A wave represented by the given equation y=A sin(10 πx+15 πt+(π)/(3)) where x is in meter and t

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A wave represented by the given equation y=A sin(10 πx+15...

A wave represented by the given equation $y=A \sin \left(10 \pi x+15 \pi t+\frac{\pi}{3}\right)$ where $x$ is in meter and $t$ is in second. The expression represents (a) A wave travelling in the positive $x$-direction with a velocity of $1.5 \mathrm{~m} / \mathrm{sec}$ (b) A wave travelling in the negative $x$-direction with a velocity of $1.5 \mathrm{~m} / \mathrm{sec}$ (c) A wave travelling in the negative $x$-direction with a wavelength of o.2 $m$ (d) A wave travelling in the positive $x$-direction with a wavelength of $0.2 \mathrm{~m}$

A wave represented by the given equation $y=A \sin \left(10 \pi x+15 \pi t+\frac{\pi}{3}\right)$ where $x$ is in meter and $t$ is in second. The expression represents
(a) A wave travelling in the positive $x$-direction with a velocity of $1.5 \mathrm{~m} / \mathrm{sec}$
(b) A wave travelling in the negative $x$-direction with a velocity of $1.5 \mathrm{~m} / \mathrm{sec}$
(c) A wave travelling in the negative $x$-direction with a wavelength of o.2 $m$
(d) A wave travelling in the positive $x$-direction with a wavelength of $0.2 \mathrm{~m}$

Key Concepts

Direction of Wave Propagation

The direction of wave propagation can be inferred from the signs of the terms in the phase of the wave equation. A negative sign accompanying the time term relative to the spatial term typically indicates motion in the positive spatial direction, while a positive sign with the time term suggests movement in the negative spatial direction. Understanding this sign convention is critical for predicting the direction in which the wave energy is transported.

Phase Velocity

Phase velocity is the speed at which a specific phase of the wave (for example, a crest) propagates through space. It is calculated using the relation v = ?/k. This concept is essential for determining how fast the wave fronts are moving in the medium.

Wave Number and Wavelength

The wave number, usually represented by k, quantifies the spatial frequency of the wave and is related to the wavelength by the relation k = 2?/?. This concept is crucial for understanding how often the pattern of the wave repeats per unit length and for determining the distance over which the wave cycle completes.

Angular Frequency

Angular frequency, represented by ?, describes how rapidly the wave oscillates in time and is related to the frequency by ? = 2?f. It determines the number of radians the wave phase advances per second and plays a key role in linking the temporal behavior of the wave to other properties like phase velocity.

Sinusoidal Wave

A sinusoidal wave is a type of periodic wave which can be described by sine or cosine functions. Its mathematical representation includes terms for amplitude, wave number, angular frequency, and phase shift, each of which characterizes different properties of the wave such as its height, spatial periodicity, temporal evolution, and initial phase at a reference point.

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