Question

A string vibrates according to the equation $$ y=(0.520 \mathrm{~cm}) \sin [(1.14 \mathrm{rad} / \mathrm{cm}) x] \cos [(137 \mathrm{rad} / \mathrm{s}) t] $$ (a) What are the amplitude and speed of the component waves whose superposition can give rise to this vibration? (b) Find the distance between nodes. (c) What is the velocity of a particle of the string at the position \(x=1.47 \mathrm{~cm}\) at time \(t=\) \(1.36 \mathrm{~s} ?\)

Short answer

The amplitude is 0.520cm and the speed is 0.00832 cm/s, the distance between nodes is 5.51 cm, and the velocity at x=1.47 cm and t=1.36 s is -96.83 cm/s.

Step-by-step solution

Identify the Amplitude and Speed

The amplitude of a wave is the maximum displacement from its equilibrium position, and is given by the coefficient of the equation. In this case, it is \(0.520cm\). The wave speed is given by \(\frac{wavenumber}{angular frequency}\), so in this case it is \(\frac{1.14 rad/cm}{137 rad/s} = 0.00832 cm/s\).

Find the Distance Between Nodes

A node is a point where the displacement of the wave is always zero. The distance between nodes is called the wavelength, and it is given by \(\frac{2\pi}{wavenumber}\). So in this case it is \(\frac{2\pi}{1.14 rad/cm} = 5.51 cm\).

Find the Velocity at Given Point

The velocity of a point in the wave is given by the derivative of the displacement function. Therefore the velocity is \(velocity = \frac{d}{dt}(0.520cm \sin(1.14 rad/cm * 1.47cm) * cos(137 rad/s * 1.36s))\), which simplifies to \(-96.83 cm/s\).

Key concepts

Amplitude

Amplitude refers to the maximum extent of a vibration or oscillation from the wave's central position. In our exercise, it is given by the coefficient in the sine function of the wave equation. Here, the amplitude is noted as 0.520 cm. This means that from its rest (or equilibrium) position, the wave on the string can reach up to 0.520 cm at its maximum displacement. Recognizing amplitude is crucial for understanding the energy of a wave, as a larger amplitude indicates more energy is carried in the wave. Amplitude is important in various practical applications, including sound and light waves, where it influences volume and brightness, respectively.
In summary, amplitude provides key insights into the wave's strength and the potential energy being transferred. Understanding amplitude allows us to further appreciate how waves operate not just on a string but in real-world phenomena.

Wave Speed

Wave speed determines how fast a wave travels through a medium. It can be calculated by dividing the wavenumber by the angular frequency, as shown in the solution with \(\frac{1.14 \, \text{rad/cm}}{137 \, \text{rad/s}} = 0.00832 \, \text{cm/s}\). This small value indicates a slower wave motion along the string.

Wave speed is a vital concept as it affects how quickly information or energy is transmitted from one point to another. Whether the wave is on a string, sound traveling through air, or light through space, understanding wave speed helps predict how long a wave will take to reach a destination.

• Key Factors Affecting Wave Speed:
• The medium through which the wave travels
• Tension in the medium (like tension in a string)
• Density of the medium
Understanding these factors can help in adjusting the parameters of a system to achieve desired wave speeds, essential in fields such as telecommunications and acoustics.

Nodes

Nodes are specific points along a wave where the wave remains stationary—meaning there's no movement from the equilibrium position. These occur at positions where the destructive interference results in a net amplitude of zero. In this exercise, nodes are found at intervals defined by the wavelength, which can be calculated as \( \frac{2\pi}{\text{wavenumber}} \). Here, it results in a wavelength of 5.51 cm, indicating the spacing between consecutive nodes.

Understanding nodes is essential, especially in the study of standing waves, which occur in musical instruments or confined spaces where waves reflect and interfere with themselves. The pattern of nodes will define the wave's shape and behavior, including its harmonic properties.

• Applications of Nodes:
• Designing musical instruments for producing desired notes
• Hydraulics and structural engineering by observing node formations to avoid destructive interference
Nodes thus play a critical role in both theoretical and applied physics.

Wavelength

The wavelength of a wave is the distance over which the wave's shape repeats, or the distance between two corresponding points on consecutive cycles of a wave, often measured from crest to crest or from one node to the next. In the given example, the wavelength was determined using \( \frac{2\pi}{1.14 \mathrm{rad/cm}} \), resulting in a wavelength of 5.51 cm.
Wavelength is a fundamental property of waves that impacts their interaction with materials, their ability to diffract, and more. It is especially significant in communication technologies, which utilize different wavelengths to transmit data effectively.

• Broader implications of Wavelength:
• The color of light depends on its wavelength
• Radio and TV broadcasts are transmitted at specific wavelengths
Having a grasp of wavelength is necessary for comprehending how different kinds of waves behave in diverse environments.

Particle Velocity

Particle velocity in a wave refers to the speed and direction in which a specific point on the wave (like a point on a string) moves as the wave passes through. It differs from wave speed as it describes the movement of individual particles rather than the entire wave front. In the given problem, particle velocity at a particular point was calculated with derivative techniques, resulting in -96.83 cm/s.

Understanding particle velocity is crucial for describing how energy is transmitted across the medium and how the particles oscillate around their equilibrium position. It's often a consideration in fluid dynamics and acoustics, where knowing how particles within a medium move helps in analyzing sound waves and their impact.

• Key Considerations:
• Particle velocity is highest at points of maximum slope in the wave
• It is out of phase with displacement, leading to dynamic behavior in wave motion
Overall, particle velocity gives insight into the micro-movements within a medium experiencing wave motion, complementing the broader understanding of how waves work.