Equation for a harmonic progressive wave is given by y=A sin(15 πt+10 πx+π/ 3) where x is in met

step 1 Hello and welcome to this video solution of numerate. So we are given the equation of a harmonic

Equation for a harmonic progressive wave is given by y=A...

Equation for a harmonic progressive wave is given by $\mathrm{y}=\mathrm{A}$ $\sin (15 \pi t+10 \pi x+\pi / 3)$ where $x$ is in meter and $t$ is in seconds. This wave is $\ldots \ldots$ (A) Travelling along the positive $\mathrm{x}$ direction with a speed of $1.5 \mathrm{~ms}^{-1}$ (B) Travelling along the negative $\mathrm{x}$ direction with a speed of $1.5 \mathrm{~ms}^{-1} .$ (C) Has a wavelength of $1.5 \mathrm{~m}$ along the $-\mathrm{x}$ direction. (D) Has a wavelength of $1.5 \mathrm{~m}$ along the positive $\mathrm{x}$ - direction.

Equation for a harmonic progressive wave is given by $\mathrm{y}=\mathrm{A}$ $\sin (15 \pi t+10 \pi x+\pi / 3)$ where $x$ is in meter and $t$ is in seconds. This wave is $\ldots \ldots$
(A) Travelling along the positive $\mathrm{x}$ direction with a speed of $1.5 \mathrm{~ms}^{-1}$
(B) Travelling along the negative $\mathrm{x}$ direction with a speed of $1.5 \mathrm{~ms}^{-1} .$
(C) Has a wavelength of $1.5 \mathrm{~m}$ along the $-\mathrm{x}$ direction.
(D) Has a wavelength of $1.5 \mathrm{~m}$ along the positive $\mathrm{x}$ - direction.

Key Concepts

Phase Velocity and Wavelength

Phase velocity is the rate at which the phase of the wave propagates through space and is given by the ratio of the angular frequency to the wave number (v = ?/k). The wavelength, defined as the spatial period of the wave, can be derived from the wave number using the relation ? = 2?/k. Understanding these relationships allows for the determination of the speed and spatial characteristics of the wave.

Angular Frequency and Wave Number

The angular frequency (?) and wave number (k) are key parameters in describing a harmonic wave. The angular frequency, associated with the time-dependent part of the phase, determines how rapidly the wave oscillates in time, while the wave number, related to the space-dependent part, determines the spatial periodicity of the wave. These parameters are instrumental in calculating other properties such as phase velocity and wavelength.

Wave Propagation Direction

The direction of propagation of a harmonic wave can be determined from the sign convention in its phase. When the phase is written as (?t ± kx + constant), a positive sign in front of the spatial term (kx) typically indicates propagation in the negative x-direction, while a negative sign implies propagation in the positive x-direction. This is due to the requirement that the phase remains constant as the wave travels.

Phase of a Wave

The phase of a wave is the argument of the sine or cosine function and includes terms both in time (t) and space (x). It determines the instantaneous state of the wave at any point in space and time. The specific form of the phase, particularly the signs in front of the time and space terms, dictates the direction of propagation and other wave properties.

Harmonic Progressive Waves

Harmonic progressive waves are sinusoidal disturbances that travel through a medium without changing their shape. Their mathematical representation involves a sine or cosine function where the argument (or phase) is a linear combination of spatial and temporal components, indicating the evolution of the wave over space and time.

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