=0.05 \sin (3 \pi t+2 x)$
where $\mathrm{x}$ and $\mathrm{y}$ are in meters and $\mathrm{t}$ is in seconds.
The speed (in $\mathrm{ms}-1$ ) of each wave is $\ldots \ldots$
(A) $1.5$
(B) $3.0$
(C) $3 \pi / 2$
(D) $3 \pi$

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Key Concepts

Wave Speed Calculation

The speed of a wave (phase velocity) is given by the relationship v = ?/k, where ? is the angular frequency and k is the wave number. This formula is essential in determining how fast a wave propagates through a medium based on its fundamental oscillation characteristics.

Standing Waves

Standing waves arise when two waves of the same frequency and amplitude propagate in opposite directions and interfere with each other. This interference results in a wave pattern that appears to be standing still, characterized by nodes (points of no displacement) and antinodes (points of maximum displacement).

Wave Superposition

Wave superposition refers to the overlapping of two or more waves resulting in a new wave pattern. For standing waves, this principle explains how the interference of oppositely travelling waves with identical frequency and amplitude produces fixed spatial patterns.

Angular Frequency and Wave Number

Angular frequency (often denoted as ?) and wave number (denoted as k) are crucial parameters in wave motion. Angular frequency represents the rate of oscillation in radians per second, while wave number represents the number of radians per unit distance. They are intrinsic in characterizing the temporal and spatial variations of a wave.

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