The equation of wave is y=2 sinπ(0.5 x-200 t) where x and y are expressed in cm and t in sec. Th

step 1 The given situation of the question is, as the equation of wave is given, equation of a wave is

The equation of wave is y=2 sinπ(0.5 x-200 t) where x and y...

The equation of wave is $y=2 \sin \pi(0.5 x-200 t)$ where $x$ and $y$ are expressed in $\mathrm{cm}$ and $t$ in $\mathrm{sec}$. The wave velocity is (a) $100 \mathrm{~cm} / \mathrm{sec}$ (b) $200 \mathrm{~cm} / \mathrm{sec}$ (c) $300 \mathrm{~cm} / \mathrm{sec}$ (d) $400 \mathrm{~cm} / \mathrm{sec}$

The equation of wave is $y=2 \sin \pi(0.5 x-200 t)$ where $x$ and $y$ are expressed in $\mathrm{cm}$ and $t$ in $\mathrm{sec}$. The wave velocity is
(a) $100 \mathrm{~cm} / \mathrm{sec}$
(b) $200 \mathrm{~cm} / \mathrm{sec}$
(c) $300 \mathrm{~cm} / \mathrm{sec}$
(d) $400 \mathrm{~cm} / \mathrm{sec}$

Key Concepts

Traveling Wave

A traveling wave is a disturbance that moves through space and time, carrying energy from one location to another without the physical transfer of matter. It is typically represented by sinusoidal functions that combine spatial and temporal variations, indicating how the wave propagates through a medium.

Wave Equation

The wave equation is a mathematical model that describes the behavior of waves. It is commonly expressed in the form y(x, t) = A sin(kx - ?t + ?), where A represents the amplitude, k the wave number, ? the angular frequency, and ? the phase constant. This formulation encapsulates the essential dynamics of wave motion, including periodicity in space and time.

Wave Number

The wave number, denoted by k, quantifies the spatial frequency of a wave and is defined as 2? divided by the wavelength. It describes how many wave cycles exist per unit length and is crucial for linking the spatial characteristics of the wave to its propagation speed when considered along with the angular frequency.

Angular Frequency

Angular frequency, denoted by ?, indicates how quickly the wave oscillates in time and is measured in radians per second. It is directly proportional to the frequency of the wave (? = 2?f) and plays a key role in determining the wave’s temporal behavior.

Phase Velocity

Phase velocity is the speed at which a specific phase of the wave (for instance, a crest) propagates through space. It is calculated by the ratio of angular frequency to wave number (v = ?/k), linking the time-based oscillations with the spatial structure of the wave. This concept is fundamental in understanding how fast the wave pattern moves irrespective of the medium.

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