91Ó°ÊÓ

A transverse sine wave with an amplitude of \(2.50 \mathrm{~mm}\) and a wavelength of \(1.80 \mathrm{~m}\) travels from left to right along a long, horizontal, stretched string with a speed of \(36.0 \mathrm{~m} / \mathrm{s}\). Take the origin at the left end of the undisturbed string. At time \(t=0\) the left end of the string has its maximum upward displacement. (a) What are the frequency, angular frequency, and wave number of the wave? (b) What is the function \(y(x, t)\) that describes the wave? (c) What is \(y(t)\) for a particle at the left end of the string? (d) What is \(y(t)\) for a particle \(1.35 \mathrm{~m}\) to the right of the origin? (e) What is the maximum magnitude of transverse velocity of any particle of the string? (f) Find the transverse displacement and the transverse velocity of a particle \(1.35 \mathrm{~m}\) to the right of the origin at time \(t=0.0625 \mathrm{~s}\)

Step by step solution

01

Calculate the Frequency and Angular Frequency

Since the velocity (v) of the wave is given as 36.0 m/s and the wavelength (λ) as 1.80 m, the frequency (f) can be calculated using the formula \(f = v / λ\). After calculating the frequency, the angular frequency (ω) can be calculated using the formula \(ω = 2πf\).

02

Calculate the Wave Number

The wave number (k) can be calculated using the formula \(k = 2π / λ\)

03

Derive the Wave Function

The wave function can be written as \(y(x, t) = A sin(kx - ωt)\) where A is the amplitude, k is the wave number, ω is the angular frequency and t is time. Note that the minus sign in the equation is because the wave is moving in the positive x direction.

04

Derive the Wave Function at the Left End of the String (x=0)

At x=0, the wave function becomes \(y(t) = A sin(- ωt)\). Replacing - ωt with φ (phase), the function simplifies to \(y(t) = A sin(φ)\). Because the left end of the string has its maximum upward displacement at t=0, therefore φ=π/2. Thus \(y(t) = A sin(π/2)\)

05

Derive the Wave Function for a Particle 1.35 m to the Right of the Origin

Substitute x = 1.35 m into the wave function to find \(y(t) = A sin(k*1.35 - ωt)\). Again because the minus sign is there due to the wave moving in the positive x direction.

06

Calculate the Maximum Transverse Velocity of any Particle of the String

The maximum transverse velocity (Vmax) of the string can be calculated using the formula \(Vmax = ω * A\) where ω is the angular frequency and A is the amplitude.

07

Transverse Displacement and Velocity

To find the transverse displacement at a specific location and time, substitute the given distance x = 1.35m and time t = 0.0625 s into the wave function. The transverse velocity can be found by taking the derivative of the wave function with respect to time and then substituting the given values of x and t.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Frequency

Wave frequency is a fundamental aspect of wave dynamics, referring to the number of wave cycles that pass a fixed point per second. This value is crucial since it determines the wave's characteristics, such as energy and pitch for sound waves. It's measured in hertz (Hz), where one hertz is one cycle per second.
When we have a wave traveling with a velocity of 36.0 m/s and a wavelength of 1.80 m, we can calculate the frequency (f) using the relationship \( f = \frac{v}{\lambda} \), where \( v \) is the wave velocity and \( \lambda \) is the wavelength. In the context of our exercise, this gives us the wave frequency, which offers insight into how rapidly the wave oscillates as it propagates through the medium.

Angular Frequency

Angular frequency provides another way to describe the oscillation of a wave. It relates to the frequency but instead of cycles per second, it measures the radians that are passed through per second. The formula \( \omega = 2\pi f \) connects the angular frequency \( \omega \) to the wave frequency f.
This value is especially useful in physics and engineering because it simplifies the expression of sinusoidal functions that describe wave motion. In our example, once we have determined the wave frequency, we can find the angular frequency, offering a more complete understanding of the wave's tempo in angular terms.

Wave Number

Wave number represents the spatial frequency of a wave, denoting the number of wavelengths per unit distance. It's defined by the equation \( k = \frac{2\pi}{\lambda} \), where \( \lambda \) is the wavelength. The wave number tells us how many wavelengths fit into a 2\pi radian span, linking the spatial geometry of the wave to its periodic nature.
With the wavelength given, calculating the wave number provides us with an insight into the structure of the wave. In practical terms, the wave number is instrumental in tailoring the wave function, ultimately describing the specific layout of the wave's sine pattern in space.

Wave Function

The wave function, \( y(x, t) \), is the mathematical description of the wave's shape at any given location and time. For a transverse sine wave, it is expressed as \( y(x, t) = A \sin(kx - \omega t) \), where A is the amplitude, k is the wave number, \( \omega \) is the angular frequency, x is the position along the wave, and t is time.
It outlines the precise displacement of the wave's medium at any point and instant, essentially mapping the entire motion of the wave. Understanding the wave function is pivotal for predicting how the wave will behave under various conditions and is the key starting point for solving many problems related to wave motion.

Transverse Velocity

Transverse velocity is the speed at which a point on the medium — such as a string — is moving up and down or side to side, perpendicular to the direction in which the wave is traveling. The maximum transverse velocity (\( V_{max} \)) of the string is given by the formula \( V_{max} = \omega A \), linking it to the angular frequency and amplitude of the wave.
Finding the maximum transverse velocity is crucial as it defines the limit of the medium's velocity during the wave's passage. This plays a significant role in applications where the durability and stability of the medium under wave-induced motion are concerns, such as in the design of strings for musical instruments.

Transverse Displacement

Transverse displacement refers to the distance a point on the medium is moved from its rest position due to the wave. At any specific time, we can describe the transverse displacement using the wave function. For example, the transverse displacement of a point 1.35 m from the origin at time t = 0.0625 s can be found by substituting these values into the wave function.
By calculating the transverse displacement at various points and times, one can visualize the wave's position and analyze its effects on the medium. This is essential in scenarios such as diagnosing the vibrations in a mechanical structure or understanding the patterns of waves on a string.