A continuous sinusoidal longitudinal wave is sent along a...

A continuous sinusoidal longitudinal wave is sent along a very long coiled spring from an attached oscillating source. The wave travels in the negative direction of an $x$ axis; the source frequency is $25 \mathrm{~Hz}$; at any instant the distance between successive points of maximum expansion in the spring is $24 \mathrm{~cm}$; the maximum longitudinal displacement of a spring particle is $0.30 \mathrm{~cm} ;$ and the particle at $x=0$ has zero displacement at time $t=0$. If the wave is written in the form $s(x, t)=s_{m} \cos (k x \pm \omega t)$, what are (a) $s_{m}$, (b) $k$, (c) $\omega$, (d) the wave speed, and (e) the correct choice of sign in front of $\omega ?$ Incident

A continuous sinusoidal longitudinal wave is sent along a very long coiled spring from an attached oscillating source. The wave travels in the negative direction of an $x$ axis; the source frequency is $25 \mathrm{~Hz}$; at any instant the distance between successive points of maximum expansion in the spring is $24 \mathrm{~cm}$; the maximum longitudinal displacement of a spring particle is $0.30 \mathrm{~cm} ;$ and the particle at $x=0$ has zero displacement at time $t=0$. If the wave is written in the form $s(x, t)=s_{m} \cos (k x \pm \omega t)$, what are (a) $s_{m}$, (b) $k$, (c) $\omega$,
(d) the wave speed, and (e) the correct choice of sign in front of $\omega ?$ Incident

Key Concepts

Propagation Direction and Phase Sign

The propagation direction of a wave is determined by the sign associated with the time-dependent term in the wave equation. For a wave represented as s(x, t) = s? cos(kx ± ?t), the choice of sign before ?t indicates the direction of movement: a negative sign for ?t typically corresponds to propagation in the positive x-direction, while a positive sign indicates propagation in the negative x-direction. This choice must also align with the initial phase conditions provided in the problem.

Angular Frequency

Angular frequency (?) is a measure of how rapidly the wave oscillates in time, expressed in radians per second. It is given by ? = 2?f, where f is the frequency of the wave. This parameter directly relates time intervals to phase changes, playing a crucial role in the temporal behavior of the wave.

Wave Speed

Wave speed is the rate at which the wave propagates through the medium, linking the spatial and temporal characteristics of the wave. It is calculated by the product v = f?, where f is the frequency and ? is the wavelength. Understanding wave speed is essential for connecting how quickly a disturbance travels with its frequency and wavelength.

Wavenumber

Wavenumber (k) represents the spatial frequency of a wave and is defined as the number of radians per unit distance. It is calculated using the formula k = 2?/?, where ? is the wavelength, which is the distance between successive points of maximum displacement in the wave. This parameter describes how rapidly the phase of the wave changes with distance.

Amplitude

Amplitude is the maximum displacement of a particle from its equilibrium position in a wave. In the context of sinusoidal waves, it quantifies the extent of oscillation and is directly related to the energy carried by the wave. It is represented by the coefficient multiplying the cosine or sine function in the wave equation.

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