A wave disturbance in a medium is described by y(x, t)=0.02 cos(50 πt+(π)/(2)) cos(10 πx) where

step 1 In this question we have given a wave disturbance in a medium and we have to find that which of

A wave disturbance in a medium is described by y(x, t)=0.02...

A wave disturbance in a medium is described by $y(x, t)=0.02 \cos \left(50 \pi t+\frac{\pi}{2}\right) \cos (10 \pi x)$ where $x$ and $y$ are in meter and $t$ is in second. Then (A) First node occurs at $x=0.15 \mathrm{~m}$. (B) First anti-node occurs at $x=0.3 \mathrm{~m}$. (C) The speed of interfering waves is $5.0 \mathrm{~m} / \mathrm{s}$. (D) The wavelength is $0.5 \mathrm{~m}$.

A wave disturbance in a medium is described by $y(x, t)=0.02 \cos \left(50 \pi t+\frac{\pi}{2}\right) \cos (10 \pi x)$
where $x$ and $y$ are in meter and $t$ is in second. Then
(A) First node occurs at $x=0.15 \mathrm{~m}$.
(B) First anti-node occurs at $x=0.3 \mathrm{~m}$.
(C) The speed of interfering waves is $5.0 \mathrm{~m} / \mathrm{s}$.
(D) The wavelength is $0.5 \mathrm{~m}$.

Key Concepts

Angular Frequency and Wave Number

Angular frequency (?) and wave number (k) are fundamental parameters in wave equations. ?, expressed in radians per second, determines how rapidly the wave oscillates in time, while k, expressed in radians per meter, determines the spatial periodicity of the wave. They are essential for linking the physical properties of waves, such as frequency, period, wavelength, and speed, across diverse applications in physics.

Standing Waves

Standing waves result from the superposition of two waves of equal frequency and amplitude traveling in opposite directions. The given displacement function can be factored into a product of a time?dependent term and a space?dependent term, which is characteristic of standing waves. This concept is crucial in understanding resonance and wave interference phenomena in various physical systems.

Nodes and Antinodes

Nodes are points along a standing wave where the amplitude is always zero, while antinodes are points where the amplitude reaches its maximum. Determining the locations of nodes and antinodes involves analyzing the spatial part of the standing wave equation, typically by setting the cosine (or sine) function equal to zero or one. This concept helps in understanding how energy is distributed in systems that support standing waves.

Wavelength Calculation

The wavelength of a wave is determined by the spatial component of its displacement function, often expressed in terms of the wave number k. In the standing wave equation, k is the coefficient of x in the cosine or sine function. The relation ? = 2?/k is used to compute the wavelength, an essential parameter for describing the spatial periodicity of the wave.

Wave Speed Calculation

The speed of a wave is the rate at which the wave propagates through a medium and is related to both the angular frequency ? and the wave number k by the formula v = ?/k. In the context of standing waves formed by interfering traveling waves, this calculation is fundamental for relating temporal and spatial variations in the wave's structure.

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