The amplitude of a wave in a string is 2 cm. This wave is propagating along the x -direction wi

step 1 In this question we will learn about some basic concept of the oscillation and waves. The questi

The amplitude of a wave in a string is 2 cm. This wave is...

The amplitude of a wave in a string is $2 \mathrm{~cm}$. This wave is propagating along the $\mathrm{x}$ -direction with a speed of $128 \mathrm{~m} / \mathrm{s}$. Five such waves are accommodated in $4 \mathrm{~m}$ length of the string. The equation for this wave is (A) $\mathrm{y}=0.02 \sin (15.7 \mathrm{x}-2010 \mathrm{t}) \mathrm{m}$ (B) $\mathrm{y}=0.02 \sin (15.7 \mathrm{x}+2010 \mathrm{t}) \mathrm{m}$ (C) $\mathrm{y}=0.02 \sin (7.85 \mathrm{x}-1005 \mathrm{t}) \mathrm{m}$ (D) $\mathrm{y}=0.02 \sin (7.85 \mathrm{x}+1005 \mathrm{t}) \mathrm{m}$

The amplitude of a wave in a string is $2 \mathrm{~cm}$. This wave is propagating along the $\mathrm{x}$ -direction with a speed of $128 \mathrm{~m} / \mathrm{s}$. Five such waves are accommodated in $4 \mathrm{~m}$ length of the string. The equation for this wave is
(A) $\mathrm{y}=0.02 \sin (15.7 \mathrm{x}-2010 \mathrm{t}) \mathrm{m}$
(B) $\mathrm{y}=0.02 \sin (15.7 \mathrm{x}+2010 \mathrm{t}) \mathrm{m}$
(C) $\mathrm{y}=0.02 \sin (7.85 \mathrm{x}-1005 \mathrm{t}) \mathrm{m}$
(D) $\mathrm{y}=0.02 \sin (7.85 \mathrm{x}+1005 \mathrm{t}) \mathrm{m}$

Key Concepts

Angular Frequency and Wave Number

Angular frequency (?) and wave number (k) are related to the frequency and wavelength of a wave through the relations ? = 2?f and k = 2?/?, respectively. They are used in the wave’s mathematical formulation to account for the cycle’s progression in time and space. Their values determine the argument of the sine (or cosine) function in the wave equation.

Wave Speed and Frequency

Wave speed is the rate at which the wave propagates through the medium and is given by the product of frequency and wavelength (v = f × ?). Frequency, which represents the number of cycles per second, influences the angular frequency in the wave equation. These parameters are crucial as they relate the spatial and temporal characteristics of the wave.

Wave Equation Form

The wave equation in its basic sinusoidal form is typically written as y(x, t) = A sin(kx ± ?t), where the sign before ?t indicates the direction of wave propagation. A negative sign corresponds to a wave moving in the positive x-direction, while a positive sign represents a wave traveling in the negative x-direction. This formulation helps in understanding the relationship between the wave’s phase and its propagation direction.

Amplitude

Amplitude is the maximum displacement of points on a wave from their rest position. It determines the energy carried by the wave and is visually represented by the height of the wave peaks. In the context of the wave equation, the amplitude is the coefficient before the sine or cosine function.

Wavelength

Wavelength is the spatial period of the wave—the distance over which the wave's shape repeats. It is mathematically defined as the distance between consecutive points in phase, such as crest to crest, and is denoted by the symbol ?. In problems where a certain number of waves fits into a given length, the wavelength can be calculated by dividing the total distance by the number of waves.

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