A SHW is represented by the equation y(x, t)=a0 sin2 π(v t-(x)/(λ)). If the maximum particle vel

step 1 In this question we have given the simple harmonic wave equation and we have given the maximum p

A SHW is represented by the equation y(x, t)=a0 sin2 π(v...

A SHW is represented by the equation $y(x, t)=a_{0} \sin 2 \pi\left(v t-\frac{x}{\lambda}\right)$. If the maximum particle velocity is three times the wave velocity, the wavelength $\lambda$ of the wave is (A) $\frac{\pi a_{0}}{3}$ (B) $\frac{2 \pi a_{0}}{3}$ (C) $\pi a_{0}$ (D) $\frac{\pi a_{0}}{2}$

A SHW is represented by the equation $y(x, t)=a_{0} \sin 2 \pi\left(v t-\frac{x}{\lambda}\right)$. If the maximum particle
velocity is three times the wave velocity, the wavelength $\lambda$ of the wave is
(A) $\frac{\pi a_{0}}{3}$
(B) $\frac{2 \pi a_{0}}{3}$
(C) $\pi a_{0}$
(D) $\frac{\pi a_{0}}{2}$

Key Concepts

Simple Harmonic Wave (SHW)

A simple harmonic wave is a periodic disturbance described by a sine (or cosine) function. It represents how a particle in a medium oscillates with a constant amplitude while the wave propagates through space. The function is typically expressed in terms of an amplitude, a phase (which involves both time and space variables), and parameters such as angular frequency and wavenumber.

Particle Velocity in a Wave

In the context of a harmonic wave, the particle velocity refers to the instantaneous rate of change of displacement with respect to time. This is obtained by differentiating the displacement function (the sine function) with respect to time. The maximum particle velocity is given by the product of the angular frequency and the amplitude (?a), representing the peak speed at which a point on the medium oscillates.

Relationship between Angular Frequency, Wavenumber, and Wave Speed

A key property of harmonic waves is the connection among angular frequency (?), wavenumber (k), and the wave speed (v). They are related by the equation v = ?/k, where the wavenumber is further defined as k = 2?/?. This relationship allows one to connect the time-dependent oscillatory behavior with the spatial properties of the wave, such as the wavelength, and plays a fundamental role in deriving relationships among these parameters.

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