Transverse waves on a string have wave speed 8.00 m / s, amplitude 0.0700 m, and wavelength 0.32

step 1 We're told that transverse waves on a string have a wave speed of 8 meters per second and an amp

Transverse waves on a string have wave speed 8.00 m / s,...

Transverse waves on a string have wave speed $8.00 \mathrm{m} / \mathrm{s},$ amplitude $0.0700 \mathrm{m},$ and wavelength 0.320 $\mathrm{m} .$ These waves travel in the $x$ direction, and at $t=0$ the $x=0$ end of the string is at $y=0$ and moving downward. (a) Find the frequency, period, and wave number of these waves. (b) Write the equation for $y(x, t)$ describing these waves. (c) Find the transverse displacement of a point on the string at $x=0.360 \mathrm{m}$ at time $t=0.150 \mathrm{s}$ .

Transverse waves on a string have wave speed $8.00 \mathrm{m} / \mathrm{s},$
amplitude $0.0700 \mathrm{m},$ and wavelength 0.320 $\mathrm{m} .$ These waves
travel in the $x$ direction, and at $t=0$ the $x=0$ end of the string is
at $y=0$ and moving downward. (a) Find the frequency, period, and wave number of these waves. (b) Write the equation for $y(x, t)$ describing these waves. (c) Find the transverse displacement of a point on the string at $x=0.360 \mathrm{m}$ at time $t=0.150 \mathrm{s}$ .

Key Concepts

Transverse Waves

Transverse waves are disturbances that propagate through a medium where the displacement of the medium is perpendicular to the direction of wave propagation. In contexts such as a string, the motion of the medium (for example, the up and down motion of a string) is at right angles to the direction in which the wave travels. This concept is key to understanding how energy is transmitted in such waves without a net transport of matter.

Wave Speed, Wavelength, and Frequency Relationship

The relationship between wave speed, wavelength, and frequency is fundamental in wave mechanics. Wave speed (v) is the rate at which a wave propagates through a medium, and it is given by the product of the wavelength (?) and the frequency (f), expressed as v = ? f. This relationship allows one to determine one property if the other two are known, making it a cornerstone concept in the study of waves.

Frequency and Period

Frequency (f) is the number of oscillations or cycles a wave completes in one second, while the period (T) is the time taken for one complete cycle. They are inversely related, with the period defined as T = 1/f. Understanding this inverse relationship is critical for analyzing the temporal characteristics of wave motion.

Wave Number

The wave number (k) is a measure of the spatial frequency of a wave, defined as the number of wave cycles per unit distance. It is calculated using k = 2?/?, where ? is the wavelength. Wave number provides insight into how rapidly the phase of the wave changes with position, an important aspect in the mathematical description of wave phenomena.

Angular Frequency

Angular frequency (?) represents the rate of change of the phase of a sinusoidal wave, measured in radians per second. It is related to the frequency by the equation ? = 2?f. This concept is important because it allows the oscillatory behavior of waves to be described in a form that is particularly useful for mathematical analysis and formulation of wave equations.

Wave Equation

The wave equation is a mathematical description of how a wave's displacement varies with both space and time. For a sinusoidal wave traveling along a string, it is typically written as y(x, t) = A sin(kx - ?t + ?) or a similar variant with cosine. This formulation encapsulates key properties of the wave such as amplitude (A), wave number (k), angular frequency (?), and phase constant (?), enabling a comprehensive description of the wave's behavior.

Phase Shift

Phase shift refers to the horizontal displacement of the wave's profile relative to a reference point, which is essential in matching initial or boundary conditions. Determining the correct phase constant (?) in the wave equation ensures that the mathematical model accurately reflects the initial displacement and direction of motion of the wave at a given point in space and time.

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