The displacement of a wave traveling in the positive x...

The displacement of a wave traveling in the positive $x$ -direction is $y(x, t)=(3.5 \mathrm{cm}) \times \cos (2.7 x-92 t),$ where $x$ is in $\mathrm{m}$ and $t$ is in $\mathrm{s}$. What are the
(a) frequency,
(b) wavelength, and
(c) speed of this wave?

Key Concepts

Traveling Wave Equation

The traveling wave equation is a mathematical representation of a wave moving through space and time, typically written as y(x, t) = A cos(kx - ?t) or a similar form. This equation encapsulates key parameters such as amplitude (A), wave number (k), and angular frequency (?), which are instrumental in describing the wave's characteristics like its oscillation, periodicity, and propagation direction.

Angular Frequency

Angular frequency (?) measures how rapidly the wave oscillates in time and is expressed in radians per second. It plays a crucial role in converting the oscillatory behavior of the wave into its frequency (f), using the relationship f = ?/(2?). This concept connects time-domain oscillations to the cycle-per-second measure that is more widely understood in frequency analysis.

Wave Number

The wave number (k) describes the spatial frequency of the wave, indicating how many radians the wave phase changes per unit distance. It is directly related to the wavelength (?) via the expression ? = 2?/k. The wave number encapsulates how the wave’s phase changes with respect to space, making it fundamental to understanding spatial periodicity.

Frequency and Wavelength Relationship

Frequency is the count of how many complete oscillations occur in one second and is derived from the angular frequency, while wavelength is the distance over which the wave's shape repeats, derived from the wave number. These interrelated concepts provide a complete description of the wave's periodic behavior in both time and space.

Wave Speed

The speed of a wave describes how fast the wave propagates through a medium, and it is related to both its temporal and spatial periodicities. It can be calculated using the equation v = ?/k, or alternatively through the product of frequency and wavelength (v = f?). Understanding wave speed is central to predicting and analyzing how waves traverse different media.

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