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) $k,(\mathrm{c}) \omega$ (d) the wave speed, and (e) the correct choice of sign in front of $\omega ?$

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) $k,(\mathrm{c}) \omega$ (d) the wave speed, and (e) the correct choice of sign in front of $\omega ?$

Key Concepts

Sinusoidal Wave Equation

A sinusoidal wave is a mathematical model that describes a periodic oscillation in both space and time. The general form, s(x, t) = s? cos(kx ± ?t), represents the displacement of particles in the medium as a cosine function, characterized by an amplitude, a spatial periodicity given by the wave number, and a temporal periodicity given by the angular frequency. This equation effectively encapsulates the essential features of wave motion, such as oscillation, phase, and propagation.

Amplitude

Amplitude is the maximum displacement of any point on the wave from its equilibrium position. It represents the energy carried by the wave and is a key indicator of the intensity of the physical disturbance in the medium. In a sinusoidal wave, amplitude directly gives the height of the peaks and the depth of the troughs, regardless of the wave’s other properties like frequency or wavelength.

Wave Number

The wave number, denoted as k, quantifies the spatial frequency of the wave; it is defined as 2? divided by the wavelength (k = 2?/?). It specifies the number of radians of phase change per unit distance, thus linking the spatial periodicity of the wave with its mathematical description. The wave number plays a crucial role in determining how rapidly the wave oscillates in space.

Angular Frequency

Angular frequency, represented by ?, describes the rate of oscillation of the wave in time and is defined as ? = 2?f, where f is the frequency. It provides a measure of how many radians the wave oscillates per second. This concept is critical in describing the temporal behavior of oscillatory systems and in relating time-dependent variations to their periodic characteristics.

Wavelength

Wavelength is the distance over which the wave's shape repeats, typically measured between successive points in phase such as peaks or troughs. It is a fundamental characteristic of a wave that determines its spatial scale and is inversely related to the wave number. The wavelength, along with the frequency, plays a decisive role in dictating the speed and energy propagation of the wave.

Wave Speed

Wave speed is the rate at which the disturbance, or phase, of the wave propagates through the medium. It is related to both the wavelength and the frequency by the equation v = ?/k or v = f?. This fundamental relationship connects the temporal and spatial aspects of a wave and is essential for understanding how quickly information or energy is transmitted.

Direction of Wave Propagation and Phase Sign

The direction in which a wave propagates is dictated by the sign in the phase of the wave equation. For a wave traveling in the negative direction along the x-axis, the phase term is typically written as kx + ?t, ensuring that the wave fronts move in the appropriate direction. This aspect of the wave equation is crucial for correctly modeling the direction and time evolution of the wave's displacement.

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